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, it
follows that
if
jr, f) is
-
an eigenstate
=
t)
-r
q with
of
another eigenstate of q with eigenvalue
(F/./J)
*>,
eigenvalue
r,
then
ipjr,
}
is
r.
Thus, in essence, parity operation is a coordinate transformation of the = 1,2,3, which is a transformation from the righttype xl = Xi, fori
handed to the left-handed coordinate system. [q,
but commutes with even powers
If
[T?, i) is
2
CP]
",
0,
(VI.1.4)
0,
(VI.1.5)
p; that
of
=
anticommutes with q or p,
is,
0,
a state of a dynamical system we can define a "mirror state" by
\f The
q and
of
Cq
= =
It
t
representative of this state in (r,
t\
t}
=
=
f)
q
=
(VL1.7)
t}.
representation
is
(r,
f 5(r
Hence
and the effect tion. Thus
of
(P
on a function
=
iK-r,
of position is to replace it
by its mirror func-
and
=
*(r,
t}
SO (P
and
its
2
=
1
(7IJ.S)
eigenvalues are dbl. In the above equations the number of position is, the number of particles) are not restricted; the parity
coordinates (that
operator will operate on equal footing on all of them. The probability of finding a particle at r in the state (P^ probability of finding
it
at
r in the state ^(r,
f).
We
shall
is
equal to the
assume that
(for
a
Symmetry
166
Principles
and Quantum
Statistics
[Chap. VI]
number of dynamical systems) (P is a linear Heraiitian operator and commutes with the Hamlltonlan (in most cases)
large it
that
:
=
[<$>,#]
(VI.L9)
0.
Equation (VI. 1.9) means that: a constant of the motion, is a scalar,
(a) (P is
(b)
The Hamiltonian
(c)
(P
can be used
to
form
H
together with
set
a complete commuting
of
observables.
Because of the anticommutation relations (VI. 1.4) and (VI. 1.5), the parity or momenof a system cannot simultaneously be measured with its position and Hamiltonian the with measured simultaneously can be tum. But parity
also for the
measured in
momentum
of the system corresponding to the "intrinsic parity" total the with also angular rest frame. Parity commutes
casep its
==
0,
operator J.
momentum of a particle that is, eigenfunctions of orbital angular also are Y eigenfunctions of parity operator lm (B, <) the spherical harmonics Hence we say that parity 1) (P (since
l
.
P=
of a one-particle state of orbital angular momentum I is given by which is usually called "orbital parity." For odd or even values of
1
(
I)
,
parity = of the are i for If parities , n, 1, 2, P,-, is odd or even, respectively. is whole the of the then system parity constituents of an n-particle system, -
defined as the product of the parities of
mentum states; that is, orbital momenta of the
P=
+^+ (-1)^
n --
1
-
Z,
-
relative orbital angular
+ 2--*, where
Zi,
k, k,
'
'
' ,
mo-
k-i are
particles.
VL1.A. The Time-Reversed State Dynamical System
of a
motion are symmetrical with respect to past and for classical future. The same applies in the form of "microscopic reversibility" is exof the reversibility microscopic property statistical ensembles. Here
The
classical equations of
as pressed in terms of "correlation functions,"
where the average value
is
calculated over a microcanonical ensemble of
are state variables of the statistical The functions/,-, for i = 1, 2, a value ff (t) at a time t and a value fj(t + r) system. Thus correlation between correlation between fi(t) and a value of as the same the is r later at a time -
systems.
fj at
a time r
earlier.
-
[VI. I. ]
Parity of a Dynamical System
167
Quantum mechanical laws also posses? similar time-reversal invariance properties. The displacement in time of the state of the dynamical system, from a fixed time
I
=
j*,
By replacing t by
to time
=
t)
t
we
f,
given by [see (V.2.9f
is
-
'*, 0>
^ fadtL
*+(t, ti)H(ti)
(VI. 1.10)
obtain the time-reversed description of the dynamical f>, which satisfies the integral ;^,
system in terms of the state vector equation I*,
By making
-*>
= y, o) -
the transformation
\* -t) 3
where the function
= |,
+(,
Hence the time-reversed I*,
If (t
=
-
0)
1)
=
+
--
*
ti
I Jj +(-*,
t)
we
in the integral,
get
-
defined by
is
state satisfies the equation
I*, 0)
-
6_(f,
#(-
fa)
I*,
-
ft)
dk
(77 J
we
require that the time development of the system from the present a future time t in a state 1^, t) be the same as its time development to 0)
from the future time
t
to the "present past" in a state !^
3
then the two
by a quantum mechanical operation
states will, naturally, be related jtfr
t),
- = t)
3*^}.
3:
(VI.L18)
The Schrodinger and Heisenberg dynamical variables are related by unitary transformations and therefore every dynamical variable (via Heisenberg' s equations of motion) has a well-defined time development. This means that a time-reversal operation will affect all the dynamical variables of a system. For example, in a state 1^, t) all velocities and spins of a dynamical system have opposite directions to those in the state l^, t).
In order to see the fundamental significance of statement (VI. 1.12) let us solutions of equations (VI. 1.10) and (VI. 1 .11). Symbolic forms of the solutions are obtainable from the generalization of the solution (1.7.9)
work with the
for a time-dependent rotation operator.
L
/6
Thus we can JU
J
write ?,0)
and
|, _f> =
p
c
exp
f^-
I Jj fl(-f )
0),
ctt^ll*-,
(VI.1.14)
168
Symmetry
where the
Principles
effect of the operator
P
and Quantum is
e
Statistics
[Chap. VI]
to order the operators in reverse order
to the ordering of operators by P, followed by Hermit ian conjugation. More explicitly, let us use the notation PLE for the operation of ordering of operators from earlier time to later time and PEL for ordering In the reverse order. Thus
we have PC = PB and PEL
The
=
=
PLE
PEL?
PLE*
PBj]0 on
the operators following it is equivalent to ordering the operators from later time to the earlier time, plus Hermitian conjugation, From (71 J J), (VI.1.13), and (VI.1.14) it follows that effect of
= =
SPiE H(f) so the time-reversal operation 3 3
PEZ&) H(-t),
is
= PELPli,
where Pj?z,Pzl is a unitary operator U. Thus for the most general form of the time reversal-operation 3, we have 3
The operator
3,
ing (VI. 1 .15}
we have used
as defined
by
=
VC.
(VI.1.15)
is
an antiunitary operator. In deriv-
(VI. Lid'),
the assumption H(t)
3H(t)3~
l
=
H(~t).
= H(t),
or rather
(VI.1.16)
was not true, the time-reversal operation 3 would not be a symin quantum mechanics. The unitary operator U in (VI. 1 .15) element metry is, further, restricted by the transformation property (the spin) of the dyIf (VI.1.16)
namical system. Statement (VI.1.16) implies that (because of the antiunitary nature of 3) the Hamiltonian of a dynamical system cannot contain any its structure. For example, a coupling constant representa particular interaction must be a real number. ing the strength of complex coupling constant would violate the time-reversal invariance of a dynamical
complex function in
A
system.
Another important property of the time-reversal operator 3, not shared by parity operation, comes from its repeated operation. Thus, for example, 32
From
the unitarity of
U it follows
U* =
at/-
1
that
= aW = a(aU-
Hence, taking the transpose of both
sides,
we
1
)
get
~VI-1J
Parity of a Dynamical System
169
or
a2
=
a
=
1
32
=
1.
1;
therefore
yields the result
(see
(VI. 1.17)
=
with spin | we have 3 2 = 1 Chapter VIII). For an n-particle system the time-reversal operator is of
For a
we have
single particle
=
the form 3
3R
3i 3s
,
where
32
3i, 3s 7
,
1
;
with spin
1
are time-reversal operators for
3
individual particles.
In accordance with the correspondence with classical theory we can assume that the position operator q commutes with 3. But from Schrodinger's representation of the
that 3 and
p
momentum operator and from the
= -p.
3p3-i
The operator
antilinearity of 3
it
follows
anticommute: (VI. 1.18)
momentum and
3 anticommutes also with the orbital angular
therefore with the spin operator itself:
= -8
333-1
p and s under 3 operation imply certain of a time-reversal invariant Hamiltonian. forms possible and 5 can occur in the Hamiltonian only in a time-
The transformation restrictions
(VI.L19)
.
on the
properties of
H
The observables p
reversal invariant combination, with themselves or with other observables of
a dynamical system. For example, the operators s-3FC, L-s, p 2 p-s, are typical linear Hermitian operators that are invariant under time-reversal operation. ,
VI.l.B. Problems 1.
Prove that at a fixed time, say
to a time-reversed state
is
just the
t
0,
the
wave function corresponding of the wave function
complex conjugate
of the original state. Prove also that the
corresponding to a time-reversed state
wave
is
function in a
<*(
momentum
space
p).
The
absolute value of the scalar product of any two states (iF) and ]$}, change under time-reversal operation, where ($1^) is called the "transition probability amplitude." If a is an observable, producing 2.
or
|
(#1^)1, does not
transition
from the
amplitude (#|a|)
state i^} to the state |#), then the transition probability
satisfies
the equation
where
=
3|*>,
I*'}
=
3|*>.
Symmetry
170 3.
and Quantum
Principles
Statistics
[Chap. VI]
The unitary operator
which connects the states the collision matrix (or reversed form
is
S
at infinite past to states at infinite future, is called
quantum mechanics. Prove that
matrix) in
its
time-
given by*
=
S-*
=
(VLL22)
ff.
VL2. The Pauli Principle a symmetry of nature that in the space and time lies in a deeper level than the symmetries contained A system is said to consist of indistinguishable parities discussed above.
The
existence of indistinguishable particles
is
N
particles
if
no observable change
interchanged.
The
basic quantities
system can be divided into two
made when any two of its members are describing each member of the iV-particle
is
classes.
of spin but Dynamical variables: all particles have the same magnitude a position, or not necessarily the same directions of spin; each member has momentum, and a Hamiltonian operator associated with it; the dynamical orbital momentum, energy, momentum, and position variables' (a)
angular
spin,
obey the corresponding uncertainty relations. of the system have the (b) Intrinsic properties: all particles
same
electric
charge and mass.
A
system of
N
particles satisfying the particles. If
"dynamically equivalent" then the total Hamiltonian
particles,
above conditions
may
be called
ignore the interaction between the the sum of the Hamiltonians of in-
we is
dividual particles,
H = H(l) + H(2) +
+ H(N),
N
to represent the dynamical variables where we used the symbols 1, 2, total Hamiltonian is symmetrical The Hamiltonian on which each depends. ,
with respect to the interchange of the particles. Since the interactions of the the corresponding Schrodinger particles are neglected, each particle obeys equation
:
ih
1
at
1*,,
*)
= H\*t,
*>,
for i
=
-
1, 2,
* L. Wolfenstein and J. Ashkin, Phys. Rev., 85 (1952), 947-949 R. H. Dalitz, Proc. Phys. Soc., London, A, 65 (1952), 175-178. E. P. Wigner and L. Eisenbud, Phys. Rev., 72 (1947), 29-41. F. Coester, Phys. Rev., 84 (1951), 1259; 89 (1953), 619-620.
,
N.
(VI.
.1)
The
[YL2.]
Pauli Principle
The assembly
an
of
171
system as a whole obeys the Schrddinger
JV-partiele
equation ih
&,
j
f}
=
// ^,
(VI. 2 J]
t).
t
^, I) is some symmetrical function of the states j^,-, t). If a stationary state for the ith particle, with energy eigenvalue E in for then the corresponding stationary state j^} of the assembly 2,
where the state i*,)
is
=
1,
n
1
j
*
,
satisfies
the equation
H\&) =
E is given by
where the total energy eigenvalue
E = Ef +
(VI.2.3)
JS7j*>,
E]
+ Ei +
-
-
-
+
ES.
where -
*W>fc>
,s
=
1,2,
,
AT
and i
^
^
j
k
y& ... 5^ Sj
so the energy of, for example, the ith particle can be
E
eigenstate corresponding to energy
so 2,
We
Ei or
?,
and so on. The
can be written as
=
s = 1, N, JV are possible eigenfunctions corresponding to the same energy J3; can interchange the particles and corresponding states without changing
any one
of these with
i
2
1,
*
?
-
,
j
JV,
1, 2,
-
-
-
,
-
,
,
the energy E. Every possible interchange of the particles leads to another way we see that to a given eigenvalue of the
E
state for the assembly. In this
states, so there
a degeneracy arising from the ex-
assembly correspond many change process. At a given time we can form certain linear combinations of the is
remove the degeneracy. There are only two possistates can be formed by adding all possible products individual eigenstates of the type (VI. .4), obtained by permutations
eigenstates (VI. 2. 4) to bilities. (a)
of
among
Symmetrical
the particles of the assembly: I*.}
=
2
P|*f>|*$>
-
-
-
(VI.2.5)
[*?),
P
where
^ means
the
sum over
all
permutations P. There
is
only one sym-
The degeneracy
is
thus removed.
p
metrical state for a given total energy E. (b) Antisymmetrical states can be
formed in a determinantal form:
172
Symmetry
The antisymmetrical the particle states
by each [*) and
particle [#&)
a, 6,
is
and Quantum
Principles
Statistics
[Chap. VI]
state !>Px) of the assembly represents a state where q are occupied and where occupation of any state ,
The occupation
equally likely.
by any two
particles,
where *&a ) and
of
&
b)
any
of the
two
states
are the same, does not
= 0. Hence we correspond to a state for the assembly since in this case \^A) see that two particles cannot occupy the same state that is, spin direction, momentum and energy eigenvalue, position, and so on, cannot be the same. ;
For example, the wave function of a twowith respect to the coordinates of the electrons, symmetrical must be antisymmetrical with respect to the spin wave function of the two-
This
is
Pautfs exclusion
electron system,
principle.
if
electron system.
Because of the symmetrical structure of the Hamiltonian of the JV-particle assembly, the corresponding state as a solution of the Schrodinger's equation will vary with time according to the particular symmetry (symmetrical or antisymmetrical) with which it started initially. This is believed to be a basic
law of nature; it is not required for the formulation of quantum mechanics but it can be brought into quantum mechanics by formulating the particular statistics required
of the states.
by the symmetries
For symmetrical states it is possible same state. The statistics implied by similarity to classical statistics It is
for
two or more
particles to
this description
where one uses the concept
important to point out that the statistics implied
states of
quantum ensembles
statistics) is
(Bose
be in the
seems to have some of Gibbs ensemble.
by the symmetrical
considerably different from
classical statistics.
VL2.A. Density Matrix According to the discussion in (V.8.17), the
classical distribution function/,
as follows from (V.8.1&), satisfies the equation of conservation,
f =-[/,#]*, where the number of similar particles
is
(rc-*-?)
given by f f dT
=
N, with
N dT
=
n <%
dp*
t=i
as the phase space element. In this description
move independently without mutual
all
N similar dynamical systems
interaction.
The
distribution function
/
can be used to calculate the average value of any dynamical variable A(q, p) according to
(A)
=
dT. f Af
(VI. 2.8}
The
[VI.2.]
In
Pauli Principle
quantum mechanics
173
the phase space at a given time
t
cannot be defined
unambiguously. For example for one-particle distribution we can, at l>est, imagine a "cell" in phase space who.se dimensions are of the order of 3
(AflApj
=
h*.
quantum mechanics we may sometimes have
In
motion of a dynamical system which
is
not completely
to consider a
.specified.
Such
possi-
example in atomic and nuclear phenomena involving polarization, spin orientations, and angular correlations. In these bilities
occur quite frequently
cases the possible information
for
is
less that
maximum
information attained in
a "pure state/' where we can talk of states of definite spin orientation, represented by state vectors. In situations where the information is less than maxi-
mum
we introduce, by analogy with classical statistical of the a "density matrix" to represent such states. The concept description, use of the density matrix is of special importance for the description of manypossible information,
body phenomena. The idea duced by von Neumann.
The
of the density matrix description
was
first intro-
states of dynamical systems introduced so far are "pure states" charac-
by the possibility of performing an experiment in the particular stale leading to a prediction with certainty. The particular state is assumed to be an
terized
eigenvector of a Hermitian operator which forms a complete commuting set, itself or with some other set of Hermitian operators. If a beam of photons,
with
for example, is transmitted
100% through a
suitably oriented Nicol prism,
then we say that each photon of the beam is linearly polarized. This is an example of a pure state. No other state of polarization has this property with respect to the same prism. In this case the information on the system is maxi-
mum. However, a beam
of photons with partial polarization is not, with or reflected by a prism. In this case the information is certainty, transmitted less than the maximum. It is possible to analyze a nonpure state into pure
We
first assume a certain probability law and consider a dynamical that can be found in one of the many possible states according to system
states.
this law.
For the sake normal states
of simplicity let us consider a complete set of discrete ortho-
|jS n )
and expand a pure
state \&) according to
where the expansion coefficients, the representatives of JTP), define the pure state completely. Let us consider various other pure states represented by l^**).
Each pure
state can
be expanded according to oo
|(>)
=
^
n=*l
a
( |
n},
(VL&.10)
174
Symmetry
and the average value states is given by
and Quantum
Principles
of
Statistics
[Chap. VI]
an observable a with respect to any one of these pure
Calculation of the probability of a certain experimental result where the dynamical system is in a nonpure state proceeds as follows. (a)
We assume a probability P
s
for each pure state
The average value of a dynamical variable a
\& (8} ).
then defined by taking (b) the average of {a) s with the corresponding probabilities P s Because of the possibility that each P is different, this procedure corresponds to an inis
.
coherent superposition of the pure state's average values of resultant average value of a is
It
a.
Hence the
can be rewritten as
nn'
where
Hence p
=
^
|*
s
is
the
quantum density
operator.
With
this definition the resultant
average
value of a becomes <*
From
=
tr
the definition of the density operator p
operator and therefore
with the
classical
its
we
eigenvalues are real.
see that
A
it is
a Hermitian
comparison of (VI. 2. IS)
average value (VI. 2. 8) shows that the phase space integraA times the classical distribution function / is
tion of the classical observable
replaced in quantum mechanics by the diagonal sum of the quantum observable a times the quantum density function p with respect to a complete set of eigenvectors. If the representation is
the discrete
sum
in (VI. 2.13}
is
replaced
by
taken as a continuous one, then
The
[VI.2.J
Pauli Principle (a)
175
=
1
J
)
d3'.
a corresponds to an identity operation, then unity. This leads to the restriction If
=
tr p
(VI.2-14)
value must be
its
1.
(17.2.1,5)
the application of the concept of the norm of an operator [see (1.5 J 1)1 follows that the density matrix is further restricted by
From to
PJ it
tr (p 2 )
1.
(VI.2.16)
If the pure states 'ip-<>) are solutions of Schrodinger's equation, then the density matrix, as can easily be seen, satisfies the equation
which
differs
from the Heisenberg equations
of
motion for a dynamical
vari-
able in the minus sign on the right. It has a formal resemblance (except in the total time derivative) to classical conservation equation (VI. 2. 7) for the classical distribution function
/.
Furthermore, from the definition of p we deduce the statement
E
<Wp!0> =
\
a"\*P*
(VI.2J8)
s
on the probability
of the observable
tion (having the values
-
9
which are diagonal
in
\/3 n }
representa-
).
w is formed from pure states \^ } for density matrix of finite order N. The conditions of Hermiticity and (VI.2.15) imply that the s = 1, 2, , 2 1 parameters. Hence a state matrix p is completely defined by 2V" density 2 1 data. For example, the pure states can be identified by 2V" possessing
N
A
*
-
N
-
N
= 11 pure states of spin orientation. is 5 and it has (2s 1) Therefore the description of a specific state of spin orientation generally re2 1 = 120 independent data. quires (II) spin of
To
+
Co 60
illustrate the
above formalism
let
us consider orientation of particles
states of opposite (up and down) and hence the density matrix is a 2 X 2 matrix. In a nonpure state the degree and direction of spin orientation is described by a vector,
of spin P,.* In this case one has
two pure
spin orientations
q
=
( ff)
=
tr (<rp) 3
(VL2.19]
where the a are Pauli spin matrices. For definite spin orientation in a pure = 1 in a state of random orientation, \q\ = 0. The fact that state, one has |q| ,
*
TL Fano, Rev. Mod. Phys., 29
(1957), 75.
176
Symmetry
the Pauli matrices
and Quantum
Principles
Statistics
[Chap. VI]
together with 7j form a complete algebra can serve to
cr*
express p in the form*
=
p
where from
tr p
=
1
and
tr
diffi
+
=
a-,
204
Using the expression (VL2.19)
aacr*
+
ajffj
+
a*,
we have
=1,
=
a*
for the vector g,
use the easily established formula tr Hence the density matrix is
(0%-ov)
p=i(I + 2
= .
i-
we obtain
-
gt
=
2a,-,
where we
25,-y.
g ).
(VI. 2.20)
This result can be applied to the treatment of Larmor precession of spin orientation in a magnetic field. The energy of an electron in a magnetic field 3C with magnetic
moment
/* is
H where
=
ft
=
-M-se,
(Vl.s.ei)
(e/2mc)h
use the equation of motion (VI.2.17) for p, with the Hamiltonian operator given by (VI. 2. 21) and the definition of p as in (VI. 2.80), we find
we
If
the classical result
9 ,,
dt
[see (1.4.^)
and
!L f. U v mc^ A
(VJ 9 $>P\ K JL.X5.X/^>) '
IP *V
I
(7.0.#0)].
A second example can be constructed from the classical anology of a system in thermodynamic equilibrium at a temperature T. The classical Gibbs distribution of such a system can be taken over in quantum mechanics in the form H T p = A
where
A
is
a normalization constant defined by -~ 1 A = e H xT (tr
H
<'
)-
quantum Hamiltonian of the system. From (VI.2.2S) we see that p commutes with H, so the energy of an assembly in a steady state can be measured, as well as the probability that it remains in this dp/dt = and
is
the
state.
VL2.B. Bose-Einstein
Statistics
A deeper understanding of classical and quantum probability and statistics concepts
may
be obtained from a comparison of the basic ideas of Boltzmann
and Bose-Einstein (and
also of Fermi-Dirac)
distributions.
The
classical
phase space can be imagined to consist of a large number of small "cells," *
H. A. Tolboek and
S.
K deGroot, Phya. Rev., 83 (1951),
189.
[VI.2.]
The
each of
size a,,
Pauli Principle where
i
=
number
numbers are that the
of particles in w,/?;,-,
where
the total volume of phase space being F. particles (or systems over the cells. Let n be
1,2,
i
,
X
Consider the distribution of the
177
17 a
=
\
states (or cells) so that the
-
In Boltzmann
*
-
1, 2,
.
mean occupation numbers
of all states are small
so the distribution of the particles over different states
another. If each one of distributions.
(ifi)"*
number
1,
particles
n,-
The
probability that
is
all
is,
compared
cell
assumed to unity,
is
independent of one
ij*
states the result is
particles will
n,-
that H 2 particles will be found in
"equal a priori probability" (that probable)
put into one of
is
occupation
statistics it is
number
2,
be found in
and
cell
so on, with
points in phase space are equally
given by
p* =
TT^'
the basis of the Boltzmann distribution for a system of particles not in equilibrium. Derivation of equilibrium state distribution, as the most
This
is
probable distribution, follows from the entropy* of the system of particles:
8=
K
log PB.
(VI. 2.25)
Hence
For very large Hi we can use the formula log nl
=
S and obtain the entropy
log
1
+ log 2 +
n
f
JQ
in the
log
a:
dx
-
n log
+ log n (VI JS Jiff)
6
form
In terms of the average occupation number,
-
ffi t
=
n^/rn,
we have
where
i
where *
E is the
total energy.
i
With these
conditions,
and from the method
of
E. Sdirodinger, Statistical Thermodynamics , Cambridge University Press, Cambridge,
1948.
Symmetry
17ft
Principles
and Quantum
Lagrange's undetermined multipliers,
we
Statistics
[Chap. VI]
find the condition for
maximum
entropy (the equilibrium state) as
+ AN + BE\=Q
Jr (~ or
n(A where A and B are
where
F
1/icT,
+
BE,
-
log
Tli)
=
0,
constants. Hence the most probable distribution
a constant; the constant B is where K is the Boltzmann constant. is
For the Bose-Einstein distribution the
identified in
is
thermodynamics as
restriction of equal a priori prob-
state. The relaxed, there can be any number of particles in each of ways of number total the become to is modified probability (VUJ4) ability is
states. In this case, instead of a large amongst number of small cells, as in the Boltzmann picture, we have a large number of cells with no restrictions on their sizes. We may envisage the phase space
distributing
n
t
particles
17 ,-
as consisting of a one-dimensional array of cells of various sizes, including the possibility of equal cells as well as cells of zero dimensions. If there are 17* 1 phase space cells are partitioned from one another by 17* further separate each of the ?& particles distributed over 77* states arranged along the one-dimensional array by additional phase space 1. The number n{ walls. Thus the total number of walls in the array is rji
states,
walls.
of
then the
We may
which one can choose
in
ways
17*
1
walls out of TH
+ +n
t
1 objects is
the
required probability for the Bose-Einstein distribution, or
The corresponding entropy, distribution,
following the procedure for the
Boltzmann
is
SB*
=
K
^
*[(!
+ **) log
(1
+ n) -
(VI. 2. 29)
Jh log nj,
i
which for
Ui <SC
17.
reduces to the corresponding entropy for Boltzmann distri-
bution. For very large
n
i7
equation (VI. 2.89) becomes
(VL2.80)
which would follow from SB by replacing n^ by m and 17* by probable distribution for a Bose-Einstein system is therefore
where
G
is
a constant.
n-.
The most
[VI.2.3
The
is
179
now a two-panicle system and
the only two possible states that can occupy. According to Boltzmann statistics, if the two-particle system
Consider
It
Paul! Principle
thermodynamir equilibrium at very high temperatures, the probability any one of the particles in either state is equal (fqual a priori prttb-
in
of finding
The
distribution consists of a probability of J that both particles are in state I, a probability \ that both particles are in state II, and a probability ability).
| that If
one particle
we apply
of the
two
is
in states I
and
//.
the Bose-Einstein distribution,
particles.
From
(VI. 2. 4)
we can
we
use the symmetrical states
form, for a two-particle system,
three symmetrical states,
(VI3JSZ)
!*s >
where a and
6 are
=
!*?>!*},>
+
!*?>]*?/>,
used to designate the two
particles.
From
the
quantum
mechanical density function (VI. 2.23} for two particles at high temperatures it follows that p = A and that the three states (VI. 2. 32) are equally probable.
= I, 2, 3, to occur is just -J. Hence probability of each state |^}, for i the Bose-Einstein probability that two particles are in the same state is greater The
than the classical probability. Furthermore, in the Bose-Einstein description of a gas consisting of particles
with mass in a volume
of the gas.
The compression
V there is a limit to the maximum possible density of
a gas beyond a certain density gives
so-called Bose-Einstein condensation
by
phenomenon. The
and
density.
the
superfluidity exhibited
liquid helium is attributed to Einstein condensation where,
tain density of the liquid, the decrease of the
rise to
volume does not
beyond a
cer-
affect pressure
The condensation phenomenon does not occur with H*. The
and f; the spin obeys Bose-Einstein statistics nuclei are Fermi-Dirac statistics. In general, all nuclei with odd mass the but spin J obeys number obey Fermi statistics and all nuclei with even mass number obey Bose statistics. Therefore a natural expectation is that there exists a close connection spins of
Hj and Hi
between spin and
statistics of
VL2.C. Fermi-Dirac
a
particle.
Statistics
Fermi-Dirac statistics is the statistical description of systems of particles obeying Paulf s exclusion principle. In this case each quantum state can be occupied by one particle at most. Therefore, the numbers n { are of the same order of magnitude as the number of states rjt. The probability PB or PBE of Boltzmann or Bose-Einstein statistics is replaced by the number of ways of
180
Symmetry
distributing in each.
n
-
t
Principles
and Quantum
individual particles over
m
states,
[Chap. VI]
with at most one particle
Hence
and the corresponding Fermi-Dirae equilibrium
where
Statistics
D is
a constant.
It is
shown
in statistical
distribution
is
mechanics that
(for
a gas of
identical particles) the decrease of temperature at a fixed density leads, in the case of Fermi statistics, to an increase in the pressure as compared to the
value for ordinary gas (Fermi degeneracy of the gas). Since a Fermi gas of identical particles has an antisymmetric wave function, we say that the quantum mechanical exchange effect sets up an additional
As was pointed out previously, the exchange effect for a Bose gas (symmetric wave function) gives rise to the 77 condensation phenomenon and hence to an effective "attraction between effective "repulsion"
between
particles.
the particles.
VL3. Isotopic Spin 7
has been pointed out in the discussion of Pauli s principle that the intrinsic properties of the particles described by charge and mass are not directly It
involved in the application of this principle, even though mass and charge possible play the primary role in the indistinguishability of the particles. answer to the solution of the problem may be found in a further increase (in addition to energy and angular momentum quantum numbers) of the number
A
of permissible quantum states of a dynamical system. Historically, the experimental observation of the charge independence of nuclear forces between protons and neutrons was the main motivation for a further generalization of
the concept of indistinguishability; that is, as far as nuclear forces were concerned the positive charge of the proton and the zero charge of the neutron did not play any role. Therefore, like two electrons, one with spin J and the other with spin J, the nucleons (protons and neutrons) could be regarded as
two
different charge states of the
spin degree of
freedom for
particles, it
"same"
particle.
follows that
By
analogy with the observ-
we may look for an
able whose two eigenstates correspond to protons and neutrons with eigenvalues describing charge and chargeless properties of the particles. The ob77 servable in question has been called "isotopic spin. If we assume that the
we assign the eigenvalue f to Thus proton and neutron corre-
total iso topic spin of the "nucleon" is f, then
the proton and the value
| to the neutron.
181
Isotople Spin
[YI.3-]
spend to
and
isotopic spin-up
isotopic spin-down, respectively.
We
can thus
introduce an isotopic spin space. For convenience we shall use the abbreviation "7 spin" for "isotopic spin." The I spin formalism can be developed in a way completely analogous to or rather, to angular momentum formalism. If the idea of freedom were confined to neutron-proton systems only, it
ordinary spin
I spin degree of
would be somewhat difficult to believe in its validity as an isolated law of We must look for a possible degree of freedom for I spin for all dy-
nature.
namical systems. Therefore we can begin by postulating the existence of a linear vector Hermitian operator T t for i = 1. 2, 3, like angular momentum -,
Ji, satisfying
the angular
momentum commutation
=
tr, TJ1
i
ijk
Tk
relations
(VI.SJ)
.
With respect to the / spin concept all dynamical systems (for example, elementary particles) can be put into two classes, (a) Dynamical systems with = and T = 1. (b) Dynamical systems with integral "intrinsic I spin," T half-integral "intrinsic 7 spin/'
T
J.
We
shall use the notation
%n
for
operators of / spin f which are of the same formal structure as Pauli spin matrices o-^ and they satisfy the commutation relation (VI. S.I). The opera,
tors of 7 spin
1,
or
,
for i
=
1, 2, 3,
have the same formal structure as
ordi-
K
matrices). nary matrices of spin 1 (that is, the The charge of the nucleon can be represented by
QN =
e+(Jr,
+
(VI.3.2)
i),
is r,-. Thus proton and neutron correspond to eigenvalues | and | of r 3 respectively. Hence we say that the nucleon is a particle with total 7 spin value |. In a similar way a typical particle of 7 spin 1 is called a T meson and its charge is defined by
the third component of the I spin matrices
where rs
Qr =
(VI. 3.3)
***,
the third component of the 7 spin matrices K,-. Its eigenvalues are (see Section 1.8), so altogether there are three TT mesons corre1, 1, + e+ and 0. We can use two representations, where in to charges e sponding each one TZ and K$ are diagonal and define 7 spin | and 7 spin 1 spaces spanned
where
+
*3 is
and
,
,
by the
eigenvectors of r s and
KS,
respectively. In this case the wavefunction
of a particle is either a spinor in space of 7 spin | (as for the nucleon) or a vector (or a scalar) in space of 7 spin 1 (or space of 7 spin 0). The ir mesons
constitute a vector in space of 7 spin
wave function
of 7 spin 1 lr>
(See Section 1.8.)
is
of the
1.
Thus the most general form
of the
form
= UXik) + 1-iX-ik) + = T+ |l> + ir_| - 1) + WO).
[o}(ok>
(VI.S.4)
182
The
Symmetry eigenvectors
|1),
A
axes, respectively.
erated
Principles
and Quantum
[Chap. VI]
and jO) are called positive, negative, and neutral rotation around the neutral axis of I spin space is gen1),
j
by the operator
*c 3 :
=
I*-')
Hence, as in Section
Statistics
1.8,
we
e***
T).
obtain the gauge transformations of the
JTo)
=
first
kind,
|TTO),
which leaves the length of the vector unchanged:
(TTJTT)
=
f
(ir'\Tr )
=
{XC [TTC).
The quantity (TTJT) is related to the probability of finding three x mesons. In this way we see that the three TT mesons can be described in terms of a single wave function which transforms like a vector in space of I spin 1. In a similar
way
the nucleon representing the proton or neutron transforms like a spinor
in space of I spin
|.
In general, strongly coupled particles interactions where electromagnetic can be related to a charge multiplet with an / spin forces can be neglected
quantum number T and "orbital / spin/
can assign
7
multiplicity (2T
+
1).
We may
use the concept of
by the commutation relations (VI. 8 J), and we where the Coulomb interaction is negligible com-
as implied
(for light nuclei
pared to nuclear interaction) a
T
value to each nuclear level. In particular,
one can use the conservation of I spin, as for the case of ordinary spin, in nuclear reactions leading to some selection rules.
The conservation
of I spin
independent of charge.
In
is,
of course, valid only for reactions that are
this sense all strong interactions leave charge
tiplets, corresponding to eigenvalues
T3 =
T,
T
+
1,
,
mul-
T, rigorously
degenerate. Hence, electromagnetic interactions, being essentially chargedependent, will not conserve total I spin. The electromagnetic interactions, if taken into account, can differentiate between proton-proton and neutronneutron interactions and hence between proton and neutron, and thus remove / spin multiplet degeneracy of the nucleon. The T$ component of the I spin,
being a function of charge, will
and a charged
commute with the Hamiltonian
particle system, since the coupling
between the
pends on charge. Hence the electromagnetic interaction conservation of
will
of a last
photon two de-
not affect the
TV
VI. 3. A. Isotopic Spin of a Meson-Nucleon System
The I
wave functions
a two-nucleon system are eigenfunctions of the third component of the total intrinsic I spin: spin
of
r
=
|(r a
+r
6 ).
(VLS.8)
[VI.3.]
Isotopic Spin
functions consist of three symmetric and one antisymmetric (with respect to the interchange of nucleons; functions (see Section I V.I. A, prob-
The wave lem
1),
Inn}
=
\pp}
=
]an)'bn),
(VI.3.7)
where the symbols p and n signify proton and neutron, respectively. Now we can generalize the Pauli principle by including J spin as an additional quantum
number of quantum states. If the I spin wave functhree in (VI. 8.7), then in accordance with the generalized Pauli principle the space and spin wave functions must be antisymmetric if the two-nucleon system is in the fourth J spin state of (VL8.7), then space number
into the possible
tion is one of the
first
;
and spin wave function must be symmetric. For the symmetric triplet state in (VI. 8. 7) we can say that the I spins of the nucleons are parallel and add up to total I spin 1. For the singlet state the / spins are antiparallel and add up to total I spin 0. For a meson-nucleon system the appropriate / spin operator meson and nucleon / spins:
T= From total
\r
the rule of addition of angular
J spin values are
(see Section
T=
f and
T
is
the
+ K.
sum
of
(VI. 3.8)
momenta it follows that the \ The corresponding wave .
resultant
functions
IV.3.B) are
7/5
l""-/!""^/
T
A/Q
laifoM&w},
(VI.S.9)
v for
J spin value
T = f and Ts (f) = -f, - J, ,
ir-p)
=
+n)
=
E
\a
-
J,
'
f
a7r+
states,
and
\air-}\b
(VI.3.10)
for
I spin value
T=
% and
-;^
T3 (i) =
i,
i states. For two-7r-meson systems
and Quantum
184
Symmetry
(as follows
from problem 4 of Section
Principles
Statistics
[Chap. VI]
I V.I. A) there are altogether 9 / spin
states corresponding to the total / spins,
T =
0,
T =
1,
and T
to Finally, formula (VI. 8. 2) for the charge can he generalized
=
2.
an A-nucleon
system by writing for the total charge
Q A = e+(T
3
where T*
is
the third component
+
JA),
of the total I spin vector operator
T=
E H).
n=I
Thus the
total I spin
T
is
the
sum
(in
accordance with the addition rule for of the nucleons. Hence total I spin
angular momenta) of the individual I spins T assumes integral values for even .4 and half-integral values for odd A.
VI.3.B.
The Conservation
of
Heavy
Particles
reactions leads Experimental evidence on nuclear and elementary particle into to the empirical statement that "heavy" particles cannot be transformed in that means This the that if any than kss whose masses are proton. of particles
will reaction a heavy partick is annihilated or is lost, then another heavy particle of heavy particles take its place. The best evidence in favor of the conservation matter. nuclear of the of stability course, is,
a heavy particle, then a semi-theoretical formulation of charge of conservation of "heaviness" must be based on the conservation is not symvia / spin degree of freedom. The definition of charge by (VI. 8.8) metrical with respect to the assignment of I spin. There is, of course, no theoretical reason for excluding the possibilities of assigning the I spin value Since the neutron
is
for a neutral heavy particle. | for a charged heavy particle and the value | In this case formula (VI. 8.8) can be replaced by
The corresponding heavy e+
particles are the
one with I spin
J
and charge
and the other with I spin \ and charge zero (antineutron). basis for these particles will be discussed in the relatheoretical The actual mechanics. In this section the experimental facts on tivistic form of (antiproton)
quantum
and antineutron are sufficient for an understanding of laws formulated in (VI. 8.8) and (VI. 8.18). of a new quanEvidently the definitions of QN and Qx imply the existence
the existence of antiproton
tum number, called "the baryon number." In terms of the baryon number we can combine (VI. 8. 8) and (VI.8.18) into a single statement where the baryon number
N=
1
N
corresponds to proton and neutron and the
185
Isolopic Spin
[VL3.]
= 1 to antiproton and antinenfron. To complete the baryon number A" picture we can define the charge of the mesons and nueleons (and antiby
nucleons)
=
e+(T*
+
|A1
(VL8.15)
?
= K3 for even odd baryon numbers with N = 1, and Tg r = A Hence the content 0. the with meson numbers is zero. bary&n of baryon The absolute conservation of Q means that the conservation of ?\ Is the same where
T = 3
ir for 3
as the conservation of
A'.
However, this last statement does not exclude the possibility that there may still exist a quantum number that is absolutely conserved. Because conservaany elementary particle reaction the number number of antinucleons remains unchanged. But there Is nothing to prevent us from generalizing the heavy particle conservation into a conservation law: in any elementary particle reaction the number of heavy particles minus the number of heavy antiparticles is unchanged (proton and tion of heavy particles implies thai in
of nueleons
minus
the
antiproton being the heavy particles with lowest mass). Thus, in addition to the absolute conservation of electric charge, one can introduce* another quan-
tum number
that
is
quantum number is
also absolutely conserved. This
called
=
+
nueleons the baryon charge n. It can assume the values 1, for baryons = antinucleons antihyperons; and for 1, for antibaryons hyperons;
+
mesons
(TT
mesons
+ K mesons).
Therefore the most general form of electric charge (after separating out the absolutely conserved baryon charge n) is a combination of three from
N
quantum numbers number
of I spin T$,
baryon charge
n,
and "strangeness" quantum
,
= T3 i<3 e where the conservation of integral
"quantum number"
(VI. 8. 16}
is
3
,
now
(VLS.16)
the same as the conservation of the
the strangeness.
The
general conservation law
and mesons. The strangeness nueleons (antinucleons) and for
applies only for nucleons, hyperons,
quantum number TT
T
+ in + iS,
assumes the value
for
^ mesons. All strongly coupled particles with The quantum number is conserved for all
particles.
tions. Therefore, for all electromagnetic is
and strong
are called "strange"
T3
conserving interac-
interactions, strangeness
conserved.
For / spin (Ts) nonconserving interactions (weak interactions) a change \AS must be the same as the change AjP3 since T3 + i is conserved. The observable T$ + i can be regarded as a further generalization of the I spin ,
* M. Gell-Mann and A. H. Rosenfeld, Annual Review Annual Reviews, Inc., Palo Alto, Calif., 1957, p. 413.
of
Nudear
Science,
Volume
7,
186
Symmetry
Principles
and Quantum
Statistics
[Chap. VI]
be compared to having an "orbital / spin* and "spin / spin," respectively. Thus the sum of T% and | is always conserved. However, the above theory of strangeness, as proposed by Gell-Mann, concept, where T$
and |
7
may
not necessarily complete, since there may still be a possibility to extract iS another absolutely conserved quantum number like Q and the baryon charge n. Furthermore, it has not yet been possible to derive the
is
+
from Tz
observationally "correct" law (VI.3.16) from a basic theory such as
quantum
electrodynamics. that the only possible total intrinsic I spin values for strongly interacting particles are 0, |, and 1 and that the baryon charge, like the electric charge, can assume the values 1. 1, and 0, then the strangeness
Now,
if
we assume
quantum number
is
limited to values
2,
1, 0,
1,
and
2.
Hence strange-
=
2 and the corresponding 21+1 values of the magnetic quantum = 2. number m, has the total strangeness number 2 = 3(3 1), where 3 that the From these arguments we reach the important conclusion concept ness, like
I
+
some important and deep-seated bearing on the problem and that it is as yet in its infancy stage.
of isotopic spin has
of the elementary particle
A law
similar to that for the conservation of the
baryon charge
exists for
light fermions or leptons (electron, muon, neutrino, plus the corresponding antlparticles positron, positive muon, and antineutrino) the law is called ;
conservation of the lepton charge. Lepton charge is defined as the number of leptons minus the number of antileptons. The fact that total / spin is not conserved by electromagnetic and weak interactions implies that the I spin concept as introduced for strongly interacting particles is not involved in the
conservation of the lepton charge. With respect to the two distinct laws of conservation of the lepton and baryon charges, all elementary fermions can
be put into two classes Baryons can further be
leptons (antileptons) and baryons (antibaryons). classified as nucleons (antinucleons) and hyperons
(antihyperons). All elementary fermions, such as particles of spin f, are to be described by spinor wave functions. All mesons are bosons with
assumed
and they can be put into two classes with respect to the strangeness = +1 The value = corresponds to TT mesons and quantum number 1 to four K mesons. All mesons are assumed to be described by pseudoand scalar wave functions. The photon is the only known particle of spin 1 and is
spin
.
by a vector wave function. The lepton conservation law is best
described
muon
GU)
decay processes.
A
by the nuclear p decay and proton can capture an electron and produce a
neutrino,
p where we use the symbols trino, respectively.
We use
+
e~
illustrated
-
n
+
v,
(VI.S.17)
proton, neutron, electron, and neuthe convention that the neutral massless particle
p, n, e~, v for
187
Isotopic Spin
[VI.3.1
in reaction {VI. 3.1?)
Is
energetically possible
From
to be called neutrino.
that
forced reaction
Is,
(VI.,3.17)
we can
infer the
from conservation of the
electric charge, >-
p where we
shall
+
n
r+
+
(VI. 8.18)
*,
assume that proton and neutron have lepton charges of
zero.
assign a lepton charge of +1 to the neutrino, then the conservation of 1 for the positron e+. In this the lepton charge Implies a lepton charge of
we
If
sense,
if
Next
Is a lepton, the positron Is an antilepton. us consider the free decay of a neutron,
the neutrino let
n where the symbol
v is
meant
*-
+
p
e~
+
(VI. 8.19}
v,
for another possible massless particle of spin
|.
Since the electron has a lepton charge of +1, then the v particle must have I and therefore v is an antilepton (antineutrino). the lepton charge of An interesting elementary particle reaction involving leptons alone is the free
decay of a positive muon:
0+ _>, e+
From
+v+
the conservation of the lepton charge
p,
it
(VI. 8.20) follows that
JJL+
and e+ have
+ equal lepton charges. Hence /i is an antilepton. According to the lepton conmesons zero lepton charge. Therefore, the reaction law all have servation -
7T+
M+
+
cannot take place with an antineutrino; but 7T-
-
IT
+
V
(VI.3.21)
v is
produced in the reaction
V.
(VI.8.82)
We reaction, with the same strength of interaction as the electron capture, occurs also for fir: must point out that a capture pr
+ p^n +
v.
(VI.8.28)
muons (/i+, yr) and elecare essentially the same. It is, therefore, logical to expect the existence of other decays that conserve the lepton charge Hence we trons
(e
+
,
see that the electromagnetic properties of
e~~)
:
IT-
and
> e~
+v
(VI. 8.24)
also
These reactions have, actually, been observed.* Another interesting evidence in favor of lepton conservation comes from the observed radioactive
ft
decays et TT mesons, > e+ v TT+ TT-
*
Fazzini, Fidecaro, Morrison, Paul,
> e-
+ + j, + v + 7,
and Tollestrup, Phys. Rev.
(VI. 8.28) (VI. 3.27) Letters, 1 (1958), 247.
188
Symmetry
where the symbol 7
Principles
is
and Quantum
Statistics
[Chap. VI]
used for a photon. Finally, the lepton-conserving
reactions
+ _>. 3+ -
+% _^ e +7 f M
(VI. 328) (F/.S.50)
have also been observed.*
VI.3.C. Isotopic Spin and Nuclear Levels
We have
seen that the degree of freedom for isotopic spin is confined only to heavy particles; leptons do not possess this symmetry. The only degree of freedom for lepton charge, with the exception of neutrinos, is invariance
under "charge conjugation." This invariance is a symmetry principle which is valid for electromagnetic and nuclear interactions but which fails for weak decay, \i decay, and so on). The charge conjugation that the laws of nature remain unchanged if all positive
interactions (nuclear
symmetry means
ft
charges are replaced by negative charges and vice versa, or, in more general form: the laws of nature remain invariant with respect to interchange of a particle
This symmetry breaks down for all processes involving the neutrino or antineutrino. The conclusion is that electromagnetic interactions (because of their charge dependence) do not conserve with
its antiparticle,
total
and
vice versa.
I spin and weak interactions violate charge conjugation
(also parity).
We may add that the absence of an / spin for leptons is also inferred from the absence of strong interactions for leptons. Thus, from the point of view of I spin, there seems to be an impenetrable wall between leptons and baryons. This situation does not arise with ordinary spin, where baryons and leptons have the same spin. But baryons and leptons do not have the same interaction properties; for example, the proton can interact strongly with the neutron
and
can also interact electromagnetically with the electron, but the electron
it
cannot interact strongly with any particle. Therefore the / spin degree of freedom of a system is closely related to its strong coupling property. Hence the interaction of baryons and mesons must be invariant with respect to
/ spin space, which is equivalent to the conservation of total spin or the charge independence of nuclear forces.
rotations in
We shall now give a
mathematical formulation
of
I
the application of I spin
to nuclear interactions. Consider the / spin operators T+
=
|(TI
+ tY
2)
and
r_
=
fr2 ),
f ( TI
*
Many physicists believe that there are two kinds of neutrino pairs, so the two neutrinos occurring in (F/.3.20) need not necessarily be antiparticles but just different neutrinos + + >- e + >- P+ -f v9 >j? TT obeying the lepton conservation law. Thus > M v^ e + -f- ve and so on, are possible decays, with ve and v^. being different frQin, " aixd *>*
+
,
+
,
[VI.3.]
189
Isotopic Spin
which satisfy the commutatloB relations
where
The charge
of the nucleon
can be expressed by
QN = If
we
replace
e+
e4
by
"
e+r+T-.
(VL8.88)
and take the Hermitian conjugate
of
Q& we }
obtain
the charge of the antinucleon:
-Q$ = -^r.r+
QN -
(FJ.^4)
The last result (the charge conjugation) can also be obtained by a unitary e+ : transformation induced by r2 and replacing e + by T^~T. f r__T2
=
T2 T+ T_T2
=
T_T+
.
The
operators r^r and r_r+ are projection operators and therefore their eigenvalues are +1 and 0. The I spin eigenstates in a representation where TS is diagonal are
The operators r+ and
T
act on the proton
and neutron
states |1)
and
|
1),
respectively, according to
r+ !l>
=
= =
r+ |-l>
0,
r_|l}=|-l),
r_|-l)
|1>,
0.
Thus r+ is an operator which annihilates a proton state and converts a neutron state into a proton state. Similarly, r_ converts a proton state into a neutron
state
and annihilates a neutron state. For the operators r+ r_ and r_r+ we have T+ r_|l>
=
=
r+ r_l-l)
|1>,
0.
Such processes of conversion occur in ft decay of the nucleus, which involves the conversion of one neutron to a proton, while positive p decay involves the conversion of a proton into a neutron. Besides
ft
decay, nuclear forces are
dependent on exchange effects (exchange forces) where the proton and neutron are interconverted one into the other; the operators r+ and r_ provide a convenient tool for the description of such processes. For a two-nucleon system [see (V7.5.7)] we define the charge exchange operator
by
P^ =
T2
_
i
=
i(i
+ Ta
.
Tb))
(VI.8.35)
Symmetry
190
=
Principles
and Quantum
Statistics
[Chap. VI]
and 1. r b) and the eigenvalues of the total I spin are J(ra Therefore the eigenvalues of the charge exchange operator are obtained from
where r
+
T = 1 for the singlet state [the fourth state in (VI. 3.7)} and for the triplet states [the first three states in (17.3.7)] of two-nucleon systems. 2 The charge states (VI.3.7) are simultaneous eigenstates of r3 and T , so where
we
T=
obtain Pi|nn>
=
metric triplet states
= [pn>, and Pl,\pp] = |pp> for the symand P^\pn) = - Jpw> for the antisymmetric singlet
jnn>, PS>!pn>
-state of (VI.S.7).
state wave phase shifts are the same for all triplet charge on the total of states behavior depends functions. Hence we deduce that the is result This very similar / spin and not on r3 (neglecting Coulomb effects) total spin s and not to ordinary spin theory where the levels depend on the the symmetry forces. Furthermore, one as spin-orbit on s so
The energy
levels,
.
long 8| character of the
The ground
neglects
wave function depends on the
state of a deuteron, for example, in a space-spin
I spin and not on r3
total
where
I
=
wave function.
It
and
s
=
1
.
(triplet
must, therefore, be
symmetric spin T = and the corresponding wave antisymmetric in the charge state; that is, in (VI. 8.7). function is given by the fourth equation state), is
By
analogy with
P& we p;,
can define a spin exchange operator by
=S 2
1
=
i(l
+
(VL3.36)
S = |(
where
exchange operator of two nucleons. In accordance with the generalized Pauli principle under the operation of interchange of I spin of two nucleons, the wave function must change position, spin, and sign:
PSWiPi* = -*
(VI.3.37)
that (PS&) 2 Multiplying both sides by PJ and P%> and noting we obtain
for all the
we can
wave functions ^
=
(P&)
satisfying the generalized Pauli principle.
=
2
1,
Hence
write
ra -n)
and
(VI3J88)
r 2 (the
of the position vectors 11 Majorana exchange) r 2 In a of the relative coordinate r = r x the to sign reversing equivalent Thus the nuclear two-particle system this is the same as the parity operation.
The interchange is
.
force depends on the relative angular momentum of the nucleons. The operation by Pab does not change the state for even I and it changes its sign for odd I.
191
Isotopic Spin
[VI.3.J
The charge independence or I spin degeneracy of nuclear removed by treating the Coulomb energy and mass difference
force? can be of nueleons as
a small perturbation. This Implies the existence of I spin multiple!?.
From
QA/* =
(VI. 3.11), for the charge of an ri-iiucleon system, and setting %, we obtain
T,
Z and
=
i(Z
N refer to the number A = Z+A
-
Al,
(VI.3.39)
and neutrons in the nucleus, the total J spin, defined by respectively, = f we have T3 = J, (VI. 3. 12), assumes half-integral values. Thus f or T y3 = J. For A = 15, the mass number, (VI. 8.89) yields the two nuclei T = 7. These are called mirror nuclei', the with Z = 7, A = 8 and Z = 8, in one of them number of neutrons equals the number of protons in the other. where
of protons
r
and where
For odd
.
A
N
The 2T
+
(A, T), (A, I
1 nuclei
-
7
1), (A, -T), corresponding to 1 values of Ts, are called isobaric nuclei; the total / spin T and the 2T they are all described by the wave function ^ for the nucleus (A, T3 ), and ,
+
have the same energy, provided we neglect the Coulomb energy. If then the original nucleus contains more protons than neutrons. If we 1 replace one of the protons by a neutron, the corresponding J spin is T3 they
all
>
jP3
and the maximum number one.
The
of identical particles (protons)
process of replacing a proton
possible value of
/ spin
less
than
T
by
decreased by a neutron should not lead to a
since otherwise there
3,
The
isobaric nuclei (Aj
T3 )
isotopic multiplet with total
when Coulomb and mass of isobaric nuclei can
is
the possibility
Thus T$ is the maximum value a proton by a neutron is allowed.
of violating the Pauli principle.
which the exchange of
is
T =
of
I spin
for
T are said to form an J spin T. According to I spin multiplet theory* with
T,
3
*
*
,
difference effects are negligible, the level structure
be (approximately) predicted simultaneously.
VI.3.D. Problems 1.
It is conceivable that very high-energy electron-electron or electron-
nucleon scattering
magnetic coupling
meson production, in which case the electronegligible compared to the strong coupling of meson
may is
lead to
production. (The production cross-section
is
very small, but in principle
it is
an isotopic spin to leptons? 2. Is it possible to develop an 7 spin theory for particles whose strong interactions are negligible compared to their electromagnetic and weak possible.)
*
Can
L. Eisenbud
1958.
one, then, assign
and E. P. Wigner, Nuclear
Structure, Princeton Univ. Press, Princeton,
192
Symmetry
Principles
and Quantum
interactions? In such a scheme, leptons
Statistics
and antileptons
will
[Chap. VI]
appear as doubly
degenerate I spin triplets whose degeneracy can be removed by their lepton charge.
Show
3.
that for a meson-nucleon system the projection operator
P,
T=
where
IT
+ K,
=
1-^ = ^7 -T-K),
annihilates
T = f
states
and leaves
T =
|
states
unchanged. that for an 4-nucleon system the antisymmetric I spin wave
Show
4.
function
is
where fa
=
^j(n,
*%
si]
52;
;
rNj S
X =
and Si TI P
=
where i = 1, 2, = 3V P = 1 and -
1,
=
Tap
and neutron I \Tip}\Tz p}
-
-
-
spins.
\TN p}
and
-
\Ti p ,
for the
2V+i>
A
=
spin directions of the nucleons; = 7U = -1 for the proton
Furthermore, the I spin states are of the form |^ n >. The sum is to be taken over all |2W>> *
permutations P of the A subscripts (including the identity permutation, which does not affect the subscripts), and ep is +1 for even permutation,
odd permutation.* 5. Prove that for a two-meson system with I spins K Q and K&, respectively, 2 the operator Ka -K b is an exchange operator. The total I spins T = 1 and T K -K and the effect of states and on to a b symmetric antisymmetric correspond 1 for
T =
2. Write down the 9 / spin it by the reasons for the antisymmetric and states of a two meson system explain and symmetric properties of the / spin states corresponding to total / spins T = 1 and T = 2, respectively. How would you reconcile these results with the fact that mesons obey Bose-Einstein statistics?
the state
consists of multiplying
*L. W. Nordheim and F. L. Yost, Phys. Rev. 51 (1937), 942.
CHAPTER
VII
HARMONIC OSCILLATOR REPRESENTATIONS The Fock Representation
VII.l.
For a direct illustration and also as an extension of the various concepts introduced in the previous chapters, we shall work out in detail the quantum theory of a harmonic oscillator. The present treatment contains also the foundations of
field quantization,
which in
this
book
applied only to mass-
is
less particles.
A
free
harmonic
mechanics.
oscillator is the simplest
The Hamiltonian
Hamiltonian of the
classical oscillator (I11. 1.8 6)
6
as a linear operator.
=
satisfies
(P
J(6*6
+
is
(VILL1)
given by
66*),
=
6]
hu.
(VII. 1.2}
H becomes H=
It is
treating the quantity
the commutation relation [6*,
Hence
by
+ im}
Thus the Hamiltonian
H= where 6
dynamical system in quantum can be obtained from the
of the oscillator
more convenient a+
to
and the commutation
=
i/ko
work with the
7=
6
=
+
66*.
linear operators
,
(p
+
relation
oj =
1.
(711.14) 193
Harmonic
194
Oscillator Representations
[Chap. VII]
The Hamiltonian assumes a simple form:
H The harmonic to Fock*
and
it
= /Wa 4 a_
+
=
i)
/to(a_a^
be
oscillator representation to
-
(VII. l.d)
|).
discussed in this section
is
due
has further been extended and applied by Diracf to the theory
of second quantization.
From
the commutation relation (VI 1. 1.4}
we can
easily
deduce a more
general commutation relation, [a_,
=
a\J
nan+\
(VII. 1.6)
where n
is a positive integer. The commutation relations (VI 1. 1.4) and (VII. 1.6) imply that the operator a_ acts like an operator of differentiation
of the
form d/da+.
The Fock representation minimum energy eigenvalue
of
E
Q,
harmonic
oscillator
H
assumes that
has a
corresponding to an eigenstate \EQ ),
=
H\EO)
(vii.L7)
E,\E,).
Hence we obtain
=
fiua+a-lEo)
(E
-
and
Using the Hamiltonian, the
last relation leads to
-
Ha^Eo} = (EQ
hc^a,\EQ}
3
which contradicts our original assumption that EQ was the eigenvalue. Therefore the state a-\EQ }
must
minimum energy
satisfy the equation '
=
a.\E
Q}
By using From
this equation
(VII. 1.8)
we can
we can
0,
calculate the
(VII. 1.8)
minimum
value EQ of the energy.
write ftaja+CL-
EQ)
=
or
Hence EQ is
the
minimum energy
(VII.1.9)
i%o>
eigenvalue of a harmonic oscillator.
tation relation (VII. 1.6)
From the commu-
we have
= *
=
~
a+a_
V. Fock, Z. Phys., 49 (1928), 339. Phys. Z. Soviet: USSR, 6 (1934), 425. A. M. Dirac, Ann. Inst, H. Poincare, XI-1 (1949).
t P.
The Foek Representation
[VII.L]
195
Operating on both sides with a_ and using the expression of the Hamiltonian we obtain
-
(H
=
r
|?toQ l
V
Operating on the state
ntiucKr.
+
taar'a-.
and using the condition
I
VII. 1.8)
we
get the
eigenvalue equation
(H
-
iftttJa^l-EV;
=
nhua n+\E<>)
+
$)\E n \
or
#!;} =
hw(n
(VILLKfi
where the eigenstates \E n }, corresponding to eigenvalues are defined
M
=
En
The vectors a^l^o) for n = a representation. To see this,
=
A
has eigenvalues
T
1, 2,
let
jJ^o) is
<4|
(VII. 1.12)
).
can form a complete set and therefore
us introduce a dynamical variable defined by
=
(VII.1.1S)
a+a.,
and the Hamiltonian assumes the form
0, 1 ? 2,
H If
-
-
N where
(VILL11)
by \E n )
T
4- |) ?
=
fto(N
+
(VILU4)
f).
a unit vector, then the length of the vector
= Thus the length
a^-lJ^o) is
n(n
given by
-
of the vector is
(EolcfLallEo)
From (VIL1.8) and
it easily
(VII. 1.15)
<[,> =
= nL
(VII. 1. Iff)
follows that if
n
7*
m.
In general we have
(Em \E n)
=
=
which proves that \En } with ra 0, 1, 2, Hence we can define a unit operator by
7
=
n!6 mn
(VILL16)
,
form a complete
set of vectors.
Xnl-EnXJU n=
where
V\ n is the normalization factor. Any oscillator state
{if}
can be expressed
as
n=0 (VII. 1.1
Harmonic
196
where
=
??
Oscillator Representations
X m(2? n j?7} and the operator jP(a+)
=
F(a+)
is
(VILL18)
in
The vector
[Chap. VII]
representing any oscillator state, can be normalized according
\ij},
to
where
nl\Tj n
state
that
2
the probability of finding the oscillator in the state where its energy is [n f j/ta.
is
\
is,
o|
}
,
aj~
,
to the case of
many
oscillators represented
by
satisfying the commutation relations oJ
+)
+) o}
a^
)
)
aj+)
ai->a}
The operator corresponding
to A^
-
+) aj
o|
+)
af-'aj+) aj
ai-
}
= = =
0,
(V11.1. SO)
0,
is
=
Ni The
nth excited
+
The theory can be extended +)
its
oj^of-^
(VILl.fl)
stationary states of the oscillators are
|#. where
*
ni,
*
7^2,
>
=
+)
)-(4
(ai
are eigenvalues of
+) n2
*
-
|#o>
)
The
-
A/i, JV2,
-
The
state of
minimum
energy \E
Q)
satisfies
oi->|
for all oscillators, so
|.Z?o}
of the
is
\EQ )
Any
=
>
(VIL1M)
length of the vector
(VIL1.8S)
-.
the equation
=
(VIL1J64)
form -
\EQI )\E
state of the oscillators represented
(VILI.SS)
.
by (^
}
can be expanded ac-
cording to
(vii.i.se)
The
state
|t?i 2
)
can be normalized by
X) h nmz
where the
coefficient
*
2
* \
2 \ynint
*
niln 2
==!,
l
(VIL1.&T)
"
|
ni!n 2
!
is
oscillators in the states corresponding to ni,
the probability of finding the
n2
,
.
The Foek Representation
[VILI.j
VILLA. An Ensemble
of
197
Bosons and the Stationary
States of a System of Oscillators Consider a system of n bosons. will be symmetrical with respect to states of the
We all
a state vector which
shall construct
the particles. Let us write the oscillator
form \
a [)
li>
= (4^)-jW>, = (c^) *}, 7
(VII.1.28)
Each of these can be regarded as a symmetrical state since we may,
for ex-
ample, write
%
the operators a{ +} are repeated times; therefore the interchange of Lhe same HI operators among themselves will not change the state \ai). If
we
w ere T
to label each one of ai +}
therefore, look
Bosons. In the
nil possible
ways
of
bosons) without changing the state |i). state |ai) as a symmetrical state of HI can say that there are n2 bosons in the state \a),
(in this case
permuting ni objects
We may,
then there would be
,
upon the
same way we
ind so on. Hence the symmetrical state corresponding to n bosons with ''occupation
2
We may
7
!
7i 1}
n^
+%=
+n +
where ni
Df
numbers
n.
regard the state \a[az
an arbitrary state ara s a*
*
+ a* +
'
given by
is
*
a'j)
as arising from the symmetrization
a q}, where
+ a = n^ + n a + z
q
-
.
z
(VII. 1.30)
Thus we can write
where the symmetrization operator
is
defined
by
S = -4=SP, Vnl with
P
n
implying nl permutations of
bhe length of the state vector \a{o&
bosons. 0.3) is
(77IJ.S*)
From
(VII. 1. 23)
just niln*!
-
we
see that
njl, so (VII.1.81)
yields
{aras a*s
-
-
-
az\S*\a r as
-
-
-
aq) =
nMl -
,
- -
n^l
(VII. 1. S3)
n bosons occupying the states n/, respectively. |ay),,with occupation numbers n n*,
the length of the state vector of
|a2 ),
-
i}
,
[ai),
Harmonic
198
Oscillator Representations
[Chap. VII]
We may
use state vectors of the form (VII. 1.31) and set up a complete set of vectors as Jo>,
-
ia r >, Slara,),
,
S\a*x.
-
-
-
(VILl.34)
*>,
\E$) Is the state with no bosons present, the a r ) are the states with one boson present, the S\ara*) are the states with two bosons present, and so forth. It is evident from definition (VII. 1.31) and from (VII. 1.29) that all these states of various number of bosons are orthogonal. For an ensemble
where
of
one boson we have \a T )
It
means that the operator
\EQ}.
The
oJ
+)
= oJ+W.
"creates
7 '
vector representing any boson state
|>
=
(VII.1.35)
one boson from the "vacuum state"
A,oJ
+
is
given
by
W.
(VIL1.36)
r
Hence, multiplying with as "\ we get (
=
\,\E
},
r
which corresponds to a state of no boson present. Therefore, the operator ~ as corresponds to an "absorption (or annihilation) operator" from the (
}
state \as }.
The operation with
a^~
}
on the general state (VII. 1.29) yields
(VII.L37)
so operation
A
by
a^~
}
decreases the
number
of bosons in the state \a s )
general state of an ensemble of bosons can be represented
in the
by 1. by (VII. 1.26)
form
where the various terms
of SI>(a+ ) are related to the
number
of
bosons accord-
ing to the degree of each term.
Let us
now assume
that the states \a r } for a boson corresponds to the sta-
tionary states with energy levels
2?i, 25*,
k> = Hence the time dependence
of o
+)
+>
=
<^
which shows
.
Thus we have
\Er)e-M**'. is
given
+ (flr
[see (111.1.35)] that the
(VII. 1.39)
by
)o- (W ^.
time dependence of a
(VII .1 40) +)
is
like that of a
The Fock Representation
[VII. L]
199
w r = E r 'ft. This means that an ensemble same of bosons have the description as a system of harmonic oscillators having frequencies E r ,'h. In terms of occupation numbers the total energy of the harmonic
oscillator with frequency
ensemble of bosons
is
H
NE
=
*
*
N
10 oj" ^"* has the eigenvalues n r where the occupation number operator r Hence a set of harmonic oscillators can represent a noninteracting assembly of bosons in stationary states. According to this picture the nth quantum state of one of the oscillators of the set represents a state where there are .
n bosons.
VII. l.B.
We may
An Ensemble
of Fermions and Stationary States of a System of Oscillators
envisage a system of oscillators obeying Fermi-Dirac statistics;
that is, we assume that it is not possible to have more than one particle in a given state. The commutation relations (VII. 1.20) for Bose-Einstein oscillators have now to be replaced by the anticommutation relations of Fermi-
Dirac oscillators:
+ o} +) oi +) oi-'o}- + aj" a^ +) + aj + M-> ai->a}
oJ
+)
+)
o}
)
)
We
= = =
0,
(VII. 1J&)
0,
**.
anticommutation relations are the only
shall see that the
possibility
consistent with Fermi-Dirac statistics.
From
the
first
two anticommutation relations we (a^>)
The occupation number
2
=
2 (o^) =
0,
operator
N
T
is
defined
find that 0.
(VII.14&)
by
= o^o^.
(VII.L44)
Hence
^ +) aJ- = = a^a^.
N* =
aJ
+)
aJ-
)
)
(4
+
\l
-
a^W )a^)
)
Thus JV?
implies that
property of
N N
r
rj
=
N
r
(VII.1JS)
or 1. (This a projection operator and its eigenvalues are required by Fermi-Dirac statistics, could not be obtained
is
from the commutation relations obeyed by aj"f) and a^ -) .) In a way similar to that used for bosons we can define a vacuum state by
=
0,
<
Harmonic
200
Oscillator Representations
[Chap. VII]
Hence we get
(EM+W1E*) = meaning that the length
}
of the vector ai~ \Eo) aJ->jjE
The vector (4 +r\E}
is
>
A
+)
an eigenstate of
iJ?o) is
zero. Therefore
0.
(EQ\EQ}
Furthermore, the occupation number
aJ
Is
we must have (VII.1.48)
of unit length;
=
Hence
=
0,
N
r
N
r
=
i.
operates on a state according to
belonging to eigenvalue
simultaneous eigenstate of all occupation
1.
number operators Ni, N%,
- -
can be constructed as
where now the states arj a$
* ,
*
are
all different
since
two
identical particles
cannot be in the same state. Mathematical expression of this fact is contained in the anticommutation relations (VII. l.J$) and what follows from them
(VIIJ^S). Because of the anticommutation relations the state represented by (VILL4&) is antisymmetric and of unit length. It can be regarded as a q), or arising from the antisymmetrization of a general state \ara s -
where r
is
^s^
5^ g,
and
+
the antisymmetrization operator with sign for even permutations and sign for odd permutations of the fermions. Ensembles of fermions can be
described in terms of complete set of vectors, |#o>,
or),
A\aras),
-
-
,
A\otT a s
...,>,...,
(VII. 1.51)
which are orthogonal and normalized, with \Eo) corresponding to a state with no fermion present, |a) to states with one fermion present, and so on.
A general state
of
an ensemble
of fermions is of the
form (VII. 1 .52)
rs ... q
The operator
function of the a (+) ,
consists of noninteracting occupied fermion states alone.
The operators
TH.l.j and
The Foek Representation
201
flr" are the respective operators of "creation"
and "annihilation" of
femiions.
VII. l.C.
The Matrix Representation of Oscillator Operators
We r
introduce normalized eigenstates of the occupation
number operator
by (VII. 1.6$)
Vnl
n where
(nln'l)
N\n)
Thus the vectors representation.
=
N=
n\n),
n) form a complete set
The following
(VIIJ.54)
a+ a-.
and they can be used
operator relations can easily
to set
up a
be established:
or
Similarly,
We may now
1).
obtain the representatives of the position and
momentum
operators of a harmonic oscillator from
P =
1)
hiiimn
(7/7J. 59)
Harmonic
202
Oscillator Representations
[Chap. VII]
Hence q and p have representatives for those transitions in which the quantum number n of the oscillator increases or decreases hv one.
YII.2.
The
Scfarodinger Representation of a Harmonic Oscillator
VIL2.A. One-Dimensional The Schrddinger wave equation
Oscillator for a
harmonic oscillator
is
Putting
we
obtain
/9E
7
,72.;,
g+(S The wave function must remain
\
)*-
finite for
x
=
.
For large x the term
2E/fua can be neglected; then the solution satisfying the boundary conditions 2 is of the form exp ( |x ). Thus in the presence of the term 2E/h^ the wave function
is
of the
form ^ = e vary 3).
Using the wave equation (VI1. 2. 2) we obtain
where the function
t]
(x)
must be
finite for all finite x,
and
for large x it
must
be such that the wave function $(x) shall tend to zero. The solutions satisfying these boundary conditions are Hermite polynomials defined by
Hn (x) The normalized
Wz) =
= (- l)-e-
(VII.24)
; (e-**).
eigenfunctions of the oscillator are 1/4
[^gTn!)3 ]
exp f- *(/&)**]#
where
The corresponding
1 = 2n. eigenvalues are obtained from setting (2E/H&) an equivalent discussion of the oscillator
It is interesting to observe that
can be given in a
momentum
representation. In this case
we
treat the
mo-
Schrodinger Representation of a Harmonic Oscillator
[VII.2."
mentum ator
by
operator p as an ordinary
its
momentum
number and we
203
replace the position oper-
representation: q
~
.
d
lh
dp'
The Hamiltonian
is
*-&-\*j? The corresponding wave
Wist)
equation,
can be written as /9E?
^2w,
\
=
(VII.2.7)
0,
by y = p/V(mc*)k). The boundary momenta of the oscillator. For must tend to zero and it must be finite for
where the dimensionless variable y
is
given
conditions are imposed for small and large large
zero
p the wave function $(p) momentum. The momentum space
elgenfunctions are given
by
[ be obtained directly from the operator formulation of the harmonic oscillator. We shall discuss it in detail. Let \p) be the nor-
The above
results can
malized eigenstates of the momentum operator p corresponding to the eigenvalue p. Consider the representatives of equations (VII. 1.55) and (VI1. 1.8) in
momentum
representation,
{p\a+ \n}
(p\a^\E
where
<
n +i(p)
=
Q)
= V(n = 0,
(p\n +
+
l)(p\n
+
1}
=
<^+i(p) V(n
+
1),
(VII. 8.0) (VIIJ6.10)
1}-
The
(p\a+ \n)
=
representative (p\a+\ri) f
f
J (p\a+ \p
can be written as
}
(p
dp',
)
where f
{p\a+\p
)
=
f
1 /
V(2muK) L
1
d
pd(p
p')
hmo>
P
p') V
6(p
J
(VII. 2.11)
Hence (p\a+ \n)
=
P ,
<j>
n (p)
J(^^\
^ n (p).
(VII.%.12)
Equation (VII. 2.9) now leads to r
V[2(n
^n
~
TlTTlCd
/I
\\\
,77
\l_2(n
+TTT 1)J
d
j
dp
/-TTT-T
/r
^
x-.\
2. 13) (VII. v
Harmonic
204
Oscillator Representations
[Chap. VII]
A similar procedure applied to equation (VII. 2.10) leads to
where fa
=
(p\EG)
=
Equation (YII.2.14)
(p\0) is the
solved
is
The
constant
A
is
normal
state
momentum wave
function.
by
=
Ae-< l 'W>' hmu l
determined from the normalization
as
A =
1M
[^J
,
SO
=
r
Ji
-
~i i/4
e -(i/2)(P v^).
(VII. ^ 2.15)J
[_2Trhmu_\
In terms of
By
y,
equation (VII.2.1S) reads as
V[2(n + we obtain
successive iteration
*
=
Using the definitions of nomials,
we
VII.2.B.
1)]
as given
by
(VII.2.15),
The Three-Dimensional
of the
Hermite poly-
Oscillator
The quantum Hamiltonian corresponding (III. 1.41)
and
obtain (VII. 2.8).
to the classical Hamiltonian
is
ff
=
iftw
1=1
where '(2muh) 1
and toT, a/]
=
$<,.
(VII. 2. 18}
Harmonic
Sehrcklinger Representation of a
[VII.2.]
The Hamiltonian can
Oscillator
205
be written as
also
H
= ho(X
+
fX
(VII. 2.1 9)
where
= A
T
A'
afar,
A'i
+ .V, + A
T
j
=
A*2
8, T A3 =
a^ai",
a^ai".
Hence the corresponding eigenvalues are
En = where n
=
ni
+%+n
+ I),
(T7/.*.?)
The corresponding eigenstates of N are
3.
AHn)
fa(n
=
nJTi),
where
ai\EQ) =
The number integers
of
ways that n can be written
(4)(n +
is
l)(n + 2),
degenerate. If the Xj,
x2 and ,
sum
of three nonnegatlve
rcg
components is
+ l)(n + 2)-fold
of the oscillations wi,
o>2,
^s are not
given by (III. 1.38} and the degeneracy
In the latter case the energy levels are
Enz = ^(ti 2 + total energy is Eni + E^ + E ny
E^ = The
as a
0.
so the eigenvalues are (4)(n
equal, then the potential energy is lifted.
= a^\E ) =
TtoiOi!
The wave
+
),
^=
i),
^3(^3
+ 4).
functions for the degenerate case are
W 3,.3
"11/4
[ TWC /?716)\1 rr
r
(T) J
\?*
For a one-dimensional (a
H
*>
l/mO)
Vu
oscillator in the
Gaussian type wave function), the
normal
state, represented
maximum occurs at the origin.
by ^
Further-
more, the probability of finding the oscillator outside its classical region (the region between the two ends of its oscillations) does not vanish, even though in such states its total energy arises
A
from the uncertainties
particle's
capability, in
is less
than
in the position
its
potential energy.
and momentum
quantum mechanics,
barrier higher than its total energy
is
The
latter
of the oscillator.
to penetrate a potential
usually called the "tunnel effect."
VII.2.C. Problems 1.
Show
that the average values of p 2 and g 2 with respect to harmonic
oscillator states are given
by
Harmonic
206
Oscillator Representations
=
2 )
Hence prove that the measure
(n
iJAwm,
and p
of spreads in g
AgAp =
Thus the harmonic
+
oscillator level
(n
n
+
=
(zero-point energy) corresponds,
its
2.
satisfy
$)h.
Gaussian type wave function, to a state of Consider the differential equation
with
[Chap. VII]
minimum
uncertainty.
+ nu = Q ~%~x^ ax dx~ and the equation obtained from
that
f
orn =
1
the equation (VII.2.24) 1
u ,
f
-
V27r./where
is
n and x by
by replacing n by
it
f + x|+tfShow
(VII.g.83)
a small positive number.
0.
ix:
(VIL2.24)
is satisfied
by
ex z
By
integrating differential equation
(VII.2.24), prove that integral (VII.2.8ff) can be reduced to x
2
f
3. Using expression (IV. 3. 17) for the square of angular momentum, show that the simultaneous eigenfunctions of L2 [see (VII.%.19)] } and I/ 3 can be expressed as
H
,
tnim
where hn are solutions i
" and
n,
4.
wave
Z,
By
2
ft
.
=
- h nl (r)Y lm (d,
of 1(1
+
l)
x
*
m are eigenvalues of H, L
2 ,
mc
and
V 2
."I
J Z/ 3 ,
^= ,
, (
w
respectively.
using a power series expansion of the one-dimensional oscillator
function,
show that the expansion
coefficients
obey the recursion formula
-
-
2(n
o 2n
-= T2E
s)
where 1
1.
[VIL2.]
Show
Schrodinger Represent a lion of a Harmonic Oscillator
that the higher terms of
^
differ
values of
tive constant, so for
from
-r
,
j
207
'
only by a multiplica$ jt will behave like ex2 Heoce the thoi?e for
c
.
(
only acceptable wave function can be obtained by breaking off the series after a finite number of terms, leaving a polynomial. The value of E which causes the series to break off after the sth term is E = h&[& II-
+
Solve Heisenberg's equations of motion for the oscillator hence show that the average values of q and p lead to the classical form of the oscil5.
lator. 6.
Use
definition (VI. 2 23] of the density operator for a
system in thermo-
dynamic equilibrium and calculate the average energy of an = hu(N + ).] Then thermodynamic equilibrium. [Hint: put
oscillator in
H
Now use (E) =
tr
(pH) to obtain Planck's formula, /T-A
7.
By udng the
1*i
ft
expansions
n=0
give expre^ions for the expansion of a delta function in terms of oscillator functions. Hence derive the expansion of e (t"/A)9p. Comment on your
wave
results. 8.
By using the theory of harmonic oscillators as developed in this chapter,
up a similar approach cussed in Chapter IV. In
set
for the representation of angular
momentum
dis-
particular, compare the set of irreducible eigen7 for the total vectors (IT 1.38) angular momentum and the complete set of .
eigenvectors (VI1. 1.34) for a system of harmonic oscillators. Use the resemblance between the operators J+ J_ and a+, a_. 7
CHAPTER
VIII
THE LORENTZ GROUP \1IL1.
The Three-Dimensional Complex Orthogonal Representation
notion of spin comes from the Lorentz covariance properties of the equations of quantum mechanics. The integral and half-integral spin concepts
The
are closely associated with the transformation property of the field representing a particle (scalar, vector, tensor, or spinor fields). Spin is an intrinsic
property of a
field.
For these and other reasons we have to discover
sible representations of the
It
all
pos-
Lorentz group.
has been shown in Section
1.9 that real
orthogonal transformations in
three-dimensional space are homomorphic (one-to-two correspondence) onto
unitary transformations in the spinor plane.
The proper Lorentz transforma-
tion (for instance, three velocity components and three angles of rotation) as a six-parameter representation of a homogeneous Lorentz group and also
improper Lorentz transformations can have a complex space representation.* Every complex linear manifold is suitable for the representation of the Lorentz group. For example, the three-dimensional unitary space cannot be used for
the representation of the Lorentz group. This is because of the Euclidean nature of this space and also because of the requirement of 8 parameters _ i = i) for the representation of the three-dimensional unitary (n
y
unimodular group.
We may
envisage a three-dimensional linear manifold spanned by the special type of complex vectors formed from space and time components of an antisymmetric tensor field.
The
representation of the Lorentz group
orthogonal group
=
is
also suggested
by
by a three-dimensional complex
the fact that the Lorentz-invariant
wave can be replaced by the two Lorentz-invariant statements that the electric and magnetic vectors of the wave are (a) of equal magnitude and (b) perpendicular to one another. At each point of the wave we can set up an "invariant coordinate system" path length ds*
*
208
of a plane electromagnetic
B. Kur^unoglu, J. Math. Phy*., 2, No. 1 (1961), 22-32.
r
Complex Orthogonal Representation
YIII.L]
magnetic vector
with the electric vector of the coordinate system
209
JC. chr>osing
the third axis
in the direction of its propagation
(or its spin
direction;.
4X4
matrix, which under a Lorentz transLet fas i>e any antisymmetric formation of coordinates transforms like the antisymmetric combination
oo
,00, two 4-vectors !
of
r
Thus, ?
jj
3
=
if
f'(x
jL;xD
;
)
is
the matrix resulting from a Lorentz transformation
wehave / = LfZ, ;
(VIII.L1)
where o
*
/
""" J
In accordance with the notation of Chapter
Cx! =
The bra (x;
We
is
defined
/ /M
o
Da,
,
II,
we
(vni.i.2)
define the bra
Cxi by
(VIII.1.3)
]
by
must now distinguish between two types of scalar products of two i%) and if) belonging to the three-dimensional complex space.
vectors
(a)
The
Lorentz-invariant scalar product of
cr]x3 = (b)
The Hermitian
In particular, for
|f)
fai
jx)
and
|f) is
+ r x2 + r^xs.
(vm.L4)
2
(or gauge-invariant) scalar product of |x)
=
\x)
and
|f) is
we have
CxlxD =
x!
+ xi + xi
= o+
2iA
(VI II. 1.6)
and <xlx)
=
2
Xil
+
2
ixsl
+
where
= which
is
a
3C 2
-
2
scalar,
A =
fi.se
which
is
a pseudoscalar, and
which
is
called the dual tensor of J>.
=
l
The
quantities
e'
1
'^ constitute a contra-
variant tensor of fourth rank, antisymmetric with respect to the interchange
The Lorenlz Group
210
two
of
indices.
The
*trf
of (1234), for
any two
1
*
Is
so the
pvpff
They transform
= LZLmLl**'** =
^ depends on
of
det
the
,
the former
fpiige-invariaiit; that
the transformation
is,
is
(I/^
ff ,
of det (I). It
Is
seen from
Lorentz-Invariant and the latter
(VIILtJB) and { VI ILL?} is
assumes the values of even
odd permutations of (1234), and like an axial quantity as
of
for
indices.
w
+1 when
[Chap. VIII]
(VIII.L7) for an arbitrary. Invariant, real phase variable All vectors of the complex three-dimensional manifold are of the form
not l(x).
(YIILL10) or its complex conjugate. The ket 'xa> can further be qualified as a complex vector by its transformation properties. For this purpose we introduce complex orthogonal transformations which leave (VI ILLS) unchanged. If R is the transformation matrix of a complex ket, we obtain another ket belonging to the same complex
by Hence
CxVD so the invariance of
R
= Cx\e*RRe*\xD - CxixD-
R
(VIILL6) under
transformations requires that
all
transformations must satisfy
RR = RR = These conditions on
e-^.
(VIII.L12)
complex or twelve real equations; therefore, of the eighteen parameters fixing R (plus the phase 5), six are independent.
The
six
-R
correspond to six
parameters together with a given for 5
^
5 will fix
member
a
of the
complex group (extended complex group We shall be interested in the two subgroups of the extended group corresponding to the two special values of (a)
6
=
0).
5.
and
0,
ER = RR = L This
is
the complex orthogonal group.
Thus we have determinant
+
tion matrices
(b) 5
all
=
The determinant
of
R
is
to distinguish between a pure complex rotation 1
and a
rotation reflection group
with determinant
the unit matrix J3 are not
(VIII.L1S)
.
1.
The
+1
or
1.
group with
which includes transforma-
identity element of the group
Because of the nonunitary nature of
R
its
is
eigenvalues
of unit magnitude.
T/2,
and
RR = RR = -L
(VIILL14)
This or
is i.
211
Complex Orthogonal Representation
FVIII.l.]
the antiorthogonal group. The determinant of J? in this case, is +i We shall see presently that antiorthogonal and improper R transforma3
tions are equivalent to antilinear operations
on the vectors
of the
complex
space.
The complex conjugate
R
jugates of
of
any ket
\x) is
transformed by the complex con-
transformations satisfying the orthogonality and antiorthogonal-
ity conditions,
R*R* = R*R* = and
** = E**
I
(VIII.L1S)
= -7,
(VIII.U6)
respectively. If (7 is it
the operator of complex conjugation (antiunitary operator), then
can operate on %} to produce x*>
=
and lx'*>
where we
=
eixO
=
PB|x>
use the antiunitary operator property*
C2 =
(VIII A. 18)
I.
For the proper complex orthogonal group a member R can be assumed to be a function of an invariant parameter r. It satisfies the equation [as follows from differentiating (VIII.1.1S)] d i
where
Z
is
^ = ZR,
a complex antisymmetric
3X3 matrix,
Z=
namely,
-Z,
j
'
where a and 3
fc
j ",
for j
=
1, 2, 3,
are real numbers.
+ ib*Kj,
Hence the
(VIIL 1.20}
six generators of
the proper complex orthogonal group are
K
i}
iK
t.
The former
generates pure rotations and the latter corresponds to velocity transformations. Typical proper R transformations are *
The subgroup
of the
extended group considered here
to four different values of the determinants of
U
is
a four-sheet group corresponding
transformations.
The Lorentz Group
212
= exp [ity + iX)XJ, B2 - exp [i(4> + i)Kd, R = exp [i(0 + ip)K ].
(VIII. 1.22)
jSk
(VIII.1.23) (VIII.1.24)
z
z
[Chap. VIII]
A Lorentz transformation
of the electromagnetic field can now be effected a matrix. Let us consider the Ei-transformation which by complex orthogonal has the matrix form
"10+
iX)
sin (^
+ i\)
cos (^
cos (^
_0
sin
+ zX) + zX)
= =
where cos (^ sin (^
(i^
cos sin
^ cosh X ^ cosh X
+ iX) + iX)_
^
sin
sinh
(VIII. 1.&
X,
+ i cos ^ sinh X,
and
=
tanhX
c
The Lorentz transformation by Ri corresponds to a rotation around the Xi direction by an angle $ and a uniform motion along the same direction with velocity
v.
From we
obtain
=
Xi
X2
= 7
=
y\
%2
X2
/
-
^
cos
(
sin
^
+
i
sin
-
^j
cos
+%
^
)
3
Xi,
(sin
+%
3
(
^
cos
+ ^ - cos ^ ^
i
-
sin
L
)
^J
(VIII.1.27)
L
where
Equating the
real
and imaginary parts
formation laws of the electromagnetic
=V 3
=
we obtain the
of (VIII. 1.27),
trans-
field as
sn
cos
+ -e Xt) cos + + -
-
+ -c X.) sin (VIII.1.28)
= y
=7
cos
sin
^
sin
^
V c 63)
which for ^
=
{
213
Complex Orthogonal Representation
[VIII. !]
=
reduce to
&
ft,
= = 7
2
+~
3
=
JC
,
-
7
,
ft V j;
VIII. 1. A.
The Transformation Energy and
of Electromagnetic
Momentum
Density
Energy, momentum, and stress properties of the classical electromagnetic are described by a symmetric second-rank tensor Ta^ defined by
field
Toft
=
lOafft
-/
Ti,
=
(6
X
3C)<,
(VIII.1.80)
Jg,
where
and the space components T/ are associated with the ponents of
momentum
current) of the
=
rjrg
+A
fijKio*
stress properties (com-
Further properties of
field. 2 )
=
i5?| x
2 2
Ta $
are
(viii.Lsi)
and 2
(|12
A
+A
^
/,C2 C J_ 1P2\2
2 )
=(
-
j
(6
X
2
3C)
.
(VIII. 1.82}
momentum vector p^ there that exists a Lorentzpi p suggests covariant representation of the energy and momentum properties of the electromagnetic field. Ihus, if we take comparison of (VIII. 1.82) with the square of the
of a material particle
(p^ =
2
)
S*
where
V
v
is
=
c
we
T^Y",
(VIII.L33)
=
(VIII. 1.84)
a unit vector, or
V then
-
V
V*
1,
obtain c2
S^ =
T^V'T^Vp
= %V'V P (#P = J& + A
+A
2
)
2
2
.
Hence the energy and momentum density vector SM defined by (VIII. 1.83) transforms Lorentz-covariantly for all types of electromagnetic waves and it reduces to
c& =
i(
=
(<S
c/Si
for V*
=
0,
F = 4
1.
2
X
+ *e 5C)i
2
),
The Lorentz Group
214
The complex space
representation of S?
[Chap. VIII]
and T^ can be obtained, by inspec-
tion, as <s,
where the ten
3X3 matrices Bu =
=
,,
(vm.1.35)
<x!^r'!x>,
are given
by
B = B
/,,
(VIII.1.37)
and, like T^, they satisfy the trace property a*'BM > If o*
=
The
B
0,
then
/Sp
=
(VIII. 1.88)
0.
can be written as
matrices, as follows from the definitions (VIII.L87),
have the matrix
forms
*-l
0" 1
00
K matrices.
the remaining B have already been defined as All B matrices are Hermitian, The matrices B Bfi
-
i:}
satisfy the relations
+ BL + Bis
=
3Z 3
(VIII. 1.41)
+ Bl + Bl
=
2k.
(VIII.1.W
1, 2, 3,
are
and B\
The
eigenvalues of any B*, for
malized eigenvectors of
J5 3
,
i
=
+1,
1,
and
0.
The
nor-
for example, are real vectors
1"
-1
1
V~2
10)
(VIII. 1.43)
[VIII.l.]
215
Complex Orthogonal Representation
Furthermore,
Bl
= B
B? Bi
= Zf, = B
=
Bi,
Bl
= 5
Bl =
m,
Bl
=
Bl
ly
3
(VIII.1.44)
and K%,
Si = -K,.
{,
From
the above properties of tion operators, such as
(VIII.l. 45)
(VIII.1.46)
B and K matrices we see that the transforma-
A (0) = e*,
G(d)
=
**,
(VIII. 1.4?)
induce the transformations
= =
Ix') |
The operator (VIII. 1.48)
is
X6)
(77/IJ.45)
"*'|x>, e *,| x ).
(VIII. 1.49)
both energy-conserving and
also
a Lorentz trans-
formation operator, since <x'lx'}
=
<xl-"*vj*-|x>
=
(xlx)
and
= CxIxD. The operator
(VIII.l. 4S} 9 because of the property (VIII.1,46}, effects only
an energy-conserving transformation:
= (xle-^e^lx) = (xix). The nonsingular matrices B^ B 5 are the generators of diagonal unitary <X6|X6>
2 2,
33
transformation operators; these transformations also are energy-conserving. The transformation properties of B matrices, under simultaneous complex
orthogonal and Lorentz transformations, can be obtained from the vector and tensor transformation properties of S^ and T^ as defined by (VIII. 1.35) and (VII1. 1.36), respectively. The tensor T^ in another Lorentz frame has the form
=
KxWx'X
(VIII.L50)
new
coordinates
related to the old coordi-
T'r
where
7
|x ) is
nates x*
a function of the
x'
and
If
x'*,
by a Lorentz transformation,
7
|x } itself is
we apply
=
Lfr',
R transformation = -R|x>. Ix')
transformed by an
according to
these transformations, together with * /W
in the definition (VI 1 1. 1.50) of
=
Ljfj.JLyl aft
T'^ we obtain (VIII. 1.61)
The Lorentz Group
216 If
we
substitute for
Ta $
transformation hold for
transformation rules of
from (VII 1. 1.36) and require that the complex vectors of the complex space, we get the
on the
all
left
B matrices: =
B*B,Jl
(VII 1. 1.52}
1*1*8.0.
The R matrix is a function of the coefficients L? alone. The symmetry properties of a space formed from \x) physical interest.
[Chap. VIII]
We shall
study some
vectors are of great
of these properties
by the application
of the improper Lorentz transformations corresponding to Lorentz matrices F, F correspond to space- and time-reflection transJ4 . But F and F, and
formations of coordinates, respectively [see (1.10.4)]- I n R space, the corresponding transformation can be obtained by replacing L" by a, which leads to
R^R = <%c&3<#. Hence various
B
(VIII. 1.53}
matrices transform according to
&R = B*K*R
R^R
= =
I,
(VIII.1.S4)
-K<,
(VIII.1.5S)
By.
(VIIL1.B6)
We shall study five different R transformations satisfying
(VIII. 1. 5Jf) an d
(VIILLSg). (a) The R transformation corresponding to the F matrix in Lorentz space can be taken to be the "parity" operator (P, which is a linear unitary spacereflection operator and transforms the B matrices according to (P
2
yCP
=
=
/,
(VIII. 1.57}
By.
(VIII. 1.59}
When
electric and magnetic vectors are regarded as and time, the parity operator (9 acts on a ket \x(r, space
ket
classical functions of f))
to produce another
7
ix }: |X'>
In the above
=
(P!x(r, *)>
we used
=
-Ix*(-r, 0>
=
-ZJlxC-r, 0>-
(VIII. 1.60)
the transformation properties of 6 and 3C under space On operating twice with (P we obtain the original
reflections (see Section 1.4). state,
The matrices
jB#,
where
i,
j
=
I, 2, 3,
remain unchanged under parity opera-
tion. (b)
The
fact that the elements of
K matrices and of 5# matrices are pure
217
Complex Orthogonal Representation
fVIII.l.]
imaginary and real numbers, respectively, suggest the choice of antilinear operator of the complex conjugation
R =
0,
R
as the
(VIIL1.61)
which satisfy (VIH.1.54) and (VIII. 1.56). At this point we note that
of scalar products,
by (xl^lx)
where the it
lowing
(VIILl.ee)
xlxl,
effect of the antilinear operator (7 is to replace the expression fol-
by
whose
Uz,,
=
its
complex conjugate.
We can also
define
an
conjugate. Thus
if
6 is
antilinear operator
the expression "preceding" a complex number, then
effect is to replace
W
=
Vb
L
=
it
by
&*
its
complex
(VIILL68)
and
= Wl =
Vb L
b*CL
=
I.
Hence
=
VI
With the use
of
&L the
1.
(VIII.1.64)
two vectors
scalar product of
|%)
and
[77)
is
defined as
and (x\VL \x)
=
=
XtXi-
= Cxi
is called The vector (x\CL <x* (c) The R transformation corresponding
(VIII.1.65)
the "adjoint" of |%}. to the coordinate time-reflecting
F) Lorentz matrix can be taken as the time-reflection operator of the complex space. We shah now prove that time-reversal operation in R space is an (
1
antiunitary operation.* To find the R transformation corresponding to shall posit
where
The its
in Lorentz space
R = W,
"F is
we
(vnu.ee)
a unitary operator,
definition of
R by
(VIII. 1.66) automatically
satisfies
(VIII. 1.54), and
substitution in (VIII. 1.55) yields
*
Space and time
formations t*
F
~
t}
|aO =
reflections of coordinates in Lorentz space are affected
^|O
respectively.
or X H
'
a;
4
and
t'
=
t,
and |oO
^loO
or
by the x'*
=
trans-
xi and
The Lorentz Group
218
VKiV* = If
R operates on vectors of the type
[Chap. VIII] (VIIL1.67)
Ki.
then the simplest choice for
\x}j
V is to
take
V=
(VIII. 1.68}
I,,
so the corresponding time-reversal operator 3 of three-dimensional complex
space has the form
=
5
(VIII.1.69)
Iff.
A more general form of the time-reversal operator which can operate on vectors of the form e fi jx) is 3
which
The is
e*Iff>
32
=
(VIII. 1.70}
operator
J
3,
33
=
33
=
7.
(VIII. 1.71}
(VIII. 1.70} for 8
as defined
by the complex orthogonal, and for
The
=
satisfies
5
=
7r/2 it is
I,
for 5
=
-I,
for d
=
=
0, 8
=
TT,
and
5
=
2x,
an antiorthogonal operator:
0, T, 27r,
|-
matrices under 3 transform according to
(VIILI.7S}
and
3-^-3 =
=
-K
(VIII. 1.73}
t-.
the parity transformation (VII1. 1.5 8) and timereversal transformation (VIII. 1.72} prove that Ki transforms like a polar vector. The vector iKi behaves differently under 3 transformations, since
For the case
5
ir/2,
and
3-HK& = so iKi for 5 (d)
=
-7T/2
behaves
like
an
(VIII. 1.7'5}
iKi,
axial vector.
A Lorentz transformation reflecting both space and time
coordinates
is
represented by
|O
=
-/JO
(VIII. 1.76}
or x'i
=
-^
^
= -.
In this case the transformation rules (VIII. 1.52} of jB^ become Xt
== Jjfjivllt
Jjp V
or
^Bv.
(VIII. 1.77}
[VIII.2.]
Complex Orthogonal Group
These equations are
219
by the successive application of time-reversal and Thus the R transformation in complex space,
satisfied
space-reflection operators.
corresponding to simultaneous reflection of space and time coordinates in Lorentz space, can be represented as
SR The
=
== 3
ea
(VIILL78)
(P.
that of Pauli, meaning strong reflection.* Actually also reflection includes, in addition to reflection of space and time costrong ordinates, reflection of the field. (e)
notation SR
is
A strong reflection transformation can be represented by Sea) = -6*E(P,
(7777 J .79)
which can operate on e\x) to produce
The change
be regarded as the change of sign
of sign of the field can
of its
source, the electric charge.
VIII.2.
The Two-Valued Representation of the Complex Orthogonal Group
The complex orthogonal transformations (VIII. LI 1) and (VIII.L1S) can be represented by a similarity transformation Q'
=
(7777.*J)
Z7-1QZ7,
where
4} =
X
The
factor
XL d=
%-
the definition of Q, as will be seen presently, is required for The complex matrix U is subject to the condition
i in
2X2
invariance reasons.
det
U=
1
(VIII JtJS)
providing two equations among the four complex elements
of
U. Thus
U
matrices can constitute a six-parameter representation of the complex or-
thogonal group.
The determinant 2
xi
+
x?
+
of (VIII. 2.f) is
x
2
=
x?
+
xi
+
xi
(VIII.24}
Furthermore, in terms of Pauli matrices, (VIIL&.1) can be written as
where the factor *
W.
Pauli, Niels
i
on both sides of the equation, because
Bohr and
the
of possible antilinear
Development of Physics, Pergamon Press, London, 1955.
The Lorentz Group
220
[Chap. VIII]
U operations, cannot be canceled out. It is also important to observe that the square of (VIII. 2.5) reproduces (VIII. 24), showing that i
a x = ZV-E .
-
a-3C
(VIII. 8.6)
We may look upon (VIII. 2. 6) as a two six-vectors (iff, a) and (8, 3C), where the transformation "1'I. 2.6) properties of iff and a are not the same, since the invariance of (VI requires that iff and cr transform like polar and axial vectors, respectively.
transforms in a Lorentz-invariant way. scalar product of
The
generators of
and iKi
of
R
U
Ki The
transformations, corresponding to the generators
transformations, are, of course,
and
i(n,
respectively.
U transformations = for i
=
1, 2,
3 (not
summed
over
exp as rotation operators,
f) }
*
dh exp
for i
=
1, 2,
3 (not
summed
over
and (VIII .2.
-,
as velocity transformation operators
t),
can
provide a double-valued six-parameter representation of the proper complex orthogonal group. Under a Lorentz transformation of coordinates the ket |x(x)} will be transf formed by the corresponding R matrix according to \x ) = R\x)', correspondingly,
a two-component ket \u(x)) of the "spinor" space U matrix according to
will
be transformed
by the corresponding
=
r
\u
)
U\u)
(VIII.2.9)
y
where
If
we
replace %< in (VIII. 2.5)
MKJ
by xi
= RVK,
and note that (VIII.2.5) must hold
for all proper
(VIII.2.10)
R
transformations of the
complex space, we obtain iff&q
= 17-^17
(VIIlJt.lt)
The U matrix is a function of the coeffican easily be verified that the six-parameter group of U
as the transformation rules of the 0. cients Rij alone. It
matrices (VIII. 2.7) and (VIII. 2. 8) satisfy (VIII. 2.11). Quantities transforming according to (VIII. 2.9), with the transformation matrix satisfying (VIII.8.11), are called two-component spinors, corre-
U
sponding to the complex vectors of will be discussed in the next section.
R
space. Further properties of spinors
Complex Orthogonal Group
[VIII.2.]
To find the U transformations It Is
221
corresponding to Improper
R transformations
necessary, because of the antilinear operations Involved, to use (YIIL2.5)
Instead of (YIII.2.11).
An
7
interesting
transformation
Is
one corresponding
to parity operation In complex three-dimensional space:
Substituting in (VI'1'1.2. 8)
we
get
-iV<x
The
left side
can be written as
where of are left-handed Paul! matrices (see Section 1.8), so space-reflection operation is not an invariant transformation in the two-component spinor "space;
that
is
U
to say, there exists no transformation corresponding to R space that can leave (VII1. 2. 5) invariant.
(P
transformation in
Next we consider the transformation *r,-xS
R=
differs
in (VIII.2.5),
= U-H*KiV, an
which, like (VIII, 2.12}, obviously implies
which
(J
and obtain (VIIL8.14)
antilinear
from (VIII.2.12] only by a plus sign on the
U
operation,
and
left.
Let us posit
u = r, where F
is
a 2
X
2 matrix to be calculated. From (VIII.2.1jf) we obtain
where we use the operator property V
The
of the a,
=
=
of
.
special matrix
/=** = [_5
(VIII. t.lg)
J]
transforms the a according to
/~W =
-<#,
(VIII.2.16*)
where
/-
1
=
/
-/,
Thus we can conveniently choose F
Hence, the required antilinear
=
2
=
-I.
(VIII.2.17)
/ and obtain
U transformation
(up to an arbitrary phase
factor) is
U=
3
= ft.
(VIII. 2.18)
The Lorentz Group
222
The
[Chap. VIII]
above has the following properties.
antilinear operator 3 obtained
(a) It satisfies
= -I.
52
(VIII.2.19)
[Compare with (VIII.L71).] (b) It
transforms a spinor
NO (c)
It
is
\u}
according to
the time-reversal operator of the spinor space (see Chapter IX).
transformations of i
The
c<
-**-[_? i][a-[_g. -
cr t
under 3 are given by
Q-Hfffl
=
(VIII A Al)
iai
and X-i ff ff
=
Hence, under time reversal operation,
an is
axial vector, respectively.
now
Thus the
(VIII A AS)
-o-,.
and
-
i
-
transform like a polar and Q matrix
choice of (VIII. 2. 2) for the
justified, since
icr-X
=
ia-S
cr-3C
shows clearly that the invariance comes from multiplying, vector
We
6 by iff and the axial vector 3FC by
U=
=
3L
scalarly, the polar
satisfied
by taking
Erf,
which operates on the bra vectors of u space according to
= VIII.3,
K
tfc]/
= K, -wj.
(VIII A.8S)
Two-Component Spinors
The preceding
discussion of spinors
and
their transformation properties
was based on the one-to-two correspondence of the transformations of threedimensional complex vectors and two-dimensional spinors. We must also study the transformation properties of three Pauli matrices a together with a 2 X 2 unit matrix 04 = I2 directly in terms of Lorentz transformations carried out in Lorentz space.
We
can begin by observing that the relativistic invariant (1.10.2) can be defined by expressed as a determinant of a 2 X 2 matrix
M
M= where 04
=
[VIIL3.]
Two-Component Spinors
Hence
223
M = cH* -xl-xl- xt
det
Unlike the case of (
2
(VIIIJSJ!)
in the present representation
?
(x^)
2
not a
is
Lorentz-invariant expression. This leads to some limitations on the transformation properties of o^ and also on the invariants formed with the use of o>.
A
1 certain class of Lorentz transformation of coordinates x^ into x * can,
formally, be expressed as a similarity transformation:
M Because
of the
f
= S- MS. 1
Hermitian character of M, however, we must use a trans-
formation of the form
M' = &MS,
(VIIL3.8)
so the Hermitian conjugations of both sides yield the
same equation.
We can
also write (VIII.8.S) as
x'^ = SVtvS,
(VIII.84}
where det
and S is not assumed to be a unitary tion (VIII. 3.5}
is
1
(VIII. 3.5}
2X2 matrix. The determinental condi-
required for two reasons:
to reduce the
(a)
S =
number
of
(b) to maintain (VII 1. 3. 2}
det (x'pffj
=
det
parameters specifying an S from 8 to 6 for the transformed coordinates x'*, since ;
S
"
1
-det
(x"
-det
S =
det
(o^crM )
or ct'*
We
-
x?
-
x?
- 42 =
cH*
-x\-xl-
xt
shall see in the following that the Lorentz invariance obtained
by a
determinantal operation alone does not [because (cr^) 2 is not an invariant] entail the entire Lorentz group as an invariance group. Let us stipulate the existence of a class of Lorentz transformations satisfying (VIII. 3.4} such that
if
L
transforms Z M according to
|O then
S
will transform
= i|O,
a two-component spinor \u')
=
where the components ua and u'a for a ,
x
and x',
respectively.
To
(VIII.8.6) \u)
by (VIII.3.7)
S\u),
=
1, 2,
of u}
and u
obtain further properties of
f
)
are functions of
S we
consider the
transformation of a spinor relation, v)
=
The requirement of Lorentz invariance dinate system we must have
(VIII.3.8)
implies that in the transformed coor-
The Lorentz Group
224 ItO
[Chap. VHI]
= '>.
(vin.s.9)
applying the transformations (VIII. 3.6} and (VII1.3. 7),
By
we
replace
(VIII. $.9) by
Hence, comparing with (VI 1 1. 8. 8) and requiring that the invariance hold for every \u)
and
\v),
we
obtain
Using
we
get
as transformation rules of o> and also as relations to define an S transformation as a function of the coefficients !/. The relations (7771.8.10) are valid for all
proper Lorentz transformations.* Because of the very important role played by
For improper (VIII.8.10)
Lorentz
transformations
is
it
convenient
to
replace
by
where the S, are either ordinary or left-handed Pauli matrices. (a) Space-reflection
transformation
satisfies
(VIII.8.1SS)
A
comparison with (VIII.
alternatives for (i)
We
S
.
16)
and (VIII. 2. 2%) shows that we have two
transformations.
can take
S
f and
set S-
=
of,
so the / matrix can be inter-
preted as the "metric" of the spinor space whose vectors transform transformations (ii)
We can also take S =
3
= fC and set 2)- =
(b) Time-reflection transformation satisfies
and S
*
t
by S
.
= S-Vi&
*
which yields (VII1. 2. 22).
S^a^S =
^
av
v
or
Z4 =
* There are quite a few differences between the two spinor spaces defined in Section VIII. 2 and in the present section. Some of these differences must be noted: (a) condition (VIIL2.S) on the determinent of U is not necessary; (b) condition (VI 1 1. 3.5) for S transformation is
U
transformations are functions of By, necessary to maintain Lorentz invariance; (c) while S transformations are directly related to Ly\ (d) the matrices
U
in
S
space and Lorentz space.
If
2,
2
=
=
f
225
Two-Component Spinors
JYIII.3.]
S
the operator of the complex conjugation C. For 1 = t and x' the Up transform like the coordinates where t = x\
then
of,
is
1
o-f,
VIII.3.A. Examples of Improper and Antilinear Transformations In a way similar to that of (VII7. 1.39) we can study the transformation of the four-vector
(VIILSJ3) which transforms as *'*
For S
=
S = "
/ and
1
f= 84
Hence
SM does
=
=
^
-/,
l<>'>
=
=
s{
^
<w|
not transform as a proper vector under
S = fC we SM
S =
/; actually it
does
all.
get 4
In this case
(VIII Jf. If)
^l&^Slu).
we obtain
$4,
not transform as a vector at
Next, for
tt/
=
l
4,
=
Si-
transforms like the coordinates themselves
that
is,
as a
vector.
With S =
We
(7
the situation
is
similar to
see that a four-vector of the
an ordinary vector under
S =
/.
form (VIII. 3.18} does not transform
like
types of Lorentz transformations. These aspects of distinguishing one set of Lorentz transformations from another set are a all
distinct feature of expressions involving o> linearly.
VIIL3.B. Further Development of Spinor Transformations The
'
"metric' / of spinor space can be used to raise or lower spinor indices.
We
can develop spinor algebra in a form quite similar to that of tensors. The metric / can be considered as a second-rank spinor //j, where a, ft = 1,2. We can also define a contravariant metric tensor fa& by the rule
/"^ = Ff* =
s?,
(vm js.it)
should be noted, the uncontracted indices a and jS are taken both as where, a first or second indices in f^ and //?7 The contravariant metric f & is not to be it
.
understood as the inverse of the matrix
The Lorentz Group
226
/=rL-i
x
i
J
oj-
a$ Actually the matrix representation of f is equal to /. The invariant scalar product of two spinors \u) and
In ket notation
defined
v) is
use the operators (?L/ and
we must
[Chap. VIII]
by
fC
to define scalar
u*v l
(VIII. 8.16)
products. Thus
=
1
[u
u} 1
,
\
JjH
= uW -
.
Also,
=
(u\fC\v)
From
a spinor with itself analogy with the proper representations of. the Lorentz group
(VIII.3.16)
vanishes.
By
we can use
we
[~u*, u
see that the scalar product of
all two-valued two-dimensional representransformation matrices must satisfy the
the metric / to infer that
tations of the Lorentz group
by S
condition
= /.
SfS
The reader can,
as a good exercise,
(VIII. 3.18}
show that all S transformations discussed U transformations (VIIL&.7) and
in the previous sections, including the
(VIII. 2.8), satisfy condition (VIII.8.18). By taking the complex conjugate of
k> = we
find that the
complex
s\u},
conjugate of
a spinor
u '*) =
S*\u*).
is
transformed by the complex
conjugate of S, as \
The metric (= fe
say).
,
(VIII.8.19)
of the complex conjugate space is the reciprocal of /; that is, / The covariant and contravariant components of u*) are related by u*
Hence
u The same connections
1
*
= t4
for \u) are
or
We
see that the
components
of \u*) are transformed as the covariant
com-
[VIII.4.]
StereograpMe Projections of Complex Representation
ponents u a of
\u}.
This result has been obtained from using
227
/ as the metric
of the complex conjugate space. But F in / In spinor space corresponds to Lorentz space. The matrix F, as a Lorentz matrix of class Lu 1, trans-
forms coordinates by reflecting time and leaving the space coordinates unchanged. Hence we infer that time-reflection transformation in Lorentz space corresponds to the operation of complex conjugation in spinor space. Spinors transforming like the product of two spinors of rank one are called spinors of rank two. The metric / is an example of a covariant second-rank spinor. Pauli matrices are spinors of rank two.
The StereograpMe Projections
VIII.4.
of the
Complex Representation The fundamental
and
its
invariant of the complex space
complex conjugate *
= xf + xf +
Q* 2 will
Q2
2
x!
=
(F777.4.2)
<xIC|x>
vanish for a plane electromagnetic wave. With respect to the point 2 2 o^ fog invariant Q* is to be regarded as the image of Q in a space con-
_
Q 2 Q* 2
sisting of the three points 0,
,
.
The functional relationship between
the points
Q z and Q* 2 is such that they
Q = 2
and from one another at "invariant disbeing related by an antilinear transformation, cannot be transformed into one another by means of a linear unitary transformation. Under R transformations the "lattice" (0, Q 2 Q* 2 ) remains an invariant structure of the complex space. A more general complex space can are situated from the point
tances."
The
2 2 points Q and Q*
,
,
be defined by including the gauge group in the complex group. Two important Hermitian operators related to complex space can be obtained by adding and subtracting (VIIL4-1) &&& (V111. 4.8) to give ,
1(0*
+
Q*
2
i(Q*
-
Q*
2
)
=
<x|F|x}
=
<x|7>|x)
=
(VIII 4.y>
and where
r=
& are linear
operators
and
)
Y=
itf?
+
=
-2*A,
&),
(VIII44) (77774-5)
= -D =
Hermitian and anti-Hermitian operators, respectively. The and D both satisfy the algebraic equations
(!)
F
7 = F
(77/7.4.7)
D =
(VIII.4.8)
s
D.
The Lorentz Group
228 Thus, both
F and D have eigenvalues
+1,
and
1,
0.
Both
[Chap. VIII]
Y and D
belong
to the complex representation of the Lorentz group.
The effect of the linear unitary operator VCi can be seen by observing that the Hennitian scalar product of two complex vectors jx) and [77} is The double-valued
representation of the rotation group can be based on the stereographic projection* of a unit sphere about the origin onto the equatorial plane x% = 0, with the south pole as the center of projection. To the
point xi, x%, x 3 on the sphere corresponds the point the formulas for the projections are
x{,
x^
on the plane, and
Xl
2a* *
3/3
~~~" -i
1
+ i
=
x-
aa* aa*
fc'
where
a
and
MI, ^2
=
x{
+ 1x2 =
Ui
are homogeneous complex coordinates which enable us to include
the south pole of the sphere in the projection. In terms of the coordinates HI 7/2
we have
Let us
now put
and
where
=
detf2
By
-(aj
using the projection formulas
&=
+ 4 + 4) =
we can
X 2 <^|[(r
X
= (N 2
+
-1.
write
-
(VIII. 4.10)
trj|tt>,
where x
The r matrices
are of the
different degree of
The form
freedom
M
2
)-
1' 2
.
same type as the a matrices but they refer to a of the sphere and therefore commute with the a.
of (VIII. 4*10) suggests that
we can complete
it
by a
vector-
matrix operator as *
See H. Weyl, Theory of Groups and Quantum Mechanics, Dover,
New
York, 1931.
Stereographic Projections of Complex Representation
[VIII.4.]
ft
and obtain three
The matrices
and
-
a)i
inYu)
f i and fs are given
(xi,
(VI II. 4.11)
ar 2 ,
x$)
on the unit sphere.
by
their determinants are
Thus the
=
X
X 2 {u\[(r
possible projections of the point
det
i
=
229
=
ft
det
& =
det
ft
=
-(a-?
+ 4 + 4).
three possible stereographic projections represented
1, 2, 3,
form a vector, and they
by
on the unit sphere. In terms of the a and
so the operator f * lies
ft,
where
satisfy
K matrices,
(VI 1 1. 4-H) can be written as
or as ft
=
(VIII. 4.1S)
-i(ff*KJ**x,;
where
Hence, each unitary transformation
on the equatorial plane sphere. In this
way we
(the spinor plane) corresponds to
obtain
a rotation
It is interesting to note that the operator tion operator
ia^Kp
is
related to the projec-
P = ^K,.
(VIIL4.14)
c
The
projection
of the
possible rotations of the sphere.
all
operator property of P
c
can be seen by taking
its
square, as
defined in (VIII. 4^4):
PI
we can
= KI -
express (
2
2
=
it/
-
2
K+
(
as
(2
Hence
-
r-K).
K+2 -
cr-K]
= P, The spin
operators
commute with
P
c
and
H^Kt + fa satisfy the usual
commutation
(VIII4-1S) relations
The Lorentz Group
230
The operators
*
can operate on the product of the kets of
[Chap. VIII]
and K-spin
spaces, respectively.
We may now jection of the
generalize the above results to the complex space. The pro"complex sphere" onto a spinor plane consists of three second-
rank spinors which together transform like a complex vector; each one by itself transforms as a second-rank spinor. For the projection of ]%) we can write
= -*V
if)
or
r
2
(VIII.4'1?)
X2
-XsJ
where the spinor indices
The formulas
and
of the f
cr
are suppressed.
of stereographic projection are
2Qb
=
X-
+
l
"~
'
ab
*
1
-
1
+ 06
ab
(VIII4.18)
where *
f
&
I
/
"L
Xi 4~ ^X2j
=
*
/
Xi
f
^X2-
We note that 6 is not equal to the complex conjugate of a. If we put a = b
=
vt/vi in
(71114.18),
X~ which
we
get
=
(VIII.4.19)
+
satisfy
x+x~
+ xi
= Q2
and X+
Xs3
/C3
/C
1=
-J
In comparing these with the projection of the real sphere stereographic projections of the complex sphere are given f*
=
y
i(<^^)* Oj|ry|ttjD,
we
see that the
by (VIII4.20)
where
in which
AFrom (VIII4.17) we have
= LX3
XX3
-x+
r L
X-l J
xs J
(VIII
[VIIL4.]
StereograpMc Projections of Complex Representation
where the f matrices
231
satisfy
+
2
1
(r
)
2
2
(f
)
+
2
3
(f
)
- Q2
(VIII.4.88)
and
=
det f l
+f
1
det (r
2
= -Q 2
det f 3
,
+ r = det (f + f - -2Q + f + r = -3Q
==
det (f det (f 1
)
=
det f 2 2
3
3
l
)
2
)
2
3
,
2
.
)
"complex sphere" can be obtained by the discussed in Section VII1,1 formations of type All possible rotations of the
U trans-
:
\
An *,
=
u\u),
=
~
Q =
we
From
are equivalent.
1, 2, 3,
x+
+
=
a x^
=
\v')
important special case corresponds to
for i
so for
u <)
U\v).
Q = 2
2Q,
|
x-
Here the three spinors
0.
(VIII. 4*18)
we can
+ b x+ =
write
2Q,
obtain
a26 2
=
1
or
-
(ab
= 0. = X2 = xa =
+
I)(a6
=
1)
0. For db 1 = 0, we corresponds to %i = 1 = 0, infer on physical grounds that Q/(db and ab 1) must, for Q be a finite quantity; otherwise the x would be infinite for an electromagnetic
The
case ab
1
+
+
plane wave. For simplicity
we
shall
+
take
Um which can be regarded as a normalization condition on the spinors u) and Thus formulas (VIII 4.19) become
x+
=
x-
u&i,
=
subject to the condition u\v\
xs == $(uivi
uivz,
+
1*2^2
=
or
VI/TJ&
a complex number. By using Vi = 0.112 and v% (VIII. 4-^) take the forms x+ = ~aw| x- = ^^f? Xs
a
is
3
Xi
X2
The vanishing
Q2
= =
la(ifc?
lia(u\
-
(VIII. 4.23)
u&z), v^/ui
= =
=
au\ "~~
y
\v).
a, say,
where
the expressions
OLU\U^.
Hence
wl),
+ ul),
(VIII.4M)
independent of a particular complex number a] it can, therefore, be absorbed into the spinor \u) and expressions (VIII. 4.84) then become of
is
X3
=
The Lorentz Group
232
The
entire
a plane containing the and magnetic vector 3C of the plane wave. We shall call this
complex sphere in
electric vector
[Chap. VIII]
this case reduces to
particular plane the "neutral plane."
We
can use the transformations (VIII.4.25) to transform the
vector Pp, defined
Because
Q2 =
momentum
by
0,
the vector
PM
is
a null-vector and the transformations
(VIII.4'^5) do not, of course, change this property of PM In terms of the new variables u\ and u^ the components of PM are given by .
cP2 =
= CP4 = cPz
J J (Wf
These can be expressed as
P^i where the spinor
|^j) is
The vector PM defined
related to \u}
by
in terms of the kets |%) satisfies, because of
Maxwell's
equations, the conservation equation,
(VIII 4.28)
The same
PM
expressed in terms of the spinor \u), will satisfy the conservation equation (VIII4-^8) provided the spinor \u) satisfies the vector
,
equation
^-L|^ = This
is
the wave equation for a
It will be discussed in detail in
VIII.5.
field of
(X
spin %h without mass
(VIII 4^9)
and charge.
Chapter IX.
The Four-Dimensional Spin Representation of the Lorentz Group
A four-dimensional spin representation of the proper Lorentz group in a C that contains no subspace other than itself and the null space (irreducibility) requires a complete algebra of four-dimensional matrices. The Kronecker
[VIII.5.]
Spin Representation of the Lorentz Group
233
m
and n X n, respecproduct of two square matrices .4 and B of order m X a matrix of mn as order mn. Its matrix X representation is tively, is defined
A @B= a nn B
between any two matrices
where the symbol
uct of the respective matrices. By using the matrices o>, where
matrices
by means
of
=
/z
the Kronecker prod-
signifies
1, 2, 3, 4,
we can form
16
(4X4)
Kronecker products
Ap t9 =
o>
(VIII.S.1)
er,,
which can constitute 16 basic matrices of a particular C* (four-dimensional spinor space). All sixteen matrices A, v are contained in Table VIII.5 of Kronecker products.
TABLE
VIII.5.
All matrices of
The matrices A^ v
4
.
can be expressed as a linear combination of the above 16
X
(4 4) matrices. Because of the irreducibility of the
4X4
cannot be transformed into one another by means of the matrix transformation rule defined in (7.4-5)
.
The matrix .multiplication of square matrices
A,
B
}
(A
where
AC and BD
of the
same
*
C,
of
D
two Kronecker products
of order
E)(C
n
D)
X
n
is
defined
= AC @ BD,
A
B
and C
D
by (VIII. 5.2}
are ordinary matrix multiplications of the square matrices
order.
For a detailed discussion of the Kronecker product in the theory of groups and its relevance to the irreducibility concept, the reader should consult F. D. Murnaghan, The Theory of Group Representations, Johns Hopkins Press, Baltimore, 1938, p. 38.
The Lorentz Group
234 If
A
and
B are nonsingular square matrices of
dimension
[Chap. VIII]
n, it follows
from
(VIII. 5 Jt) that
(A
B-
B)(A~l
=
1
)
In
= hn
In
and
B=
A
!)(/
(A
(VIILB.S)
B).
Hence det (A
We
can
from the
now show
4X4
that
all
det
The matrix
=
o-i
5)
(J n
=
4)(det 5)*
(det
elements of Table
@
(ert
0-1)
det (AB)
(VIII. 54)
VIIL5 can be constructed
I2
(72)
^2) =
(VIII. 5.5}
Jj.
a and
(0i
0"i0"2) (0"i0"3
=
/3
multiplication of the
=
=
/ n ) det
(.4
matrices a*
(a)
= =
B)
is
(0i
cr 3
(/2
i
I2)
)(cr 3
i
0-3)
-
Thus
=
as
The matrix
(b)
multiplication of the = =
ia$
=
(I
JL
2) (0*1
i(T2
/2)(0"3
1
2f
ffi)
ffi
with
= =
0*2)
Antisymmetric combinations of the
(c)
(VIII. 5.6)
a and
(B J-2)(
(^0"2
(
h.
0-2
-^2)
a
0-3
0i,
0*3
=
yield
5
^
/T7TTT r /y\ \y 111.0.7)
0*2j
01
ffi
/2-
give
1 J
=
(0*1
fry)
2z
^e<
Hence or
= -ita X
(d)
From antisymmetric
~ OS By
combinations of
-
cr
and
(77/7.5.8) ft
= -t
we
obtain
(VIII. 5. 9)
and
it is easy to show by means of the the anticommutation relations satisfy
using the definitions of the
Kronecker products that they
=
aft)
QJ
a.
ofca,
QJ
+
av a^
where /4
=
/2
=
25^/4,
(VIII. 5. 10)
Spin Representation of
[VIIL5.]
tlae
Lorentz Group
235
and
=
04
We
.
shall construct a different set of
7 = where the
7.-,
for
f
=
1, 2,
Dirac matrices by defining
=
T4
zjto,
3
7l
=
7*,
7l
=
7l
=
7*
(VIII.5.11)
ift
3 are Hermitian and 74
is
anti-Hermitian; that
-74,
is,
(VIILS.12)
and
The anticommutation
=
=
7
rules for
74
^4,
7
=
/4-
matrices can be worked out from the
Kronecker products:
On comparing
(VIII.5.18)
+ 7/>v =
-2
7*7.
(VIII.5.1S)
.
see that
7
matrices are
more
suitable for relativistic considerations. In particular, the square of the operator 7^, as follows from using (VIII.5.1S), is
a Lorentz-invariant expression. The
last result implies that the rationalization
of the Lorentz-invariant square root,
in terms of
7
matrices requires the use of the Hermitian
7*,
for i
=
1, 2, 3,
and the anti-Hermitian 74 (or vice-versa). We cannot, because of (y111.5.1$), at the same time have all 7 matrices Hermitian and also maintain Lorentzinvariance.
In
7 representation all
16 matrices of the complete four-dimensional matrix
algebra assume a more symmetrical form than in 75
=
71727374
=
J&*5
=
iffi
a.
representation. /2
Thus
(VIIL5.14)
has the matrix form ili\
I"
Lil*
J
(VIIL5.1S)
where 7!
The
75
matrk can
also
-75.
(VIII.5.16}
be written as
7!
= -I*
(VIILS.18)
Because of the vector transformation properties of the 7, the operator 75, as the product of four 7, changes its sign under (a) reflection of coordinates,
The Lorentz Group
236
and
(b) reflection of time. It transforms like a pseudoscalar
[Chap. VIII]
under reflection of
both space and time.
A very important property of
75 is its
anticommutation with
all
the other
four 7:
+ 7,75 =
757,
The anticommutation
relations (VIIL5.18)
0.
(VIIL5.19)
and (VIII. 5.19) can be combined
into a single set,
+
T A TB
A
where, for
TA
s
(
7w
The square given
(7/71.5.00)
1, 2, 3, 4, 5,
=
MfS
T5),
A/5/.
=
Mm =
0,
1,
M^ =
a,*.
of the four-dimensional operator
= YP
TA p A is
= -2Af x*/4,
TB T A
by (T
The remaining members
A
+ imcy*
p -
pA y =
2
pi
+ mV.
(VIIL6.81)
7 matrices include the matrices 757^, which transform (to be seen later) like the components of a pseudovector. The spin tensor operator 7/w is directly related to (VIII.5.8) and of the
(7///.5.P), since Jffi
7f7j
=
otictj
~
ajcti
and
We define the
spin matrices
by (77/7.5.**)
where the space parts of o>, 0*23
=
-17278,
=
0*31
17871,
or
(77//.5.*5)
LO are Hermitian,
and the time 014
=
parts,
7l&
0"24
=
0"34
72/3,
=
or 0"t4
are anti-Hermitian.
=
r
o
L
to"*
The squares
014
-itrri
.
=
of
024
=
J
a matrices
=
034
=
~^at, satisfy
(777/.5.*5) /4
Spin Representation of the Lorentz Group
[VIII. 5.]
Some
237
most frequently used commutation and anticommutation
of the
relations are
(VIII. 5.26) (VIII. 5. 27) 7/y/Yp
|[0>,
l^ 75] ft
By
= = = =
<W*757
7sw#
ff
+
CW,
+ <Wk/j
flj7p
a^ara
ppT/x,
(VIII. 5. 28)
,
0,
(VIII.5.29)
iOwGtf
+ iapv&an
idavPrt
ia
(VIII. 5. 30)
iaapSrf
ia>rfSav .
(VIII. 5.32)
introducing
from (VIII 4.30), we obtain [5^,
jSoftl
It is interesting to
commutation
rules
=
idarSrf
+ iopJSap
this
with the
compare (yIII. 5.2 6) become
classical relations (III. 1.25).
ittpvyp
(VJTT
The
% ^^\
^757^
M
of the proper Lorentz transformations Also note that the generators (M by (1.10.23) satisfy the same commutation relations as S^ defined ,)
as defined
by (VI 1 1. 5. 31)\
+ iapJMtob
faaiMw*
iarfM(ay).
(VII 1. 5. 34)
all X 4 matrices with imagiFinally we nary elements are those containing the Pauli matrix o-2 only once in the Kronecker products of the
observe from Table VIII.5 that
4
column contain tr2 and, with the exception of the element
if
U is the unitary transformation U*y*U
The unitary operator where a and
=
operator, then
require
72.
U must obviously be U = a + 672,
& are constants to
we
a linear operator of the form
be defined from the unitarity condition 72. This results in the
WU = I or from the transformation requirement on definition
U=
-7=
v2
[I
+ nj-
(VIII.5.35)
Application of the unitary transformation operator (VIII.5.35) onto the elements of Table VIII.5 transforms all of them into pure imaginary and anti-
The Lorentz Group
238
[Chap. VIII]
symmetric matrices, except for a matrices, which are transformed into symmetric matrices with real elements. This particular representation of Dirac
"Majorana representation" and is quite useful in the symmetry elements of the elementary particle physics. In the Majorana representation the matrices ft, a^ and 75 of the usual
matrices
is
called the
study of the various
representation are given as
= c$i
ar = a&, 7f = *7572.
-as,
=
<*,
ar (VIILB.Se)
The Lorentz Transformations Induced by 7 Generators
VIII.5.A.
The
spinors of four-dimensional space, whose complete matrix algebra is determined by the 16 matrices /4 75, 7M 757^, 7/y* are defined by their trans,
,
formation rules. If time,
it
IT/')
is
a four-dimensional spinor, as a function of space and
transforms according to
W) =
(VIII. 5.37}
SI*>,
where
i/\
=
x'*,
respectively.
Wf
and
|^'}
are functions of x*
and
The
coordinates x
f
by a Lorentz transformation. The transformation matrix S space is a 4 X 4 matrix and is a function of the coefficients Lf
and x*
are related
of the
spinor
alone.
In order to find a two-valued four-dimensional representation of the Lorentz group we consider the transformation of a spinor |$) defined by l*>
= rpM,
(VIIL5.SS)
where 0.
In another Lorentz frame the spinor 7
I* )
=
\
must have the form (VIILB.SS)
rj>il^>-
(VIII. 5. 37} and the corresponding Lorentz obtain the conditions
By applying the transformation transformation,
we
S-^vS = for the matrix
S and
The "metrical"
by a
special
L*7v
the transformation laws of the
(VIII.540) 7.
properties of the four-dimensional spinor space are described
S operator
corresponding to the Lorentz matrix F.
The "metric"
[VIII.5.]
Spin Representation of the Lorentz Group
by
(VIII.S.40)
on the
by replacing Z
of the spinor space follows, therefore,
239 right side of
a;:
&-i yfS
= <7,
(VIII.5.4D
or
jiS
Thus the metric (up
+ Sji =
-
74#
0,
an unimportant phase
to
74
=
0.
factor) is uniquely defined
by
taking
SP = It
0.
(VIIL5.4&)
transforms 7 matrices according to
p-W = which
is like
l
f- 74p
-7i,
F transforming the
=
p-
74,
l
7$ =
-75,
(VIIL6.4S)
coordinates according to
or
x
The S transformation
7
=
*"
=
a/ 4
and
x*
cc
4 .
in spinor space corresponding to
F transformation
in Lorentz space is
S-SvS
=
-<%
(VIILS44)
or
7*S
- Sr =
T4S
0,
+
^74
=
0.
Hence
ST and
it
=
/3
T5
(VIII.5.45)
transforms the 7 according to
= = which are
-74,
-75,
like the transformation of coordinates
by
F, as
or
The antilinear operation of complex conjugation can be used to construct a symmetry operation in four-dimensional spinor space. From (VIII.5.S9} and (VIII. 5.37) we have For
If
L =
a and
S =
we operate on both
3
=
sides
/itysC? it
becomes
by C we obtain *).
(7/77.5.47)
The Lorentz Group
240
From
this result
we
[Chap. VIII]
infer that the time-reversal operator of the four-dimen-
sional spinor space is given
by 3
=
|Sye,
(VIII. 5.48}
-1.
(VIII. 5. 49}
where
3=
=
For the identity transformation Lf
Sf the
corresponding
S
is
just the
unit matrix of C*. to the 1, corresponding For the Lorentz transformation of class Lu ^ = 5 40} in set we ^ Lf (VIII. time coordinates, reflection of both space and
and obtain
Sj,
Hence the S transformation
=
0.
in this case is just
SR =
(VIII. 5.50}
T5 ,
a further result indicating the intimate relationship between the 75 in spinor space. operator and improper transformations The scalar product of two spinors |^> and \<j>} in terms of the metric ft of
which
is
spinor space, follows as
a Lorentz-invariant expression (a scalar). In particular, the invariant square of the spinor
which
is
(+W} =
|^} is
<W>,
(viiLe.se)
dfl/j
(VIII.5.5S}
where {^j
=
It is instructive is called the adjoint spinor corresponding to the spinor ]^}. to compare this with Cp|^ or the contravariant components p of the mo-
mentum Pft. The S transformations
for the two-valued representations of the Lorentz
for Lorentz space. Let group satisfy a condition similar to (1.10.9} spinor in the transformed coordinate system, so
If
(VIII. 5.52}
is
I^'}
to remain invariant under Lorentz transformations,
be the
we must
have (VIIL6.S4) satisfied for
any spinor
[^};
that
is,
S*pS
where
+ and
signs
on
=
ifcft
(VIII.5.55}
the right apply for proper (and for the class
1/44
^
1)
[YIIL6.]
Infinitesimal Lorentz Transformations
241
44 ^ 1) Lorentz transformations, improper (and for the class the For S transformations 5 = $75 and S = 75, correexample, respectively.
and
for
sponding to Lorentz transformation matrices satisfy (VIII. 5.55) with the minus sign on the
F and
I,
respectively, will
right.
From a comparison of the commutation relations (VIII.5.32) and (Till. 5.84) we easily infer that we can use the spin tensor o> to set up a sixparameter two-valued representation of the homogeneous proper Lorentz group. The S transformations corresponding to L-transforaiations (1.10.27)
and
(1.10.28) are given
5(23)
= 5iO) =
exp
5(3i)
= &(<) =
exp
=
exp
=
Sz(6)
by I
-~
=
a-28
J
=
0-31
|^-|-
-
I
J
=
(7is
J
-
exp
I
exp
I
exp
I
727s
=
cos
J
-- 737i =
|
cos-
-
J
-- 7x72! =
cos-
-
7273 sin
7371
|
sin|>
7i72 sin-
(VIILB.S6) as rotation, transformations,
5 i4) =
5(X)
=
=
S(e)
=
C
5(24)
5(34)
=
exp
=
5(p)
exp
exp
and
-~ o-u =
I
J
^
I
I
-^
cr 24
exp
=
exp
=
exp
I
J
7174!
1 7274
J
0-34
--
I
-| 7374]
=
cosh -
=
cosh
=
cosh
-
-^
7274 sinh -'
|
|
7374 sinh
|
(7//7.5.J7)
as velocity transformations. It of
pure rotation
(o- t-)
is
interesting to observe that the generators
transform like an axial vector, while the generators
ion
of velocity transformation transform like a polar vector.
Finally
we
state the
fundamental Lorentz-covariant quantities
of spinor
and Lorentz space: (a)
M0|*> =
(b) {$[75 1^}, (c)
$1^>, a scalar,
a pseudoscalar, a 4-vector,
<^JT/iI^X (d) {^(757/il^), (e)
(^IO-MV |^),
a pseudovector
an antisymmetric
VIII.6. Infinitesimal
Lorentz Transformations
All possible motions of a rigid
one point of which
is
tensor.
fixed,
body
in three-dimensional Euclidean space,
can be described by the three-dimensional
The Lorentz Group
242
[Chap. VIII]
orthogonal group. If we introduce a time coordinate as a group parameter, then the position of the body (or its orientation) at any time t can be obtained of an element of the group. It has been shown in Section 1.7 a vector specifying the orientation of a rigid body at time t Q , orientation at time t is obtained by operating on ]a(fo)) with
by the operation that
if ja(
then
its
)) is
A(t,
fc)
= P exp [-i f Jts
produces a continuous motion of the body from 4 to t. The form of A(t, tQ ) is such that the entire history of the motion can be
so A(t,
o)
followed by infinitesimal steps; that
is,
infinitesimal rotations
sum up
to
finite rotations.
In
relativistic
The concept
theory the analogy with rigid body motion cannot be applied. be reconciled with the fact that in relativity
of rigidity cannot
simultaneity of events does not have an invariant meaning. The rigidity of a body requires an instant transmission (that is, with infinite velocity of light) of any type of disturbance at one of its points to all others. This feature of
imply a "point" description for elementary partithe or "point" nature of elementary particles cannot "finiteness" Indeed, be given any physical meaning; this feature is not what is implied by special
special relativity does not cles.
relativity theory.
tion of physical
More
precisely, special relativity implies a "field" descrip-
phenomena.
An infinitesimal
element of the Lorentz group can produce an infinitesimal can only be connected by light signals. Such
change of "field points" that infinitesimal Lorentz
motions of the four-dimensional events are generated
by the operators M^ V defined in (1.10.22) or, more generally, by those satisfying the commutation relations (VIII. 5. 34). All proper transformations = OF [see (1.10.20)1, can be generated from the infinitesimal elements )
M
where
is
a small antisymmetric matrix [in the sense defined by (I.5.D)]. of at any proper time TO allows us to construct by integra-
The knowledge
M
tion, as in Section 1.7, the transformation at
L(r,
The
TO)
definition of L(r, TO )
=P
time r as
r
exp [-i f Q(n)F dn]. JTQ
by (VI11. 6.1) contains a certain
(VIII'.6.1) restriction
on the
possible infinitesimal changes of field events: only those changes are allowed
which the positions of field events at an arbitrary proper time T are obtained from their positions at a time r by a transformation L(r r), generated for
,
according to the infinitesimal steps implied in (VIII. 6.1). All infinitesimal Lorentz transformations lie in the "small" neighborhood of the unit element I4 of the Lorentz group. For a time-independent generator Q(r),
an
infinitesimal Lorentz matrix
is
given by
Infinitesimal Lorentz Transformations
[VIIL6.]
L oij
=
J4
L= where 2
is
a real
4X4
we
TO) Into 0,
absorbing the factor i(r
-
-i(r
J4
-
TO)
243
OF
write
(I7II.O)
SF,
small antisymmetric matrix. In tensor notation
we
can write
U = V - S^aw, Z^ =
where
3V
.
If
we now put Z^pO"
and observe that the superscript
we
write (VIII. 6. 3) as
Hence x'*
=
(7///.0.S)
T
in
v
= ^,
U=
to be regarded as the second index,
is
oe
-
S?
(VIII .$
o#.
infinitesimal Lorentz transformation of coordinates is given
by
Lfx r or
=
x'
x"
-
afx*.
(VIII.6.5)
VIII. 6.A. Infinitesimal Complex Orthogonal
Transformations The
derivation of a complex orthogonal operator describing finite rotations a of complex three-dimensional manifold, as a sum of infinitesimal rotations, is a straightforward generalization of the method discussed (Section L7) for
the real orthogonal group. It follows from equation (VIII. LI 9} as T
R = P exp
[-i
f Z(n) drj.
(VIII. 6.6)
JTO
For a time-independent generator #(n), an infinitesimal complex Lorent transformation operator has the form
R = where
Z
factor r
is
-
I
i(r
-
TQ }Z,
defined as in (VIII. 1. Iff) with first-order quantities a* and TO can be absorbed into a* and 6*, and we can rewrite it as fi
=
I
-
tZ.
6*.
The
(77/1.0.7)
Hence, the infinitesimal complex orthogonal transformation of
|%) is
given
by
By to
using the transformation rules (VI1 1. 1.52} relating
L transformations, we can express the coefficients a
1
and
R
transformations
6* of
the
in terms of the infinitesimal Lorentz transformation coefficients stituting
from
(VIII. 6. 7)
second-order terms in Z,
and (VIII. 6 4)
we obtain
in
Z operator
COMV .
On
sub-
(VIII. 1.52) and neglecting
The Lorentz Group
244
- B^Z =
Z*B^ The
condition det
R =
tr
most general form
+
(VI II. 6. 9]
wtBpp).
on the proper complex orthogonal transformations
1
implies so the
i(v>B p ,
[Chap. VIII]
for
Z
Z=
(VIII. 6 .10)
0,
is
Z=
q*Kt,
where the q are complex numbers. A simple calculation for for ju = i and v = 4 from (VIII..6.9) yields l
6iJk
Kj(q
+
k
where *
The
expression for g"
can be q
=
k
O3 k
w>4*)
=
iua
=
v
=
4 and
0,
= **,/.
satisfied
\L
(VIII. 6.11)
by taking
Wfc
=
iejb
cofc,
where we take *
024k
Hence
= c-K -
iZ
and the change
of [%}
due to infinitesimal complex rotation
x'(*')>
The
total
change of
iw-JC
-
=
lx<*)>
(e-K
-
i-K)|x>-
is
(VIII. 6.1S)
however, is the combined effects of an "intrinsic" degree of freedom of the field) and coordinate discan further express Ix'C^O) i n the original coordinate
|x),
rotation (through the
K
placement. Therefore,
we
system x^ by
-
x '(x))
The ket its
x '(x
tfx
A
contains the effect of the infinitesimal transformation, so in expansion the second term of (VI11. 6. 18) is of the first order in the a?. Thus ]X (^)}
(VIII. 6.13} can be replaced lx'(*')>
The second term on the ket
by
=
Ix'(*)>
right
is
\x(x ))j since higher terms in
-
<&> -^
\x(x)).
(VIII. 6 .14)
now
expressed in terms of the original |x'(#)} contain terms in co. Using the
abbreviation
&-{*"
(VIII. 6.15}
<\
the operator <$x v - * can be expressed as ox 1
where
L =
r
Xp
is
the
quantum mechanical angular momentum
operator.
Infinitesimal Lorentz Transformations
[VIII.6.]
With the above
=
A;x)
;
lx
results,
W) -
(VIII ,6.12) becomes
ix(*)>
where is
245
J = L
+ hK
(VIII.6.17)
the total angular momentum operator of the electromagnetic field. The above discussion can easily be carried through for the two-valued
representation of the complex group. In this case the infinitesimal transformation operator is given as
U=
I
- Ji(*W +
(VIII. 8.18)
-a).
In deriving (VIII. 6. IS) we used (VIII.2.7) in (VIII. 2.11), together with the definition of the infinitesimal R transformation by (VIII. 6.7).
The change
in the spinor
\u),
induced by (VIII. 6. 9) and the corresponding
transformation of coordinates, is A| W>
=
\u'(x))
= I
-
\u(x))
(-/)
u)
+
e
where is
+ I (x*e-p +
a
[|
J-
the total angular
VIII.6.B.
momentum
The of
)]lti>,
+ ir
operator of
(VIII. 6 JKS)
a
field of
Infinitesimal Transformation
S = 7
operators (VI11. 5. 56)
coefficients
and (VIII. 5. 57)
+ i^o>
as the operator of infinitesimal transformation for
The
spin fA.
Four Dimensional Spinors
From the six spinor transformation we can easily infer the operator
space.
(VIII JS. 18)
r-ep 4
q^ are antisymmetric
in
/*
a four-dimensional spinor
and
v.
Substitution of
S
in
(VIII.5.40) for the transformation of the 7 matrices provides the four
equations If
we use the commutation
rules (VIII. 5.26) for the
rxi,
Hence &>
Thus the required operator S
=
=
i
F
y and the
we
obtain
^.
iw^
for infinitesimal transformation of spinors
is
(VIII. 6. 21)
The Lorentz Group
246
The change
in the spinor
transformation,
is
|^),
induced by
S and
[Chap. VIII]
the corresponding Lorentz
given by
(F777.ff.0S)
>,
where J
= L+
instead of
|7io-
and the second term on the
a matrices
right contains
as occurred in (VIII. 6. 19).
VIIL6.C. Problems Prove that the fourth-order group consisting of 7, 7 75, 0, morphic onto the eighth-order group 1.
7
7, 2.
7M
75,
Prove the following properties
=
2
T'Z' Z? where F^ = /^ + i^. (c)
we
F is homo-
Show
also that
of Dirac matrices:
^
^
,
[^
If
75.
F,
7/yv form a group of order 32.
7s7w
,
7,
(FJ/7.^3)
[See (7777 J 9) J .
set
T, then
=
r^rM =
and
where
^
is
defined
(d)
(4**'o>)*
(e)
From
by (7777 J. 33).
=
-
275A.
the identity
=
7^ it
757/Y5
follows that tr (7.) 3.
The determinant
M
of the tensor
=
g
Show that
=0,
detfl^
g^
=
= -(1
=
1, 2, 3, 4.
+ f^ is given by + -A
a^
2
).
the expression
V~g =
V(l
+
Q
-A
2
)
can be rationalized in terms of Dirac matrices either as Li
= C7^M +
^75&
+ ^75
or as
L2 = 0757^ where
^
is
defined
by (VIILI. 33).
+
4*760
+
^
(7777..6. 24)
(VIII
.25)
Four-Dimensional Orthogonal Group
[VI II. 7.1 4.
247
Discuss the transformation properties of the expressions
L2
<#L2i*>,
by (VI II. 6. $4) and (VIII. 6.85),
are defined
respectively. 5.
Show
that Maxwell's equations can be written as
A where the
"^->>
=
(VIII. 6. 26}
>
4X4 matrices are defined in terms of generators M^ of infinites-
imal Lorentz transformations by
A 4 = J4 A2 = -If (31)
Ai
,
and the vector
\TJ)
is
+
iM
=
Jf (23)
A3 = -If
(24 ),
(12)
+ iAf + iM
( i4),
(7177.07)
(34),
defined to be
X2
(VIII.6.28}
.
X3
L>J Prove that current
is
corresponds to a free
derivable from a scalar function
^= then a complex
field. If
<j>
we assume the
electric
as
--TlTl! 47T
(VIII. 6.29}
C/3y
would imply the existence
of
magnetic charges.
IV represents the generators of infinitesimal Lorentz transformations, and JT^IV are called the invariants of the homothen the operators iF M 6. If
T^
geneous group, where *]>* is the dual tensor to IV. Show that for fourdimensional spin representation the corresponding group invariants are 74
and
75.
The corresponding transformation
operators
(VIIL6.SO) 7
be called "invariant gauge' transformation operators, where X is an invariant function of space and time. Prove that (VI 1 1. 5. 38) is invariant with respect to $5 transformation only if the spinor |0) vanishes.
may
VIII. 7. Representations of the Four-Dimensional
Orthogonal Group The
invariant form of a four-dimensional Euclidean space
0|f |0 = 4
x\
+ xl + xl + 4
is
(VIII. 7.1}
The Lorentz Group
248
4X4 unit matrix 7
[Chap. VIII]
the metric of the space. A transformation which leaves the form (VIII.7.1) unchanged constitutes a four-dimensional orthogonal group. The commutation relations for the six generators of infinitesimal rotations can be obtained from (VIII. 5. 34), by replacing the
where the
Lorentz metric a^ by
As a
is
4
8^:
E^ we
special example, for
shall take
E,,
=
- x&,
xfi,
(VIII.7.S)
where
""^
*'-'"4: and
=
Lxn SVJ
The
operators
tum.
By analogy with the
E^
-#.
(7771.7.4)
constitute a four-dimensional Euclidean angular
three-dimensional case,
we can
momen-
construct the eigen-
(E^y. These eigenfunctions can form the basis for irreducible representations of the Lorentz group. In some cases one can obtain solutions
functions of
wave equations from the corresponding Euclidean equation by means of an analytic continuation. of relativistic
We introduce four-dimensional polar coordinates Xi #2
x3 #4 r
where
=
R*
solutions
+ xl +
xl
the
by
= r sin 6 cos <, = T sin Q sin <, = r cos 0, = .R cos \, = R sin \, x\
of
(77/7.7.5)
+ xi
In these coordinates we obtain d
-r:
+
d \
cos
sin
& =
i cot \( cos d cos
<j)
&
cot X( cos 6 sin \
i
== if
\
cos
OA
V
cot 6
\
cot e
(VIII.7.6)
~
\
& =
<j>
4^
dv
<
o(p
sin u
+ ~Sin V
sin ^ cot X OC/
/
^
)
o
TT
+
sin 6 cos
+
sin ^ sin
/
)
a a
>
d\ *
o\
Four-Dimensional Orthogonal Group
[VIIL7.]
249
Hence
L 2 is the
where
usual three-dimensional angular
momentum
operator in polar
coordinates.
The is
eigenfunctions of
some
integral
number
%,, corresponding to its eigenvalues 2/i (where /i to be determined), are of the form
= Y lm (0, <)*?nz(X),
Vnlm(0, *, X)
where Yim (6, <) are eigenfunctions
of
ZA Thus we
-
a
2
can write
-
n +iX 2 cot
-
2 cot x
*
or
=
27^<
=
Putting x
cos
X,
Plh&r
-
I}-
the eigenvalue equation (VII 1. 7. 8) becomes ;
We
This can be solved by standard methods. Vnl
and obtain (1
-
Now (i
x 2)
for
^
setting
-
Z
Fn
-
2
(X
-
first
0.
make
the substitution
i
iy WFnl
(VIII. 7. 10)
the equation
i
(21
=
^f
x*)
=
=
+
S)x
^+
in (VIII.7.ff)
-
( 2Z
+
3) x
and
&f
-
[21,
+
1(1
+
2)]Fm
differentiating it t2/i
~
z (z
I
=
times,
+ 2 W8 = )
(VIII. 7 1
0.
>
we
find
(via. 7. 12)
where
Hence we
infer that the required solutions for (VIII.7.11) are proportional
to 77$ that ;
is,
Fni
The equation
solved *
by an
Tnz
dx
l
(VIII. 7. 13)y v
for ^ n0 ,
0>vU
is
=
CtCC
ultraspherical polynomial,* d/d(cos X)(cos?iX), provided
we
G. Szego, Orthogonal Polynomials, American Mathematical Society, Providence, 1939.
The Lorentz Group
250 choose 2/i
=
7i
2
1.
Hence the /\\
*i(A)
=
*? n
.,
i'
j(X)
are given
(sinX)*rf*
-jr-
+1
[Chap. VIII]
by
(cosnX) '
fvrrr >? t n (F///.7J4)
d(cosX)W
where r,
forn
The
==
0,
1,2,
=
2
[> (n
-
2
P)(n
2
-
2 2)
-
-
-
(>
2
-
Z
2
)]
1'2 ,
(VIII. 7. 15)
-
.
eigenfunctions
are normalized according to rfO = / VnlmWVZrntW
Snn^fa',
(VIII. 7. If)
where
dO
=
sin 2
X sin
^
dX dd
(VIII.,7. 18)
d
the solid angle element of the four-dimensional space. The expansion of the function exp (ik^x^ in terms of the eigenfunctions *&nim, like the expansion of a plane wave exp (ik-r) by (IV.3. @4), can be is
by using a procedure analogous to that used in Section IV. 8. A. This can be done by first separating out the radial part of (E^y. Thus we carried through
write
Using commutation relations (VIII. 7. 4) and /!
=
R*S*
+
3R-
+R
$#, 2
-j^y
RJ
i,
we
obtain
(VIII. 7. 19)
where
The operators S 2 and
(E^ commute and the two together can form a complete
we assume that the eigenvalues of S 2 are given commuting 2 by fc (= kl + A-| + fc + fcl), then the radial eigenfunctions satisfy, as follows from (VIII.7.19'), the equation set of operators. If
dR* This equation
is
solved
'
/
by
Four-Dimensional Orthogonal Group
[VIII. 7.]
where we take
/i
=
+1) and n =
2j(j
method very
following a
= 8x Y)
eik^
vectors
fc M
and
(VIII. 7. 5),
it
A, 0,
(-1)'"
<1>
?
cos
o>
7
The results obtained
From
k^ =
cos X cos cos
cos
fix
<
X, 0,
x^, respectively.
cos
Hence the required expansion, is given by
A KK /^i(fcK)%(A0*)^
and
follows that
= =
1.
similar to one used in deriving f IV. 3. 24),
jlm
where the angles
+
2j
251
(^),
(VIII. 731)
the directions of the four-dimensional the definition of polar coordinates by
\k\R cos
,
where
A + sin X sin A cos % 9 -j- sin 6 sin cos (
hi this section will
infinite representations of
n zw
<).
be found useful in the discussion of the
the Lorentz group.
VIII.7.A. Group Invariants If
we put EM = Mi,
ESL
and use the commutation
=
M<>,
Eu =
M
z,
relations (VIII. 7.
My] = NjJ = Nil =
E = 41
),
we
NI, J?42 obtain
= N* E&
N
z,
ieaiMt,
(VIIL7.fg)
ie inMi,
(VIIL7.SS)
ie^Nt.
(VIII. 7 Jt$
Hence we observe that the commutation are the same as those between Ji and p 3
relations
between the
Mi and Nj
of the three-dimensional
inhomo-
geneous rotation group discussed in Chapter IV. In this case Ni correspond to p^ except that the Ni do not commute among themselves as do the p^
The two
invariants of the group are
I,
where
M- and
JVi are
= M-JV =
(VIII.74B)
JV-M,
regarded as three-dimensional vector operators. It
JV t
-
=
stitute
M
is
and Ni is Mi = Ki and = and JV- = the choices Ki. Specifically, Mi (l/K)Ji (l/K)Ji cona special solution for the commutation relations (VIII. 7. 82) and
interesting to note that a particular choice for
%
(77I/.7.&4).
In order to establish a complete analogy with the three-dimensional homogeneous rotation group, we introduce the operators
Q=
KM + #),
and obtain the commutation
relations
zjl LQ Qyl [Zi, Zy] IQi,
= = =
o,
in-
The Lorentz Group
252
Hence we
see that the four-dimensional rotation group can be regarded as a
direct product of the
The
[Chap. VIII]
eigenvalues of the
two homogeneous three-dimensional rotation groups. Hermitian operators Q- and Z2 are of the form
=
i(i
+
Z = 2
l),
so the degree of the representation
is (21
+
+
s(a
l)(2s +
group invariants Ji
=
(*
+
2
+Z
2
and
It
=
2
(YIII.7.27)
1),
1).
The
-Z
eigenvalues of the
2
are
=
Ji
Z(Z
+
1)
+
I2
1),
Each representation may be
=
labeled
1(1
by
+ the
~
1)
*(
number
+ 1). pair
(VIII.7.88) (I,
$).
The
tion of spacial coordinates correspond to the mirror representations (s, I)
;
the second one comes from
The
operators
commuting
M
2 ,
JV 2 ,
and
changing the sign of
M* commute and
set of Hermitian operators.
The
(I,
reflecs)
and
N.
they can form a complete
eigenvalues of
M
2
and
M
3
are
j^m^
+ 1) and m, respectively (with f). We use the analogy between the present case and the inhomogeneous three-dimensional
known
as j(j
rotation group already discussed in Section IV.5. In place of the operators pi
we here have N. Therefore the diagonal matrix elements of [#+, N-"J = 2$f3 as in (IV. 5.2}, can be expressed, and we obtain the relation 0,
,
(VIII.7.29)
where &(j) is equal to the term
=
M + JV 2
2
(7I7I.7.SO)
Equation (VIII. 7. 29)
is
solved by T2
^0")
+ jT + f =
Putting this result in (VIII.7. 30) 2Ii
To
we
+
find the possible forms of Ii
1
constant.
(VIII. 7.31")
find that
=
constant.
and the nature
(VIII. 7. 30) of the representations,
we
consider the following two cases. (a)
J2
assume a
we have $0") = constant j 2 since 3>(j) maximum value j = n, say. Thus for ^ = n 0:
must vanish:
;
is
nonnegative, j must
+ 1 the function
Four-Dimensional Orthogonal Group
[YIII.7.]
+
$(n
Hence constant
=
+
(ra
2
I)
1)
=
^
j
0,
253
n.
and
= =
$(j) 2Ix
+
1
(
n
(n
+ +
- j2
2
I)
(VIIL7.SS)
,
2
I)
(VIIL7.84)
.
This corresponds to a finite representation of the group, (b) I2 T^ 0: in this case, from 7)
we
see that j can
=
n
. , constant
., 2
?
=
assume a minimum value j
for which the right
jo 5^ 0,
side does not become negative. Thus we must have
jn
Jo
(VIII.7. 35)
and
= Hence
it
$(n
+
=
1)
0.
(VIII.7.36)
follows that
(n
+
(7II/.7.S7)
and
These results yield the relations
= jl(n + I) 2 2h = (n + I) 2 + j8 II
This representation
=
(n
(VIII.7.39)
,
is
and
also finite
+
its
1
=
n(n + 2) + &
degree
is
given
(VIII.7.40)
by
- jo).
2
I)
The
degree of this representation, as defined by (VIII.7.41), must be 1) of the representation found previously. l)(2s equal to the degree (21 The same applies to the invariants of the group. Thus we have the equations
+
2/x
+1 2I2
= =
+
+ 1) + 21(1 + 1) 2*0 + 1) - 21(1 + 1) 2s(s
= =
+ I) 2j8(n + I) 2
(n
2 .
Equations (VIII.7.4&) are solved by
n = From
(VIII.7. 39)
classified
cases:
we
5
+
Z,
j
=
I*
-
(VIII.7.43)
l\
see that the finite representations just found can be
according to the sign of the group invariant
/2.
Thus we have
three
The Lorentz Group
254 (a)
72
>
0, so
(b)
I2
<
0; in this case
s
I2
=
0,
where
s jo
=
=
(n
-/
Z
),
=
4(*
+ Jo).
[Chap. VIII]
(VIII. 7 45}
0,
s
The mirror representations
=
and
(s, Z)
=
Z
(Z,
(VIII. 7.46)
in.
have different signs of I2
s)
.
VIIL8. Infinite Representations of the Lorentz Group VIII. 8. A.
The in
The Homogeneous Group Lorentz transformations, as shown
six generators of the infinitesimal
(VI11. 5),
commutation
satisfy the
relations
'
A particular case for the generators J ^ is contained in j
v
= L =zxp v
x vp
v
(VIII 8
.
2)
This is the relativistic definition of orbital angular momentum. By analogy with (IV. 2.12) the relativistic form of the orbital angular momentum tensor operator can be written as
where
F is as
defined
by
(1.10.4) "Pi 1
\P)=
and the
4X4 matrices Af
My
!/
w pJ
= iaw
(VIII.8.4)
As an interesting exercise by (VIII. 8.3) and (VIII.8.4), (VIII.8.1). The form (VIII.8.8) of LMJ
are given
the reader should show that satisfies
;
by
(1. 10. 28).
as defined
the commutation relations
,
be used to develop a discussion of the Lorentz group, similar to the representation of Ji [see (IV. .14)]- The basic difference between the four-dimenwill
and the Lorentz groups arises mainly in the respective and L^. The commutation relations (VIII.7.22) and (VIII.7.24) are now replaced by sional orthogonal
definition of jB4t
-
i,
N,l
=
-ihtijkLk,
(VIII. 8. 6)
Infinite
[VIIL8.]
change occurs only
so in this case the
the components ==
jV3
43.
of
Representations of the Lorentz Groop
Ar where LI = ,-,
The group invariants
of
respectively.
z'/a,
where the nonnegative
number
and
=
/2
The
J'O"
=
2
is
+
iw,
A'i
between
A"2
=
i42,
form r2
(YIII.8.8)
?
(VII1. 8.9)
N
?
&(j)
is
-A
2
*-
replace <(j) and J2 obtain*
-
= i,
of the
we
We
relations
=
L-N.
the correspondence
the Lorentz groups,
commutation
isi,
ii^ = L h=
Now, because
=
is
now
are
=
2/x
in the
Z, 2 s,
255
~~
iN between
the orthogonal and
in (VIII. 7. 31),
,
(2/i
defined
+
by
ty(j)
and
(VIII. 8.10)
1),
by
DfaWlTj).
(VIIL8.12)
expression (VIII. 8.10) for ^(j) shows that the requirement that ^(j)
be a nonnegative number does not limit the possible values of j. Therefore is there is no upper limit for j and the representation for the case J2 9* infinite. Thus we have three cases to consider. J2
(a)
T* 0:
because of definition (VIII. 8. 12) for J2 j must have a minimum ,
nonzero value jV For j
=
j*
^(j) must vanish
,
2Ji
=
jg
-
1
^(jo)
=
0.
Hence we get
-
(VIIL8.13) Jo
and
which
is
(b) /2
nonnegative for
=
0, j
^> ^ 2l!
j
In
=
g:
(c)
j*
.
this case also
-
jg
j has no upper bound. I2 = j ^ 0. In this case
where
is
all
0.
;
we must have
*(j)
1,
^f(j) is
not required as a condition. Thus
=
2 ^'
^(jo)
- JS,
=
0, so
(VIII. 8.15)
always positive definite and ^(j)
we have ^(j) >
for j
=
1, 2,
,
= and
Ii<0.
We may thus form
a complete orthogonal
2I3 for jo (a)
*
=
and
W.
0, 1, 2, 3,
(b)
Pauli,
-
-
or
=fQ
J,
f
,
-
1
,
X 2,
where X
above and also part of case
set
with
J,
= >X
is real.
This
series contains cases
(c).
Continuous Groups in Quantum Mechanics,
CERN 56-31
(1956).
The Lorentz Group
256 If
we use
the case Ii
<
we
0,
=
get 2/i
1
+
2 ??
[Chap. VIII]
=> =
with J2
?
and
<
< V) 1. This representation is not used in physics. Further discussions of these points, to cite only a few, can be found in various interesting papers on the subject.*
VIIL8.B. Eigenfunctions
of the Representation
We introduce polar coordinates by = = =
Xi
#2 x$
r sin 6 cos <, T sin 6 sin <, r cos
8,
(VIII.8J6)
'BcoshX, sinh X,
fB sinh
=
T
X.
<
IB cosh X, where the alternatives r
R
(#4
= R cosh X,
= R sinh X)
and
(x4
= B sinh X,
cosh X) correspond to timelike and spacelike intervals, respectively.
Thus, for the
first choice, r>2 Jti/
is
r
a timelike interval. If
we
=
^
rf
2 *> r 1
JLr^.
replace X
by
\J
iX in (VIII.8.16),
both cases would
lead to equivalent representations of the Euclidean group. The eigenfunctions of the irreducible representations of the Lorentz group
can be obtained from the irreducible representations of the four-dimensional orthogonal group. Thus if in equation (VIII. 7. 16) we simultaneously replace X by ^X and n by in, where ^ n ^ oo we obtain one of the infinite-dimen?
sional
representations of the Lorentz group.
(VIII. 7. 16) yields
where rr/ n z
For the timelike
intervals,
the eigenfunctions t
^=
(n. X)
Qi
(sinhX)^d u is
7:
=
2
[n (n
<),
+
Z
(VIII. 8. 17)
m (cosnX)
,,
Qz
(0,
\
%Nyr+1? .i,
d(coshX) 2
+
I 2 )(n 2
+2
(VIII.8.18) 2 )
(n
2
1'2
2
)]
.
* E. Wigner, Ann. Math. 40 (1939), 149-204. Harish-Chandra, Proc. Roy. Soc. London, A, 189 (1947), 272-401. V. Bargmann and B. Wigner, Nat. Sci. Sci. Proc., 34 (1948), 211-230. V. Bargmann, Ann. Math., 48 (1947), 568-640. H. Weyl, The Classical Groups, Princeton Univ. Press, Princeton, 1939. I. Gelf and M. Neumark, J. Phys. USSR, 10 (1946), 93-94. t A. Z. Dolginov and A. N. Moskalev, Sov. Phys. JETP, 37, No. 6 (1960), 1202. A. Z. Dolginov, Sov. Phys. JETP, 3, No. 4 (1956).
Infinite Representations of the Lorentz
[VIIL8.]
The spaeellke ease follows from replacing X by X The functions SI>/ nm (D) are normalized according to rf(Q)*r'(0) dQ =
+
257
Group
(&V/2) in (YIII.8.18).
5n'5zrm',
(VIII. 8.19)
where
dO = sinh 2 X
A
dX dB
sin
comparison with (VIII. 7.8) shows that
(VII 1. 8. 20}
d
tyn im
are eigenfunctions of the
operator
2coth \--
-
dX
for
E2 >
0,
for
R <
0.
2
and
_ sinh 2 X
of
In both cases we have a2
a2
The
total volume of the Lorentz group is infinite, so it is not possible to an operation of group integration using only finite-dimensional out carry to oo .) representations. (The parameter X has its range from
VIII. 8. C. Rotation of the Eigenstates of the Representation
We
shall base
our discussion on the observation that the definition
(VIII.8.8) of the angular
momenta
suggests a correspondence of the type
momentum J^ by
discussed for the three-dimensional angular
This correspondence
is
the respresentation of
Jv where the spinor operators mutation relations S,
(VIII. 8.21)
>-|<^^Ig),
sa
and
ff'
=
q a for a
=
,
1, 2, 3, 4,
=
',
in Section IV.2.
are subject to the com-
74,
and where s* and $4 are complex conjugates of si and s2 respectively. It is easy to show that J^, as represented by (VIII. 8. 21) and (VIII. 8. 22), satisfy the commutation relations (VI 1 1. 8.1). In this formalism the role of the ,
Lorentz metric
(a^ v )
= F is played by
The operator m, corresponding
j3,
the metric of spinor space. is defined by
to (IV. .18),
m=
:
*
(VIII. 8. 23)
The Lorentz Group
258
The corresponding
eigenstates, as in Section IV.2,
\yJjMm) where the
coefficients
A$m =
[(J
= A;3^ + "rf-*4+
must be w
w
sJ-
[Chap. VIII]
of the
form
(VIII. 8 4)
,
Ai}m are given by
+ M) !(J -
M) !]-i/*[(.7
+ m) l(j - m) I]"
(VIII. 8 Jo)
1
/*,
where
J The Skj
for k
=
^J
and
j
^m ^
j.
l)-dimensional. A "rotation" of the (2J l)(2j" a transformation of the spinor can affected be by \jJjMm) to the Lorentz group, the 2. refers Since the representation 1,
representation
eigenstates
M
-
+
is
+
and the
relevant transformation can be obtained as a product of rotation
Lorentz transformation operators for a two-dimensional spinor. Thus, from the analogy with Section IV.2, we write
_
i'(*, 0,
M'=~J
]m )
where si
e
COS
7: e~~
=
w*A**-iw
^
gm
<
(VIII.8.%7) i sin
~
e (1/2) (
^+^
cos -
and tanh = v is the velocity of the new reference frame. We note that 3 and $4 are to be transformed by the complex conjugate of U. Substituting from (VIII. 8. 87) in (VIII. 8. 6) we find that the representations D<j n of the Lorentz group are related to Z) (J) and D^ of (2J + 1)- and (2j 1)T/'
+
dimensional representations of three-dimensional rotation groups
U
(VIII.8.%
"
\
by
/-I
-"
\
/
where
-
r
IV(J
+M
/
-|2J+Jdf-Af
7
-2Z
r sin |
,
0,
til*
=
+ MQ!(J - MQ M + !(/ - M' -
M)\(J Z) IZ!^
(-l)
1/ 2
I]
Z)
a~]2l-MM' Q-m-M-M' eiM*> |J
M - M'D^,. M (^,
6,
*).
(VIII. 8.29)
[YIII.8.]
Infinite Representations of the Lorentz
259
Group
TVe shall not elaborate these representations any further. For more detailed treatment the reader should consult the papers by Dolginov and Moskalev. In recapitulation:
An
irreducible representation of the Lorentz group
characterized by two numbers,
dimension of
and
j"
17.
The number jQ determines
the representations of the three-dimensional
The number
the given representation of the Lorentz group.
complex number and
is related to
may
be
the lowest
subgroup contained in 17
may
be
an
arbitrary
group invariants.
VIIL8.D. The Inhomogeneous Group The inhomogeneous group is a ten-parameter group. Its representations require the study of the commutation relations (VI II. 8.1) together with the quantum form of the classical relations (II1. 1.24):
=
Uw P J P
ih(ap,pp
-
(VIII. 8 30)
a^p v)
and
IP P J
=
(VIII. 8.31)
0,
The
invariants of the group that is, the operators commuting with all the can be constructed in a simple way. By multiplying the commutation relations (VIII.8.80) from the left by p p repeating the same from the right, and adding the resulting expressions, we find that others
,
Lp
2 ,
J=
/
(VIII. 8.32)
0,
where
& is
the invariant of the group.
To
mutation relations (VIII. 8. 30). tion of the resulting expression,
IM vp
=
JpvPp
=
pw
(VIII.8.33)
find the second invariant
By
we
cyclic permutation of &vp
use the comand the addi-
we obtain
+ JvpPn + JppPv =
PpJ^v
+ PfiJvp + PvJpn*
(VIII. 8.34)
Because of the complete antisymmetry in /x, v, p, the quantities I vp have only four non vanishing components: I2 34, Isu, Iiu, Im- The dual of the tensor
Iw defined by o! is
(VIII. 8.35)
a 4-vector.
From the commutation S^p, we have
relations
(VIII. 8.1), denoting the
right side
by
The Lorentz Group
260
we use
If
(VIII. 8. 30) to express Ppj ap,
-
-
ing expressions and multiply through by
Uw Ajj
=
in terms of
-
e
ih(apv
we
1
',
J a0pP
*
*
,
,
add the result-
get
- aw A,)
^
[Chap. VIII]
(VHL8.86)
and
IA
=
0.
(VIII. 8.37)
pfl>'S,
(VIII.8.38)
p,]
Hence the second invariant
A2 = where the tensor operator
_
CM &r
TauT J
ISHTceBT *vv /a^
/
"2
fVTTT 8 ^Q\ \ ' JLJLJ. .O.O&)
av
obtained from
is
1
36'
By
using the commutation relations (VIIL8.36)
and these lead
to
(VIII. 8. 38). The trace
homogeneous group
we obtain
S? yields the
of
first
invariant of the
:
*
irSf
=
*SS
= y^M, =
(VIII. 840)
Ii-
Because of the Lorentz invariance of the above operator relations it will be In such a frame
sufficient to consider their significance in the rest frame.
p
-
t
=
o,
where
invariant,
i
=
1, 2, 3,
and the
first
invariant
is
just
pi For the second
from (V 1 1 1. 8. 38) we get ,
where Si
=
Hence
=
M
A2 =
M pl
A2 =
h*s(s
-
2
]V2
+ N*
==
2
or
where
M
is
From
the definition of
the rest frame and
we
+ I)
A we
(VIII.841)
l)pl,
the three-dimensional angular
therefore the eigenvalues H 2 s(s alone.
+
M
2
momentum
in the rest frame;
expressed in terms of spin s see that the component Im vanishes in of
are left with I2 34, Izu,
is
Im
/ JM 3 i,
,
components, which yield 0].
Infinite Representations of the Lorentz
[VIII .8,]
For a
The
particle with
eases s
=
0, s
=
261
Group
mass there are, for a given momentum, 2s + 1 states. and s == 1 correspond to the transformation prop~i,
wave equations (and therefore of the wave functions) of particles wave functions of the particles are |fi, and ft. The corresponding
erties of the
of spin 0,
scalar (pseudoscalar), spinor, vector (pseudo vector) quantities, respectively. see that representations of the inhomogeneous group do not exclude parti-
We
with spin higher than h. For a massless free particle with integral spin, the number of states is not 1. For example, 5=1 for the photon, but the number represented by 2s cles
+
of possible states
is
just 2 instead of 3 (see Section II.2). If
m=
0,
then for
a free particle (not in the rest frame, of course) from p*p*SS
we
see that for
p
2
AX
we must have A = p?Ay =
=
2
we infer that for p =
2
Pi/43
+ P4/23 = = #8/41 + P4/31
3
02/41
Pi/42
= A = A = A4 =
Using the identity
0.
(VIII. 8.44)
A 7 we have
the definition of Al
(7III.84S)
we have
2
From
=
p 3/42
Pi/23
(p
+ P4/12 =
(p
P2/43
+ P2/31 + P2/12
(p
X N)l + X JV)2 + X JV) + 8
= P'M.
Hence we write
A where p4
A4 = p M = + pJM = op, =tV(pf + pi + p|) = =fcp; therefore
= - (p X N)
=
ap,
(7111.8.48)
P
Comparing this group,
we
result
with (IV. 5.12} of the three-dimensional inhomogeneous
see that the quantity
a in (VIII. 8. 47)
is just
the
minimum j value
possible in the given representation. Thus for a given momentum vector there are two possible independent states for a -^ 0, which correspond to
For a
=
only one state; it corresponds to the transformation properties of a spinless and massless particle. For nonzero spin we shall consider two particular cases.
two
different states of polarization.
(a)
there
is
M = L + hK, the total angular momentum of the photon. We have a=zb-p
'
+ np g '
(VIIL8.48)
262
The Lorentz Group
[Chap. VIII]
In this case the two states can be transformed into one another by a space reflection; this holds also for higher spins. The two states remain unchanged under a time reflection. (b)
M=L+
^Tkr,
the total angular
momentum
of the neutrino.
We
have
a=P- L+^ Here tion,
ff
-P-
(VIII J 49)
it is not possible to transform one state into another under space but the two states remain unchanged under time reversal.
reflec-
CHAPTER
IX
SYMMETRIES OF MASSLESS PARTICLES
IX.l.
Angular
The photon
is
Momentum
of the Photon
a vector particle and therefore we need a complete set of its wave function into its "multipoles." Such a complete
vectors to expand
can be formed from the simultaneous eigenfunctions of The special class of eigenfunctions obtained by taking j\ = I and
set of eigenfunctions
J and Jz 2
j2
=
1 in
.
the expansion formula (IV. 3.4} are called "vector spherical har-
\ylljM}
From
=
for j%
to j
=
I
Z
= m=
; I
s=
the addition rule (IV. 8.8} of two angular momenta, it follows that 1 there are three kinds of vector spherical harmonics corresponding lj
I,
I
+
1
or
I
=
j
+
shall write (IX. 1.1} in the
|yjS(,*)>-i where
j
1, j,
different vector spherical harmonics.
We
(IX J.I)
C(nj;msM)\yllm8).
1
1.
For each value of j we have three
*
form
E
C(llj;mSM)Y lm (e,
M=m+
s
(JZJJ8)
(IX.1.S)
and where we use the correspondence \ylljM}
=
\Y&(e, *)>,
The eigenvalue equations *
J.
|
7 Zlnw>
= Y lm (6,
f)\s).
are
Blatt and V. F. Weisskopf, Theoretical Nuclear Physics, Wiley,
New
York, 1952.
263
Symmetries of Massless Particles
264
[Chap. IX]
jy#>,
(J
Y1
4)
of K* as defined In The vectors |*>, where a = 0, 1, 1, are eigenvectors scalar usual the are spherical harmonics. The (1.5.7). The functions Fzm(#, <) the of photon; they span the "spin vectors ]*) are the spin wave functions
space" of spin 1. For j = o there exists only one vector spherical harmonic. For j
have
and
I
m
=
0,
=
we
and (IX. 1.2) gives
(47T)
in the radial direction opposite to the spin s. Because of the completeness of F Zm (0, <>) and of |s> the vector spherical
which
is
harmonics constitute a complete
set. Since there
to spherical harmonics corresponding
complete set
of the
I
=
j,
j
-
1,
are three kinds of vector j
+
1,
the unit operator
is
i
z i i
=
i^jw^ui,
(ix.i.6)
j=QM=-js=-l of Clebsch-Gordon coefficients and the where, as follows from the definitions harmonies satisfy the orthogproperties of |s) and 7 Zm the vector spherical ,
onality
and normalization conditions
:
sin
Any
vector function
|r?
)
w> =
of xi,
eded^^
(IX.L6)
^MM>f>ii>.
z 2} ^ 3 can be expanded according to
E ^ E^
+ ..iiyw>
y=o M=-ji*=-i
where the expansion
=
coefficients
A^ +s
,i
are defined as
the expansion coefficients are three functions of the radial coordinate r, depending on the quantum numbers j and and It follows from (IX. 1.7) that a photon state represented by a given j has a wave function consisting of linear combinations of three types of
For
0, 1,
1
M
.
M
vector spherical waves
(I
= j, j
1}
j
+
I):
subject to the conditions
r>? =
0.
(IX. 1.10)
Angular
[IX.l.]
Momentum of the Photon
Hence for a given j and of two states only.
The vector
jFj/i) for
M the state of the photon =
j
I
265
Is
obtained as a superposition
can be generated from the scalar spherical har-
monics Yim. By representing the vector operator L in the form of a column vector we can project it into the spin space [we use the unit operator (1.8.8)]
by
Hence \L)Y lm
S=~l
\-l)L+ Y lm lm~1
*!-!>[<
By
using the properties of spherical harmonics
\L)Y lm
From
we can
further write
=
this it follows that ]
=
(IX.1.1S)
0,
an arbitrary function of r and p = iKV. Application of the above method, together with the use
where F(r)
is
of the properties of
spherical harmonics Yim , enables us to obtain vector spherical harmonics of the
form _1_ 1
[7 ^7jJ where
If)
r
""11/2
|yun-w>
7
"11/2
+ L2ZTTj
1F"- ia)
'
IX L1 ^
(
-
represents the unit vector r/r. also construct a vector spherical harmonic orthogonal to the
We may
momentum
vector
p
and to
|Fzzi).
It is given
by
cp
= \XiM )
The vectors |FM) and |ZV) are called transverse and they satisfy the normalization conditions
where
a,
/3
=
0, 1.
Z
spherical vectors
266
Symmetries of Massless Particles
DLL A.
The Parity
of the
Photon
We may now use the transverse spherical vectors function
\rjf)
in the
[Chap. IX]
\XJM ) to express the wave
form
hf > =
(IX. 1.17}
FfM(r)\Xfx),
a=0
where the functions FJM are arbitrary. The complete wave function of the free photon can be written as
where the j = state, as shown before, does not correspond to a transverse mode and is therefore excluded from the summations. Because of the invariance of the electromagnetic field under parity operation
we
shall require the
operator
(P.
The
latter requirement will
definition of the functions* FMJ>
tion of the state of a photon
to be
Hence we infer that we need to specify
referring to its energy E, its angular
and
an eigenstate of the parity impose a further restriction on the
wave function (IX.1.18}
for a complete specifica-
four
quantum numbers
momentum quantum numbers j and M,
its parity.
From
the parity relation <$>Y lm (8, <)
=
Yim (ir
-
6, TT
+ <) =
l
(~l) Yim (e, <)
and If, LtJ
=
0,
where the Li are the components of angular momentum,
it
follows that
and
We thus see that for a given j and M there exist two different parity states. In the expansion (IX.1.18) the states with a = 1 and a = correspond to electric
and magnetic
netic states to electric
states, respectively.
The
and magnetic dipole
relation of electric
and mag-
(or multipole) radiations will
be
considered in later chapters.
the eigenfunctions \rjf} we can construct a probability function (*l\n}t determining the probability that the photon is moving in the direction
From
of its spin
with a polarization direction [s) (where s = 1 or 1) in an angular state (j, M) and parity a. Because of the definitions of \XfM }, the
momentum *
A.
I.
Akhiezer and V. B. Berestetsky, Quantum Electrodynamics Oak Ridge, Tennessee.
Technical Information Service Extension, 1957.
,
Part
I,
USAEC
[IX.L]
Angular
probability of the type its
is
Momentum
independent of
Hence
\Yjfi).
the
and depends only on In the state j = I we have
we can
using (IV. 8. 17)
and depends on vector
its
the
spherical harmonics
photon
is
independent of
angular momentum.
write
J2 =
r
V-
rp*r,
2
because of transversality of the waves, can be replaced the square of the photon Hamiltonian
where the operator p
by
267
angular distribution of
parity
By
a.
of the Photon
Hence
(1'X.I.$1)
,
becomes
AW +
1)
r
fi
= (r*
f
or dr*
AM. and k = E/hc = co/c. The solutions are asymptotically given by waves* representing outgoing where we take
(1/r)/,-
u
=
exp
(r )
^'^"JJj
|_z^/C7*
tor
of (IX. 1.22)
/CT*
which results from the combinations of the exact solutions
Then the
solutions of (IX.I. 22) are in the
where J 2+1/2 and in
A^z+i/2
are the Bessel
form
and Neumann functions
as defined
Jahnke-Emde.f
The complex conjugate function u\~
}
of z4 +) corresponds to ingoing spherical
waves. In accordance with the time-reversal operation of the electromagnetic field [see (VIII. 1.69)], we deduce that ingoing and outgoing spherical waves are related
by a
time-reversal operation.
Blatt and Weisskopf, Theoretical Nuclear Physics, Wiley, New York, 1952. fE. Jahnke and F. Emde, Funktionentafeln, B. G. Teubner, Leipzig, 1933. [English *
edition (4th), Dover,
New
York, 1951.]
Symmetries of Massless Particles
268
IX.l.B.
The
Partial Polarization
of a
Beam
of
[Chap. IX]
Photons
A definite state of polarization for photons is represented by either
(a)
two
plane polarized waves with the polarization planes orthogonal to one another, or (b) two right and left circularly polarized waves (Fig. 9.1). In terms of
a density matrix
where the vectors
a state of polarization corresponds to a
[see (VI.2.A}},
pure state (Fig. 9.1) represented
II)
and
1)
|
by a plane wave,
are orthogonal. Since there are only
number
the
N* is
=
1
a 2
two
states,
data required is 3 and the density matrix of
7
X
2 matrix. For a totally
beam of photons the can be calculated matrix density polarized
directly |0>
(b)
the
from (VI.2.20} and from
direction
of
the spin of the
photon. The direction of the spin in terms of two complex numbers ui
FIG IX .
.
1
.
The pure
skate, (a)
cir-
Right
cular polarization, (b) Left circular polari-
Under space and time inversions and (b) do not change.
zation. (a)
is parallel (or antiparallel) to its
and w2 in (IX.1.25} can be expressed by the definition of momentum of the wave, as given by
(VIII 4.26}. The spin of the photon momentum; therefore (VI11. 4.86) yields
=
q
(IX. 1.26}
(u\r\u),
where we use r in place of Pauli matrices in order not to confuse them with the
The normalization Hence, using
of
q
(VI. 2, 20},
M, LUtJ
=
reduces the
number
of
(u\u).
parameters from four to three.
the density matrix in a pure state assumes the form
1
q+ It can further be simplified into
qzJ
Lu*U2
M
2
(IX. 1.27}
J
an operator form p
=
(IX. 1.28}
\u}(u\,
whose matrix representation with respect to eigenstates |1) and 1) of 0-$ form and the from the of normalization (IX. 1.28} yields (IX. 1.27}. We see |
condition (u\u}
=
1 that p is
a projection operator:
The Two-Photon System
[IX.2.]
P
Hence the density matrix of a pure
269
=
2
(IX.1.2S)
p.
state under a unitary transformation can
(see Section I.5.B) into
be transformed
(IX LS -
-[J o)
where p is the density matrix for totally polarized beams. For an ensemble of states the corresponding density matrix can be written as
=
P
(IX.1.S1)
J,
[,
where we can assume that \i ^ X2 ^ 0. Using the condition tr p = 1 for a X 2 = P, we can write* p as an incoherent density matrix and putting Xi superposition of two beams,
and second terms represent a totally polarized beam with weight P, superposed incoherently to an unpolarized beam with weight 1 P, = 1 we again get the total polarization state. Hence P, respectively. For P
where the
first
^ P ^ 1, represents the degree of polarization. In general, for a nonpure state (a mixture of states) the density matrix of the form
with
P
where q
is
=
not a unit vector, but
<
same
From
2
(2p'
algebraic equation
(1 X.I. 38} I)
=
g
2 ,
it
where
we can
cast it as a diagonal matrix,
also satisfied
the eigenvalue of
p' is
X2 so the degree of polarization
is
= =
P! p'2
by the
eigenvalues of
p.
= id + = 1(1 -
p.
Hence
Igl), Ifll),
given by
p =
\!
-
X2
=
(IX.1.8S)
Igl.
The Two-Photon System
As a system
of
two noninteracting
photons can be obtained *
is
(IX.1.S4)
follows that the eigenvalues of p satisfy the equation
Xi
IX.2.
the inequality
1.
deriving an algebraic equation for p
since the
(IX.L8S)
*-g),
satisfies Iffl
By
+
4(l
is
by
particles, the
wave equation
noting that the total Hamiltonian
H. A. Tolhoek, Rev. Mod. Phys., 28
(1956), 278.
is
the
for
two
sum
of
270
M assless Particles
Symmetries of
The Schrodinger equation
the Hamiltonians of the single photons.
photons
is,
therefore, given
[Chap, IX] for
two
by
= * jJw) ot
(1X3.1)
ffka>,
where
H = H + Kb? Ha = H = a
b
and the ket
represents the two-photon tion (IX.$.1) must be supplemented by the IT/^}
wave function. The wave equatwo conditions of transversality
of the photons: Va"*lat>
=
V&"Ha6
0,
=
(IX. 2. 2)
0.
The wave function of the two-photon system must be regarded as a tensor, so each transversality condition in (IX. 2. 2) corresponds to three equations. The physical interpretation of \7j ab) can be based on the quantity the probability of finding one photon with a momentum p a and polarization in the direction a and another photon with momentum jp& and polarization in the direction 6. Condition (IX.2.2) eliminates the possi-
which
is
bilities of polarizations of the first photon along the direction p a and of the second photon along the direction p b} respectively. The wave function of the photons must further be qualified by the symmetry properties induced by the
The photons as particles wave function and the momenta pa p&.
obey the Bosemust be symmetrical
identity of the photons.
of spin 1
Einstein statistics and therefore the
|??
with respect to
a, b
,
The ordinary coordinate concept pointed out before, but physically relative
momenta
of
and write the Hamiltonian
H is
it
which, in
it is
=
P =
+ Pb, in the
i(p
- p>),
(IX.2.8)
form
+
cftS-P
(Ka
-
K
b)
-p],
(IX.24)
the total spin of the two-photon system:
S = Hence
not well defined for photons, as was meaningful to introduce the total and
is
two photons by
P = Pa
where S
a &}
K + Kb a
follows that the eigenstates of
momentum
= K.
H are of the form
space can be written as fort)
=
F(P)\ ab (p)}.
(IX.2.5)
The Two-Photon System
[IX.2.]
In this
way
271
"center of mass" mot-Ion
Is
separated out.
The
transversality
conditions (IX. 2. 2} become 3
a= l
j>-*5(p) =o,
3
]CP6-*&(P)
=
0,
6=1 or s
Pi&Ap) = ]C i=I
=
for j
0,
a, &,
where, because of the symmetry with respect to an interchange of the photons,,
we have
A
system in momentum space is obtained by taking (provided pa and p b are not parallel). In this case the momenta of the two photons in the coordinate system where "center of mass" is at rest are special coordinate
p =
represented by
Pa
=
P6
P,
= -p.
(IXJS.7)
IX.2. A. Spin Eigenstates and Parity of
From
Two-Photon System
momenta it follows that the total spin s, in the values 0, 1, and 2. The correcan assume s(s 1)] [S (IX$.5)j sponding nine eigenfunctions are also eigenstates of the exchange operator 2
the addition of angular
=
+
K K
Pab (see
problem
4,
=
The
five
five
(IX. 2.8)
b
Section IV.3.B). For the antisymmetric states
corresponding eigenvalue of Pab is
For the
a
symmetric states
1.
=
symmetric spin states are
1)
the
The three antisymmetric spin states are -1}
-
-1>
-
Cl7a, l)l7 6 , 0}
(s
(s
=
|
l7,
-1>U 0}],
|7a3
-1>K
Ta,
1)],
0>lr*, !>]-
2) the corresponding eigenvalue of Pab is
1.
Symmetries of Massless Particles
272
*,
0}
=
[Ira, I)!T &?
!7a7 &, 2}
=
Finally, the one
and |
is
DK
I7,
+
1>IT, 0>
[iTa,
+
-1>
!
IT.,
-I>!T,
Ta, 0)^,
+ 2I T
1)
,
[Chap. IX] 0}! 76? 0>],
1)3,
(/X.&J0)
I}-
symmetric state
=
(s
2 of
0) belongs to the eigenvalue
P ab
given by
TaT&j o)
=
[iTa, 1)|7 & ,
- 1) +
- 1)1%
IT-,
+
1}
0)J7 & 0)].
l7a,
,
(IX. 2.11)
nine spin wave functions are simultaneous eigenstates of the = as bz From the addition b y and of Sz value S 2 = (Ka
The above
K +K
+K
total spin
rule for angular
momenta we
J-L are given
j j
where S =
From
+ hS
(IX.2.12)
by j
= = =
l\,l
for s
l
for s
I
dh
.
find that the j values corresponding to
1,
Z
it 2,
for s
1
= = =
1,
(IX.2.13)
0,
(IX.8.14)
(IX.2.1%)
follows that the
it
wave function
of
structed as linear combinations of the products of
(IX.8.1S)
2,
the Jf^ + X&, and L is the angular momentum of
"relative motion/'
two photons can be con-
Y im (e,
<), eigenf unctions
the spin eigenstates (IX.8.&) and (IX. 2. 11). The parity of of the wave function will be determined from its orbital part. However, under
L and L z and 2
,
Bose-Einstein statistics the symmetry \nab(pa, Pb)}
of the
wave function
=
in terms of relative
I**(p)>
=
and therefore it involves the symmetry The statement (IX.2. 16) means that
Hence the parity related
|l?&a(p&,
of a state
Pa)}
momenta assumes
the form (IX.%.16)
!**(-*)>, properties of the spin
and the symmetry properties
wave
functions.
of spin states are
by I7a7 & ,
M.)
= (- 1) '1767*,
AT,),
(IX. 2 .17)
variables meaning that even states (even I) are symmetric with respect to spin and odd states (odd I) are antisymmetric. The only spin states that are possible for odd parity can be antisymmetric in spin variables. Such states are repre-
sented by (IXJ.9).
The Two-Photon System
[IX.2.]
273
For the antisymmetric spin states (IX. 2.9) to correspond to transverse waves they must also satisfy the transversality conditions. For the states = 1, I = j 1, j, so for odd Z, j must be even, j = 2m, (IX. 2. 9) we have s / = 2n 1. The case therefore j = I is not allowed, since I is odd. The say; states (IX. 2.9) can be regarded as the
second-rank tensor in three dimensions.
Ay = =
5).
We set - aA-,
labels the quantities a-
1, 0, 1
(IX. 2. 18)
and
-
6y
t
,
Hence
[see (1.4.15)].
Ay The
an antisymmetric
and where each a or & i represents a ket vector (spin eigenfunction of the photon a or The tensor Ay can represent an axial vector A f defined by
where t, j for a given photon
a t*y
of
components
6i,tAk-
transversality condition of the
spin state (IX. 2.9) can be expressed Atfpj
=
wave function
in the antisymmetric
by ewpjAjk
=
0,
or, in ordinary vector notation,
PXA=
(IX Jg .19}
Hence for odd parity states of two photons the axial vector spin wave function must be parallel to the relative momentum p that is, the spin wave function must be longitudinal. For a given j and M, as follows from (IX. 1.1 4), there is
only one such state, I
where F(p)
a function
is
of
p
and I
with the two
Z
values
Z
=j
1
db 1
pX
^4
1 /2
= 2n db 1. Condition = p X (a X b) = 0,
(IX.8.1ff), in the
form
imph'es perpendicularity of the polarization states of the two photons. Thus, two photons in odd parity state are polarized perpendicular to each other (Fig. = 1 of the photons, the only s value is 8 = and 9.2). For the spin state s
m
M
the spins are antiparallel.
The antisymmetric of
two photons.
A
superposition of (a) and (b) is the odd parity state 7r-meson as a pseudoscalar particle (that is, of odd parity)
with spin zero can, in
its rest
frame, decay into the odd parity state of two
photons.* *
H. A. Bethe and F. DeHoffmann, Mesons and
1955, Vol. II, p. 12.
Fields,
Row, Peterson, Evanston,
111.,
274
Symmetries of Massless Particles
^c/V
-P
(b)
FIG. IX. 2. and
spin zero
is obtained
The odd parity
state of
two photons, (a) The two-photon state with total
right circular polarizations perpendicular to one another, (b) This state
from
(a)
polarization of the
by spatial reflection and corresponds
to
perpendicular
M values M =
to
M
s
=
0,
left
circular
two photons.
For even parity states the only possible total 2. Other even parity states corresponding to
and and
[Chap. IX]
with
s
of spin are 0, 2, 1, 1,
with
s
=
2,
=
0, do not satisfy the transversality condition. The has been disthe vector spherical harmonics for j =
s
longitudinal nature of
s
cussed in Section IX. 1. In general, there exist no even and odd parity states with = 1. As a result, a two-photon system can never be in a state with unit angular
j
momentum.
We may also infer that a system with unit angular momentum cannot
decay into two photons.
Figure 9.3 illustrates the possible two-photon states with parallel polarization.
The
three symmetric superpositions of the states in
spond to even parity states of
(a)
-
(b)
-
(c)
-,
and
(c)
corre-
^/VM = 2 S
FIG. IX. 3. right (c)
(a), (b),
two photons.
and
left
The even parity
states of
circularly polarized, (b)
Obtained from
(a)
by space
two photons,
(a)
The
state
where a and b are
Both a and b are right circularly polarized.
reflection.
The Wave Equation of the Neutrino
[IX.3.]
275
IX.2.B. Problems 1. The positronium atom known that in the singlet S
a bound state of electron and positron. It
is
=
state (s
= spin What can
7
0,
positronium can deca3^ into two photons. fact on the parities of electron and positron? 2.
Can a
scalar
one conclude from this
x-meson decay into two photons?
The Wave Equation of the Neutrino
IX.3.
The
massless particle of spin
|,
described
by the wave equation
where H, defined by H Hamiltonian and the two-component spinor is
is
state) the ground-state
called a neutrino,
neutrino.
The
derivation of the neutrino
theoretical point of view. In this chapter
\v)
ca-p and is
the
p=
wave
wave equation and
been discussed
tion to photon theory has
=
its
in Section VIII.4
we
ihV,
is
the
function of the
formal rela-
from a group-
shall discuss its physical impli-
cations.
Fourier series expansion of I
wave 1
\
function can be written as
X"^
-T
a time-dependent two-component spinor and the wave number subject to the conditions (II. 1.33) used for the photon. From the
where
\Uk) is
vector
k
is
its
expansion (IX. 3.8) and the wave equation (IX. 3.1) we obtain ih
where
H
=
chk
-
-
\u k)
= H\u k),
(IX.3.3)
In order to find the eigenvalues of the operator state solutions of (IX. 3. 3). We can posit \u k )
and obtain from (IX.3.3) the
for
H
\v k
)e-W,
result
Ek Hence the eigenvalues
=
H we look for stationary-
\v k )
=
H\v k).
follow as in the case of photon; for each
Ek =
=b hc\k\.
The apparent negative-energy solution can be understood for the
photon
polarization states) in
/c,
(IX.34) (like
the situation
terms of positive-energy solutions with
opposite directions of spin. In other words, there are two positive-energy
276
Massless Particles
Symmetries of
particles,
which
differ
from one another only
in
[Chap. IX]
having their spins pointing
We can say that both particles are polarized in the but that for each particle there is only one spin state. longitudinal direction,
in opposite directions.
The two
particles of this theory are distinguishable, since, as
shown
in Section
VIII.4, the neutrino equation is not invariant under space inversion. Hence, the neutrino or its antipartick would not conserve parity. Space reflection for the is an invariant operation, so photons of opposite spins are not to be reas garded particle and antiparticle systems. For this reason we say that the photon has no "handedness." But a neutrino can be right- or left-handed, although one
photon
cannot be transformed into the other (Figs. 9.4, 9.5). Neutrinos which appear in the decay
n
>p
+e+
v
are right-handed; from the conservation of lepton charge they will be called antineutrino.
^\y^7^^\yiv^/\y FIG. IX. 4.
left
circularly polarized photon, where
where v is right-handed.
v is left-handed.
y
exists
S
always a transformation
}
S-lyS but the same transformation does not that
The neutrino and
right circularly polarized photon,
There into
FIG. IX. 5.
The antineutrino and
7,
exist for the neutrino
is,
S- vS l
=
such that y can be transformed
^
and antineutrino;
v.
(IX.3.5)
Experiments on weak interactions prove that the neutrino
is
described
by
the two-component theory contained in equation (IX. 3.1). After the breakdown of parity became evident, equation (IX. 3.1) was proposed as a two-
component theory of the Salam, and by Landau.*
neutrino, independently
*T. D. Lee and C. N. Yang, Phys. Rev., 105 (1957), 1671. A. Salam, Nuovo Cimenio, 5 (1957), 299. L. Landau, Nuclear Phys., 3 (1957), 127.
by Lee and Yang, by
The Wave Equation of the Neutrino
[IX.3.]
The
states represented
states. If
we
by
Figs. 9.4
9.5 are the only possible neutrino
replace a left-handed neutrino
which does not
handed
antineutriiio,
we say
that charge conjugation
neutrino.
and
277
by its antipartiele we get a lefttwo-component theory. Hence
exist in a
is not
However, charge conjugation
an
invariant symmetry operation for the
together with parity operation is
an
in-
variant operation since in this case the left-handed antineutrino arising from the charge conjugation of the left-handed neutrino becomes, under space ,
a right-handed antineutrino. Theoretically the distinction between neutrino and antineutrino is based, essentially, on the validity of lepton charge conservation. The latter is, cerinversion,
not independent of charge conjugation symmetry. But charge conjugation breaks down for interactions involving the neutrino. This situation is remedied for weak interactions by imposing a restriction on charge conjugatainly,
tion:
it
holds with reflection of charges provided that we interchange left and an additional symmetry operation. Thus the ordinary concept of
right as
charge conjugation for mass particles tions of charge conjugation
law of parity conservation
and space is
is
replaced
reflection,
replaced
by the simultaneous operaand vice versa. The normal
by the simultaneous operation
of
parity plus charge conjugation. If we represent charge conjugation operation by 6, then, in order to maintain our invariance principles on a universal basis,
we may,
in regard to the neutrino, drop our separate requirement of
invariance under 6 and (P operations and replace it by the invariance under the joint operation e
discrete
Instead of saying that the laws of nature are invariant with respect to the group of discrete transformations affected by I, G, 3,
is
not equivalent to unit operation),
we
postulate the invariance with
respect to a restricted group of discrete transformations affected by I, 6(P, 3, <3
by processes involving the neutrino do not appear as isolated phenomena that admit only a restricted transformation group; therefore these jugation
* J, Schwinger, Phys. Rev., 74 (1948), 1439. H. A. Kramers, Proc. Acad. Sci. Amsterdam, 40 (1937), 814. W. Pauli, Niels Bohr and the Development of Physics, Pergamon Press, London, 1955. G. Liiders, Z. Physik, 133 (1952), 325. G. Ltiders, Kgl. Danske Videnskab, Sekkab Mat.-Fys. Medd., 28 (1954), 5. E. P. Wigner, Rev. Mod. Phys., 29 (1957), 255.
278
Symmetries of Massless Particles
processes
may be compared
[Chap. IX]
and
to the processes involving electromagnetic
strong nuclear interactions.* It must be pointed out that the adherence to I,
6CP, 5 7 (PCS operations as the universal symmetry principles does not necessarily solve the problem of distinguishing between the neutrino and its antiparticle, the antineutrino.
Later in this chapter
it
will
be shown that the methods of conventional Dirac
theory for particles of spin J allow for the possibility of describing v
and
v
as self-conjugate particles.
As a preliminary
orientation
we
shall recapitulate
some
of the results ob-
tained in Chapter VIII in connection with the discussion of the complex orthogonal group. The simple observation was made that for plane electro-
magnetic wave the invariant
CxlxD =
x?
+ xl + xt =
Q2
vanishes everywhere, and this was interpreted as an invariant statement for a massless and chargeless field. All three stereographic projections of the com2 = 0, to projections on a plex sphere (VIII4.1) reduce, in the case where Q
=
=
0, which is the plane wave itself. three the possible stereographic projections (VIII4&1} as repreregard three states charge (positive, negative, and zero), then we see that senting
single plane, If
where
\6\
|9C
and 6- JC
we
for
Q2 =
in the
they are equivalent and lead to the transformations of the
form (VIII'4.25). Since the transformation (VIII 4-^5)
%-field
yields,
as
shown, the neutrino equation (FII/.4.##), we infer that all three charge states satisfy the same neutrino equation. Hence a chargeless neutrino is described by a complex wave function \v) and not by a real wave function.
A
real
wave function
are identical.
is
We must,
need not describe two distinct tion
is
and therefore corresponding particles it clear that a complex wave function particles. For example, the photon wave func-
self-conjugate of course,
make
complex, but the corresponding particles differ only in their direction
of polarization.
The
neutrino
wave equation (IX. 8.1)
is
not invariant with respect to anti-
linear transformations: |/}
Hence the wave function
\v)
=
Cll?},
V
=
for the antineutrino
(IX.8.6)
erf.
is
not just a complex conju-
while for the photon (77) and (?|T?) describe its states of opposite To obtain the wave equation for the antineutrino we must consider the projection of |%*) onto the neutral plane Q 2 = 0. This may be
gate of
|P),
polarization.
regarded as a process of stereographic projection of the *
left circularly polar-
However, future investigations, especially in the region of very high energy, to further modifications of our presently well-established symmetry principles.
may
lead
[IX.3.]
The Vavc Equation
279
of the Neutrino
ized photon; from the point of view of projection we have the correspondence that the right-handed antineutrino v corresponds to the right circularly polarized photon,
and the left-handed neutrino corresponds to the
polarized photon; the photon
itself
left circularly
has no handedness since both directions
of polarizations can be transformed into one another. It is important to point out that the acceptance of equation (IX. 3.1} as
a
for the neutrino was a natural result of the unexpected hyadvanced by T. D. Lee and C. N. Yang,* that nature, by way of potheses, discriminate between left and right. One of its most can weak interactions,
wave equation
in weak decays are interesting consequences has been that the electrons test of the Lee-Yang hypothesis was carried polarized. The first experimental
by Wu, Ambler, Hayward, Hoppes, and Hudson, t The basic features of the experiment consist of the observation of the angu60 cooled to a very lar distribution of the electrons emitted by radioactive Co low temperature (0.01 K). The low temperature is required in order to eliminate possible thermal motions of nuclei. Under these circumstances the out
electrons. When the cobalt experimenters observed an isotropic emission of coil the electron a solenoidal source was placed within a current carrying emission had a higher intensity parallel to the axis of the coil than in the field aligned the spin vectors of the opposite direction, since the magnetic 60 electrons occured in the direction emitted of numbers nuclei. Greater Co 60 Under a space reflection the spins the Co of opposite to the spin direction there of nuclei and electrons remain unchanged but the current changes sign .
of electron reversing of the current. As a result, the intensity pattern emission did not change; that is, the reversed and unreversed experiments maintained the same intensity and direction of electron emission. In this experiment the main role of the neutrino, as one of the four fermions of the Co 60 nucleus, was to provide -> p Qv) partaking in the decay (n the necessary balance of angular momenta for the polarization of the electrons. is Since there is an electron-neutrino correlation, the neutrino emission also
is
+
+
not isotropic; this demonstrates the handedness property of the neutrino. In the same experiment it was observed that the y radiation that follows the 60 Therefore no violation decay of Co had complete left and right symmetry. interactions. for occurs of parity electromagnetic than nuclear decay, Parity nonconservation in weak interactions, other has also been observed^ in the decay of a muon 0^-meson).
The longitudinally *
of the neutrino will effect the decay of the spinless
T. D. Lee and C. N. Yang, Phys. R&>., 105 (1957), 1671. Wu, E. Ambler, R. W. Hayward, D. D. Hoppes, and R. P. Hudson, Phys. Reo.,
t C. S.
105 (1957), 1413. t R. L. Garwin, L.
M. Lederman, and M.
Weinrich, Phys. Rev., 105 (1957), 1415.
Symmetries of Massless Particles
280
[Chap. IX]
x^-meson in the form of a polarization of its decay products ju+ and v\ that is, in its own rest frame x+ -* M+ + v. Conservation of for a TT* decaying
spin
requires that the decay products
/i
+ and
v shall travel in opposite directions;
thus the neutrino's spin angular momentum |& about its direction of motion be balanced by a M+ also carrying a spin angular momentum \h about + is 100 its direction of motion. This means that in TT + decay the jz percent
will
J
The same applies for a \r in the decay of a TT~. The precise experimental measurements of yr polarization are made especially difficult because the muons can be depolarized before they decay. The helicity (handedness or correlation of the direction of motion and direction of spin) of a decay product
polarized.
decay and in weak decays of pions are the same. For found to be right-handed and \r left-handed.
in ordinary nuclear
example,
+ /i
is
IX.3.A.
The
ft
Positive and Negative
Energy
States of the Neutrino
The neutrino, absence of mass,
The
like
is a purely relativistic particle. Because of the construct a nonrelativistic theory of the neutrino.
the photon,
we cannot
Hamiltonian of a neutrino can be formulated as
classical relativistic
pi
+ pl),
(IX. 8.7)
By using the rules of quantum mechanics we can write down a corresponding wave equation
for the neutrino, as
-
[cp 4
where the
p
cV(pf
+ pi + p|)]|p> =
(IX.8.8)
0,
are to be interpreted as operators, in accordance with the
Schrodinger representation. Equation (IX.8.8) is unsymmetrical between p* and the momenta p^ where i = 1, 2, 3. From the analogy with the photon
we should
expect that the equation for the neutrino ought to be a linear it is linear in p. (The latter is also a natural relativistic
equation in the p, since requirement.)
The only
irreducible linearization (from the group-theoretical
can be obtained in terms of Pauli spin matrices. point of view) of The result is the wave equation (IX. 8.1), obtained from an entirely different point of view. However, there is some virtue in the present approach. The (IX. 8. 7)
process of rationalization of (IX.8.7) could be avoided
if
we
used the square
of the relativistic relation,
rf
=
rf
+
pl
+ rf,
and we could write down a second-order wave equation in place one.
The relativistic requirements are still
satisfied,
of a first-order but the second-order wave
not equivalent to (IX.S.8) every solution of (IX.8.8) is also a solution of the second-order equation but the converse is not true; every solution of the second-order wave equation is not a solution of the first-order equation
is
:
[IX.3.]
Tlie
Wave Equation of the Neutrino
281
>
wave equation. Only those solutions of the second-order equation where p* are also solutions of (IX. 3. 3). Hence we see that the first-order wave equation have twice as many solutions one group of solutions with positive values and another group with negative values of the energy. In classical theory an Initial positive energy will always remain a positive energy. In quantum mechanics an initial positive energy state need not remain
for the neutrino will
positive at all times. There is
always a probability of transition
to negative
energy
and negative energy states can be separated completely and no transitions between them can occur. In the presence of an interaction the positive and negative energy states cannot be separated and a transition between the states can occur. The negative-energy solutions of the wave equations can be interpreted as referring to a different particle than the one described by the positivestates.
However, for
free particles, positive
energy solutions. According to Dirac's interpretation, "marly all the negativeenergy states are occupied with one neutrino in each state in accordance with
Pauli s exclusion principle." An unoccupied negative-energy state will appear as a particle with positive energy, since its disappearance requires the addition 3
of a neutrino with negative energy. These unoccupied negative-energy states, which actually have positive energy, describe antineutrinos. A state where all positive-energy states are unoccupied and all negative-energy states are occupied shall be called
"vacuum
state."
For a particle with
electric charge, for example,
vacuum is a state of zero "electric charge density." The electric field of a vacuum must satisfy the Maxwell equation V-6 = 0. For a neutrino we replace the concept of electric charge by the concept of lepton charge and define vacuum as the state of zero "lepton charge density," where the infinite distribution of negative-energy neutrinos does not contribute to a "field"
produced by lepton charge density. However, we must note that the ordinary defined with respect to electric charge alone is also a state of zero
vacuum
lepton charge density. In general, any departure from the vacuum state will contribute to lepton charge density, and we shall find for each occupied state of positive energy a contribution
state a contribution of
IX.3JB.
The
+ 1 and for each unoccupied negative-energy
1 to lepton charge.
The Velocity
of the
Neutrino
velocity of a massless particle of spin %h, described
equation (IX. 3.1),
is
by the wave
obtained from Heisenberg's equations of motion, ac-
cording to
F= f ^fr'^'"U/l/
Hence we
see that the spin operator
(IX 3 9} -
-
bit'
a and velocity
operator
so for a fermion moving with the velocity of light there is
Fare
"parallel,"
a complete longitudinal
Symmetries of Massless Particles
282 polarization.
Under the
[Chap. IX]
by 3
time-reversal operation affected
=
icr2
C
the
velocity changes its sign, but under parity operation, for reasons discussed previously, the transformation does not lead to the expected result instead L where the ccr L are leftof reflection of the velocity operator we obtain C<J handed Pauli spin operators [see (1.9.19)]. All other results obtained for the ,
velocity of photon can be extended to the present case.
IX.3.C. Angular Momentum Eigenstates of the Neutrino
The
neutrino
is
a two-component spinor particle and therefore
complete set of spinors to expand
The
its
wave function
special class of eigenfunctions obtained
we need a
into its "multipoles."
by taking
/i
=
I
=
and /2
I in
the expansion formula (IV. 3. 4) are called "spinor spherical harmonics." 2 %h
+
(IXJS.10)
s),
where
s
=
1
1,
and
M=m+
= Y lm (e, The Clebsch-Gordon ji
=
I
in Table IV.2.
*)|>,
coefficients in the
expansion (IX.3.10) are given for
We use the notation
and rewrite (IX.S.10)
in the
form
(IXJS.11)
m
where the only nonvanishing C coefficients on the right side refer to values = m | in the first sum and = m f in the second sum. for which For ji = I Table IV.2 can be reconstructed in the following form.
M
TABLE
M
IX. 1.
The Clebsch-Gordon m,
j
coefficients for ji
=
|
+
= m,
I
C(l%j,
=
mm,M).
J
= l+$
n 20-
J
The Wave Equation of the Neutrino
[IX.3.]
For
=
I
we
IX. 1,
j = |, 3f obtain
1,
= -|, -1, i
283
| values from (IX. 3.11) and from Table
-
=
sn
-
sin 6 e^\l).
For
Z
wave
=
1,
j"
=
I,
M=
J values,
i,
we
obtain the total angular
momentum
functions: =
-^=
Yi(6, 0)1
1}
cos
1)
0|
l
V(f) Yi
(0,
sin
.
<)[!)
e~*+[l),
(4?r)
(1X3.18)
sin
e*|
- 1)
= cos
0| 1).
V(47r)
We now
give some further properties of spinor spherical harmonics. Consider the operator r
where
=
1.
In polar coordinates
cos 6
e~
itf>
=
I '
r it is
given
sin 6
v Y
cos0
The
by
eigenfunctions of o> corresponding to
its
1 i
eigenvalues
~~
/-
+1
and
1
are
284
Symmetries of Massless Particles e
M
cos
K
1}
e~~**
.
sin
= &
*r[S
(IX.3.15)
cos
sin r
On operating by
[Chap. IX]
on the spinor spherical harmonics
= - V(4^
m,
C-CZJj,
2
\Si$),
we obtain
-i
Hence we obtain (IX.3.16)
IX.3.D.
The Neutrino Hamiltonian in Polar Coordinates
We begin with the operator relation (
=
=
r-p
Hence
=
O--JP
(fX.8.18)
where
The Hamiltonian can now be written
= In this form of
T,
angular
C&rpr
~T"
as
T
momentum
is
(IX.8.19)
rC
not separated out. For this pur-
pose consider the operators
J2 = (L
+
2
ifta)
= L2
+ ha-L + f
and 2
(cr-JL)
=
ff&jLiLj
=L z
ha-L.
Then we obtain f
where
2
= J2
+.
P
2 ,
(IX.8JBO)
[IX.3.] It is
The Wave Equation of the Neutrino
IH, a
a*L
+
antleominiites with
=
H: (ix. s
o.
M}
H
To h
+
easy to show that the operator h
2&5
+
obtain an operator that commutes with and depends linearly on cr-L, we introduce a linear operator with the following properties:
E, A
+
cr
EC,
We may now define a linear We can take
= ] = = r] +
0,
(IX.S.&R
0,
(IXJ8M)
0.
(IX.3.26)
Hermitian operator that commutes with the
Hamiltonian.
=
+
(IX.3.27}
where f now commutes with the Hamiltonian
just as its square (IX. 8 Jiff)
f
and
does <7r ,
it is
e (h
The operator f commutes
a constant of the motion.
also
with
for I>r,
fl
= E^ = 2o- f + (
r
+ cr-L)]
o-JLj
eEcrr,
The second term can be
= [
calculated:
[ay,
=
I
,
X
a- (r
r.
L)
e#
+
7 [(""Ofr-lO
I
eft
"
+
Using (IX.S.17) we obtain -
r,
Hence
it
o-L] =
20v(l
+
cr-Ir).
follows that
= Furthermore,
crr
commutes
also
with p r r,
pr
The eigenvalues
of f can
C
(pr
-
(IX.SJ88)
ift/r:
-
= ~"]
The Hamiltonian (IX. 3. Iff) can now be
H=
0.
^\
0.
(IX.8.29)
written as
ar
+
crr ef .
be obtained from the eigenvalue equation
(IX.S.80)
286
Symmetries of Massless Particles
[Chap. IX]
Hence
rw =
^ 2 o*
+
*)
and
Thus the eigenvalues zero.
of f are all positive
The anticommutation property
# -i 6
and
it
shows that
e is
and negative
(IX. 3. 24)
=
integers, excluding
can be written as
_#
(IX. 3. 83)
just the parity operator
itself.
The
general state of the
M
neutrino for a given j and can be expressed as a linear combination of two possible spinor spherical harmonics: !"f >
The wave function of depends on
the parity).
= yjf(r)|^o/j>
+ &W|Sa/iw>.
the neutrino is not
The two
an
(IX.S.S4)
eigenstate of the parity (since
states of polarization,
H
one parallel and the
other antiparallel to the direction of motion, correspond to two distinct particles. thai the
Furthermore, the dependence of
the
Hamiltonian on
the parity implies
angular distribution of the neutrino will, in addition to angular
tum, depend on
its
neutrino emission.
parity; hence there will be
momen-
some directional preference for
CHAPTER X
THE WAVE EQUATION OF PARTICLES OF SPIN PI WITH MASS
X.I. Dirac's Equation In accordance with the four-dimensional irreducible spin representation of the Lorentz group, the wave function of a particle with mass and spin ^ must transform as a four-component spinor. particle with mass and spin can
A
assume two spin
states in opposite directions.
for a free particle of spin | with
of energy:
E = cV(p + m c ). 2
2 2
To
obtain the
mass we must use the
The arguments used
wave equation
relativistic definition
in the derivation of the
neutrino equation can be extended to particles with mass. In this case, however, because of the mass term in the square root, we cannot rationalize it by using Pauli spin matrices. For particles with mass, as follows from the discussion in (FJJ/.5),
we must
use four-dimensional Dirac matrices. Thus
the relativistic Hamiltonian operator of a free particle
H where
a.
and
ft
=
given
by
+ pmc*,
cot-p
(X.LI)
matrices are defined by (VII 1. 5. 10). Hence, the Dirac wave
equation for particles of spin J
is
given by
where the four-dimensional spinor
|^> is
Introducing y matrices (VIII.S.ll), manifestly covariant form as (yp,,
where the 7M
is
satisfy the
-
as defined
we can
wnc)|*>
=
by (VIII. 5. 37). write the Dirac equation in a
0,
(X.1JS)
anticommutation relations (VIII. 5.1$) and 287
The Wave Equation of Particles of Spin |h with Mass
288
~ a
<**-** 5? The Lorentz-co variance
p,
of the
=
^ .,
a
wave equation
=^ .,
a
[Chap. X]
.
(X.I. 8) has already
been
dis-
cussed in Section VIII. 5.
7^
The operator bra
(\f/\
(lM01#)
is 'not Hermitian and therefore the equation for the not be the same as (X.I.,8). We shall now show that the quantity {$!#}, as the fundamental scalar of spinor space, implies that the
will
=
the right spinor, satisfying the same equation as (X.l.S). If we take the Hermitian conjugate of (X.I. 8) and multiply by ft from the right, we
bra
{$j is
obtain the
wave equation <#I(7*P,
where the
differential operators
?>
-
iroc)
=
(X.14)
0,
are assumed to act on the bra {$( to their
left.
We now use the wave equations (X.l.S)
and (X.I. 4} a&d derive a conservation law of probability and current. If we multiply (X.l.S) from the left with (#| and (X.I. 4) from the right with |^}, add the results, and recall that Pn in (X.1.8) acts to the right and p? in (X.I .4) acts to the
left,
we obtain
where the components of the current vector
are to be interpreted in accordance with probability interpretation of
mechanics.
From
we
J4 is
see that
a positive definite quantity and can therefore be interpreted
as a probability density.
Thus the quantity (t\t)d*x
is
quantum
the probability that at time
t
(X.1.8)
a particle or antiparticle can be found within
the volume element d*x with the spin of the particle or antiparticle pointing "up" or "down." The integral of (X.1.8) over the whole of space should be unity;
Note that 4he 'probability density {^1^} is not the same as the scalar quantity ($|^X The -l&tter- need not be positive definite. The result (X.L9) can also be given in a manifestly Lorentz-covariant form by introducing a hyper(surfac$
element by
Dirac's Equation
[X.I.]
and writing (X.I. 9)
289
form
in the
where the integration extends over the hypersurface gral
is
the same regardless of the hypersurface x^
=
=
2*4
constant.
The
inte-
constant, over which
we
integrate.
X.I. A. Particle and Antiparticle States
The
general solutions of the free particle Dirac wave equation are of conwrite equation (X.I. 3) in the form
siderable importance.
We
We
introduce the anti-Hermitian operator
where ^J
=
yp and 7!
=
-1.
(X.1.13)
We further introduce the transformation p = me tan X and write the wave equation (X.l.ll)
in the
(X.l.lJf)
form
wc(l or
me cosX
(cos
X
+ TP sin
Because of (X.I. IS), we can treat the operator y p as the relations cos X
+ 7P sin X
cosX
=
=
^
ex
and
Hence the wave equation becomes \\h\
From
=
the relation
and (X.1.14) we obtain 2
1 ~~
cos X
^
ex
+ e~
= ~~ x
1
me
2 = V(P +
""
me
'
i
V
1
and write
290
The Wave Equation of Particles of Spin ft with Mass
Hence the wave equation
[Chap. X]
(X.1.15} becomes
Let us now introduce the unitary transformations induced by
U=
We
e- (1 /2)x^.
(X.I. 17}
write
and pi in the
wave equation
=
(X.1.16}
W&U = p*
(X.1.19)
and obtain
Hence, using the operator relation 0-(l/
we
get finally the
wave equation
where
Ep = and where we assume that a Fourier transformation in 3-dimensional mo-
mentum
space has been carried out; that
By using the
matrix form of
/3,
is,
equation (X.I. 20) can be
split
up
into
two
two-component spinor wave equations in the form
(X.1.21) ft
! feu) --,!*.>,
where the two-component spinors are defined by
A more
convenient notation
is
to write ~*i~
fll*>
=
%
,
(X.l.
[X.I.]
Dirac's Equation
291
(X.1.28)
where are projection operators.
From
we
see that the lower
two components (< 3 and 4 ) refer to and negative-energy solutions of (X.I.21) can be obtained from the representations (X.I. 22) and (X.I. 23) in the form (X.I. 21}
the negative energy state.
A
<
set of special positive-
(X.1.2S)
where the spinors |lt>, |li>, |-lt>, I-1-J-) constitute an orthonormal of the
set
form
-
These four solutions can be expressed in the form of a 4 -/ft)JV
X
(X.1.26)
4 matrix
o e -a/VEpt
000
o
eW*** e c*A)
where the columns represent the solutions. The most general solutions can now be obtained from (X.1.18) in the form of a 4 X 4 matrix, [*]
The unitary matrix
#[*]-
(X.1JB8)
U can be expressed in a simple form. On writing sin ;in
we
=
|X
=
e
=
sin
L
1 | tan"
(
=
)
\mcyj
sin
0,
mc
find that sin 6
=
1/ 2
cos ^
=
+ me
,2-11/2
292
The Wave Equation of Particles of Spin ih with Mass
[Chap. X]
Hence
where
J7 satisfies
TJ\
+ _L
/*?
=i
(X.1.29)
the equation
HU The matrix form
of 17
=
(X.I 80)
is
-
Ep + mc2 Ep + me
U=N
2
-cpcp*
Ep + me
cp-
(X.1.81)
2
Ep
cps
where
N
= me 2 )]
In matrix form the Hamiltonian
is
H= U
We
can show that the
first
[Ep
= cV(p
solutions of the Dirac equation.
2
+mc 2
2
)]
are the negative-energy tions, as represented
two columns
Ep =
[
of
cV(p
2
by the four columns
system. In the rest frame of the particle (p U reduce to the solutions (X.I. 26).
Next consider a case
we
particle
having
its
correspond to positive-energy
+ m c )] 2 2
The
of U, constitute
=
an orthonormal
0) the solutions
momentum p
represented
by
along the Z-axis. In this
find that the spin operator
s =
two columns
last
solutions. All four solu-
,
a
v (X.1.8S)
assumes the value JR with respect to states represented by the first and third columns of U and the value %h with respect to states represented by the second and fourth columns of U. Hence we see that the four solutions of the Dirac equation refer to two positive energy states with "spin
down"
two negative energy "spin These states will be represented by to
'E,p
\E+,
=
+ me
N
states with "spin
'
2
* -*> = cp*
up" and and up" "spin down."
Ep + me
2
(X.1.84)
Dirac's Equation
[X.I.]
for positive eneigy states
i)
- A
293
and
cp*
Ep
(X.1.SS)
me*
for negative energy states.
The
operator of the spinor space spanned
operator
/
=
by (X .1.34) and (X.I.35} and therefore we can form a unit
states represented
constitute a complete orthonormal set
these four spinors.
by
Thus the unit
is
\S+ , JX^-B
il
+
\E+,
-4Xf,
-II
+ 1^ iX^ + \E~, -JXJB-, -J|. *l
(X.1.86)
Any
state represented
by a spinor
of four states i*>
=
\E+ ,
=
a+\E+, |)
)(E+ ,
where the expansion
m+
+ 6+|jE+
[^)
-
\E+ ,
,
can now be expanded as a superposition
~i)
+ a-\E-, i) + &-|JSL,
coefficients are probability
-|),
(X.I.37}
amplitudes for the particle
having positive or negative energy with spin up or spin down. a solution of Dirac's equation, then we can express it in the form
in the state [^} If [^} is
W)
=
f
+
H,
(X.LS8)
where |^+). and |^_) are two two-component spinors formed from the upper and lower two components of |^}. Thus from the Dirac equation (X.1.8)
we
obtain (p 4
where cp*
=
E.
From
(X.L39)
the second equation i
r
\
C <*'P
we i
,
obtain \
mc
E we can look for the nonrelativistic limit of (X.1.40). limit we have E = me and p = mv, therefore
For the positive energy In the nonrelativistic
2
This result shows that in the nonrelativistic sense we |^_) as large
and small components,
respectively,
where
may
regard 1^+} and
|^+) refers to positive
and [^_) refers to negative energies; [^-} is of the order of v/c (in the nonrelativistic limit) compared to |^+). It is interesting to note that in the extreme relativistic approximation,
energies
294
The Wave Equation of Particles of Spin fh with Mass
that
is,
^cp, where mc/p s
for
solutions
[Chap. X]
and (X.I. 35)
(X.1.34)
become "
p
p pLP+J
'-P-
-p+
show that
It is easy to
in the
extreme
relativistic states
practically parallel or antiparallel to its
is
the spin of the particle
momentum,*
X.2. Charge Conjugation The concept of charge conjugation as an important symmetry operation can be used to establish a relation between negative-energy solutions and removes the association of antiFor the sake of convenience we shall particles with negative energy use Dirac's equation describing the motion of a particle in an external electromagnetic field. Such an equation follows from replacing the momenta P by Pft (e/c)Afi and hence writing Dirac's equation for particles interan with external field A^ in the form acting
antiparticle states with positive energy. It states.
0.
(XJS.1)
A
displacement of a particle in time multiplies its wave function by the factor e~ <*/*>#* and the complex conjugation operation changes the sign of the energy. It
is
natural to expect a relation between charge conjugation that
replacing the charge
Theorem.
e
by
6,
and sign
is,
of energy.
Charge-conjugate states oj a particle of spin \ are related ty
an
antiunitary transformation.
be the wave function of a particle satisfying the wave equation obtained from the wave equation for \\[/} by replacing the electric charge e Let
by *
and
e
\\l/c )
Thus from
The
states (X.1.4&)
zero
the neutrino states,
we write
have resulted by setting m = in the states (XJ.84) mass here does not imply that the states (X.l*4&) are the same as since the neutrino is not described by a Dirac equation where m = 0.
We note that the (X.1.&5).
(X.8.1)
[X.2.]
We
Charge Conjugation
shall write
It
in the
295
form
+^A )
The complex conjugation
+
7
4
[74 (?4
(jp
-
^)
imcj
I&>
=
0.
(ZAS)
of (X.2.1) reads as -
y*
imc]
|**>
=
0,
where
r* and according to the
=
definition of the
^ ~ From
(VIII. 2 .16}
it
y we may write
LiaL
J
follows that
/~Wj J
f-Haf
oJL-zV where / where 6
OJL/
=
icr*
is
called the charge conjugation matrix
01
Hence
e = ro /i If or
=
OJ
is of
the form
*
r Lioi
e =
and
O
^72.
It has the properties of (a) unitarity,
ef e (b)
=
i,
antisymmetry,
e = -e, (c)
e 2 = -i, (d)
e-iT/I e
= ~fM
.
Thus, using the expression (X.8.4) for y* in the complex conjugate equation,
we obtain p4
+ -A +
er e
.P +
A+ me
|^*>
=
o.
The Wave Equation of Particles of Spin |n with Mass
296
Multiplying through with
[74 (pi
6 and
+ ~ W) + Y
using 6746 -
(p
=
74,
we
-
+ -C
A)
tmc]
[Chap. X]
get !**>
-
0,
from charge conjugate equation (X&.S) in the positive sign of the second term. Multiplication with ft and use of 0e|^*} in place of
which
differs
leads to the equation ~
A4 - 7
+ *^) -
(p
)
= nc]0el**>
or
= Comparing (XS.11) and are related by
we
(X.2.SS),
find that the charge conjugate states
where the "charge-reversal operator" 3 e
=
5e
and
it is, like
0.
is
given
by
zt^eU,
(X.8.18)
the time-reversal operator, an antiunitary operator;
it satisfies
the relation
The
adjoint charge conjugate state ($ c
= |
($t \$
is,
as follows
from
by
given
<&I
=
where
By operating with \E+ , |>
Thus
5e
=
on the negative energy states (X.L 85} we obtain
3 e |^_, -i>,
|?+J -i)
= -3e |^_,
|>.
3 e transforms a particle of negative energy with spin state
particle of positive energy with spin state up,
and
down
into
a
3 e transforms a particle
of negative energy with spin state up into a particle of positive energy with spin state down. Hence negative energy states, actually, correspond to antiparticles of positive energy. The change of the sign of energy is expected also,
because 3 e anticommutes with the Hamiltonian operator of a free
particle:
[3
The
reversal of the spin
is
fi
,
#]+ =
also a
0.
(X.8.18)
consequence of the anticommutation
relations:
De, OjJ + Furthermore,
if
the
momentum
=
0.
eigenvalue of the state
(XJ1.19) |^> is
p, then the
The Four-Component Neutrino Wave Equation
[X.3.]
momentum momentum
eigenvalue of the state
p, since 3 e anticommutes with the
$ } is e
operator:
COJS0)
t3.,p3+-0. The
297
anticommutes also with the time-rever-
"charge-reversal operator"
sal operator of Dirac theory. The time-reversal operator 3, as defined 1. 5. 48), together with 5 e yields the anticommutation relation:
by
(VII
C3
31+
=
(X3M)
0.
This result implies that particles and antiparticles can be described as moving in commutes with the total reflection "opposite directions" in time. However, 5* of the space
and time symmetry operation:
=
75?,
r
IX The
X.3.
current defined
=
(X.I,ff) remains
by
o.
(X.*.**)
unchanged under the
3* operation:
The Four-Component Neutrino Wave Equation
A description
of the masslessness of the
photon by a guage-invariant
field
new
gauge-invariance should constitute an important guide to discover of a massless fermion field. A detailed discussion the for description principles was of the relation of the complex orthogonal group to the neutrino field
The approach
given in Section VIIL4.
and
also
its
was based on a three-dimen-
A
four-dimensional representation of the complex relation to Maxwell's equations have been discussed by Moses*
sional representation.
group and
there
by Lomont.f
We
shall not consider here
a direct group-theoretical
base our arguments on simple isoapproach but Lorentz group. Such correspondthe morphisms of various representations of ences between representations of spin i and spin 1 have already been estabwill find
it
sufficient to
lished in the previous chapters.
can begin with the four-dimensional form of Maxwell's equations, where it is not necessary to separate out the transversality condition of the 8. 8) of electromagnetic waves. Thus we shall use the wave equation (VIII.
We
problem 5 in Section VIIL6.C.,
*
H. E. Moses, Nuovo Cimento, Suppl 1 (1958), Lomont, Pkys. Rev., Ill (1958), 1710.
t J. S.
1.
298
The Wave Equation of Particles of Spin |n with Mass
where the
4X4 matrices A A4 = A2 =
The
J4
are given
M
by
= A = A!
,
If (3i)
+ Of
4X4 matrices M^ are
-31(23)
defined in
(12)
ket
[17}
(X.3.2)
(i4),
(34)-
they are generators of
(1. 10.23)]
The
+ iM
M + Of
3
<24),
finitesimal Lorentz transformations.
[Chap. X]
is
the
wave function
in-
of the
photon as a four-component complex vector,
toEquation (X.8.1), if written explicitly in terms of real and imaginary parts of \n) as electric and magnetic vectors, respectively, leads to Maxwell's equations for free fields, including the transversality condition V-TTJ
=
0.
we must
In order to obtain an equation for the massless use spin f matrices and o> matrices of spin J reprethe correspondence between sentation. There is no other justification needed, since such a correspondence field of
M^
already exists between the Lorentz matrices (1.10.87}, (1.10.28) and the transformation matrices (VIII. 5'.56), (VIII. 5. 57). Therefore, to the above A matrices, in spin \ representation, there will correspond the matrices
Ti
= =
T3
=
T4
J4
,
-io-23
J
where the a commute with
+ i
0-34
=
-
+
(X.S.3)
75),
0-34(1
[See the definition of spin matrices in
The neutrino wave equation follows from the wave photon equation (X.8.1), replacing the A by the T and [77} by a fourcomponent spinor |^). Hence we obtain (VIII. 5.23) and (VIII. 5. $4).]
+
where i(l be written as
75) is
>
It is
now easy
to the
|
a projection operator; therefore the wave equation can
(1
+ *Y5)M = -o-V[i(l + ns)M.
to see that this four-component
wave equation
two-component wave equation (IX. 3.1). From the form
tion operator,
(X.34) is
equivalent
of the projec-
The Four-Component Neutrino Wave Equation
[X.3.] it
299
follows that
which can be written as
4(1
We thus see that the wave equation tion (IX. 3.1). There
is
some virtue
(X.3.)
is
reducible to the
wave equa-
in the above derivation of the neutrino
equation. Most importantly, it leads to a new type of gauge-invariance principle for a massless fermion field. The masslessness of field corresponding to the wave equation is guaranteed from the method of derivation. The equation
is left
unchanged with respect to a gauge transformation
W) = where X
is
some
fixed
e*iys\j),
(X.S.5)
number. The Dirac equation, for example,
is
invariant
under the gauge transformation (X.3.5), provided the mass m = 0. (The reader should show it for himself.) The above gauge-invariance requirement
was the
two-component theory proposed by Lee and Yang, Salam, and Landau. However, the derivation given here gives a clear insight into why a gauge transformation of the type (X.8.5) associates a rigorous masslessness with the neutrino field. starting point of the
analogy with electromagnetic interactions we can say that the neutrino will interact with matter in a 75 gauge-invariant way; that is, the neutrino
By
+
occur only in the combination 4(1 iys)\v) and its Hermitian conis the basis of vector and pseudo vector interaction in weak interaction theory. As we have seen, the projection operator |(1 iy&) strikes
field will
jugate. This
+
out two of the four components and hence also one of the two spin states. Physically this means that the neutrino is produced or destroyed in one spin state only,
and
this, in turn, is
the basis of the breakdown of parity conserva-
tion.
X.3.A. Problems 1.
4(1
Show iy 5 )
that the normalized eigenvectors of the projection operator constitute a complete set. Consider a wave equation of the form .
.
lyp,
- mi(l -
and expand the wave function
|^}
*y B
)M
=
0,
with respect to the complete set
(X.S.6) of eigen-
300
The Wave Equation of Particles of Spin |h with Mass
[Chap. X]
functions of f (1 iy$). By interpreting the four eigenstates obtained in this as mass and massless states of particles and antiparticles, discuss the solutions of (X.8.6).
Compare
the results with solutions of the Dirac equation
with mass and also with the solutions of the neutrino equation.* 2.
Show that the free particle Dirac wave equation
is
invariant with respect
to gauge transformations of the form |^>
where S
is
a
=
e
real function of space
(X.3.7)
and time and
satisfies
the relativistic
Hamilton-Jacobi equation
3.
Show
+Mc = 2
(i)'
2
0.
Q6 =
that the eigenvalues of the operator
i(5
+ 3z,)
are
+1,
-1, andO.
X.4. Physical Interpretation of the
Dirac
Wave Equation
invariance property, the Dirac equation is gaugewith the gauge invariance of the classical relativistic By analogy Hamilton-Jacobi equation (III. 2.10) [see (III.2.11)], the gauge transforma-
Besides
Its relativistic
invariant.
tion for the Dirac equation
where
G is
is
defined
by
a function of space and time and
is
restricted as a solution of the
equation
The
restriction (X.4.S) is
a consequence of the so-called Lorentz condition,
gf The wave equation [(7%, is
-
(X44}
0.
- wic)M =
.
obviously invariant under the gauge transformations (5T4-0
where
*
B. Kuriinoglu, Nuovo Cimento, 15 (1960), 729.
(Z4.5)
Physical Interpretation of the Di-ae
TX.4.J
Wave Equation
301
However, the preservation of gauge invariance in specific calculations has met with some difficulties. The requirement of gauge invariance is, of course, verv important, since it implies a definite link between the two unobservabies
A
and
ty)
of electromagnetic field
and matter waves. An important example
of the consequence of manifestly gauge-invariant calculation has been dis-
cussed by Schwinger* in
quantum
field
The gauge group should be
theory.
to the Lorentz group and hence must play a expected to have some relation of dynamical systems. Its signifiinteractions the in role determining basic at present. A very interis understood than level a much deeper cance lies at of gauge invariance connecting the Lorentz esting approach to the problem has been proposed by Salam and Ward.f It leads to a genand
gauge groups
and
eralization of the electromagnetic field
description of
weak and electromagnetic
to
some
possibility of a unified
interactions.
Further properties of the Dirac equation can be seen by comparing it with the Klein-Gordon equation, which for free fields follows from the discus2 sion of the transformation function of a relativistic particle Fsee problem (V.BJSS)'].
In the presence of
fields it is
given
by
a scalar (or pseudoscalar) wave function. Let us the easily derivable operational statements
where
\
is
where the electromagnetic
We
also
have
field
tensor /^
+ 7*7")
is
defined
first
write
down
by
W+
Multiplying the Dirac equation on the obtain
left
by
the operator (7^^
=
+ imc), we
0.
This equation differs from the Klein-Gordon equation (2T4-7) in the term
if *
t
Schwinger, Phys. Rev., 82 (1951), 664. A. Salam and J. C. Ward, Nuovo Cimento, 11 (1959), 568.
J.
The Wave Equation of Particles of Spin Jh with Mass
302
The
physical meaning of the two terms in (X.4-1%)
wu
^
[Chap. X]
^ e discussed
in the
following section.
X.4.A.
The Low-En ergy Limit of the Dirac Equation
The study
form of the Dirac equation
of the low-energy
is
based on the
assumption that the kinetic energy of the particle is small compared to its rest energy. This is equivalent to the expansion of the interaction energy and the wave function into a series in powers of the parameter p/mc. There are various methods* and almost
of
all
them lead
to the
same
results within
the range of energies that are of practical interest. The simplest way to study the low-energy approximation to Dirac's equation
is
to extend the free particle solutions (X.I. 17) to the general case of
interaction with
an external
field.
We
must
first
express the unitary operator
(X.I. 17) in a suitable form, one that allows the replacement of the
vector p
^
T jv
by
Tt
and
p*
by
= = 7^ ,tan" (V\ -^-) \mcj i 1
The operator \y p
7r4 .
Tp
in (X.I. 17) can
1 fP\"^ fjL\ +c()
rfP\ (--)
l ~ ( 3 \rncj
L \mcj
5
5 \rncj
momentum
be written as *
1 J
...-= ^'tan-
....
1
Using the definition of jp as given by (X.I. 12), we obtain XT-D
Hence the unitary operator
=
i
tan" 1
me
/
(X.I. 17) becomes [/"free
=
6Xp
(ijS ),
where \
me
/
a Hermitian operator. The form (X.4-14) of the generator of the unitary operator does not involve the magnitude of the momentum operator, as is
only the components of *
p
appear in
/So.
The corresponding unitary operator
G. Breit, Phys. Rev., 51 (1937), 248; Phys. Rev., 51 (1937), 778; Phys. Rev., 53 (1937),
153. "
L. L. Foldy and S. A. Wouthuysen, Phys. Rev., 78 (1950), 29. G. Breit and R. M. Thaler, Phys. Rev., 89 (1953), 1182. B. Kurunoglu, Phys. Rev., 101 (1956), 1419.
Physical Interpretation of the Dirac
[X.4.]
for a particle interacting with
to
an external
field
Wave Equation
303
can be inferred from (X.4.14)
have the form
U=
exp
(X4.15)
(zS),
where
The new wave
function corresponding to (X.I. 18} !<>
If
we
write the
we can in the
wave equation
=
in the
is
-
form
derive the corresponding equation (X.I. 15) in the case of interaction
form
where we use the transformation (X4-17) and the operator property
To compare it is
the approximation to be made with the nonrelativistie theory convenient to transform the equation (X.4-18) into two two-component
wave equations, where the upper and lower components Thus,
let
of |<) are separated.
us write equation (X4-1&) i& the form
where
M = 2mcj3(U* R= L= operator
U
R, and L are Hermitian. 1 changes to C7 "; therefore, while
operator
L
changes
The operators M,
M
,
JK,
**
of sign of
m the the
In order to see more explicitly the structure of for small p/mc, we can write
and L
=
Under a change
M and R remain unchanged,
its sign.
+ c^,
53
-
~
Hence 2!
L U =
ifiTA ^L71^,
St J
~ o i
l>4, -si,
304
The
The Wave Equation of Particles of Spin |h with Mass generator
S
is
defined
[Chap. X]
by
S= It is
now
convenient to
up equation
split
tions,
M\4>+)
=
Hence
where
Bl$+>
(M (M !<+}
and
M
+ Li*-),
R)\4>+)
fi)|^>
- L(M - L(M -
|<_) are defined, as in
(X.4-&0) hito
\4>-)
R)-
=
R\
= =
l
L\
B)-lL|->
*
wo coupled
equa-
+ L\+).
0,
(X4.26)
0,
(Z.4.07)
(X.Lgty and (X.1.88), according to
Because of the unitary property of U the expectation values of the observables in both the old and the new representations are the same, provided the definition of an observable
new
representation
in the old representation
by
=
J'
Thus the
replaced in the
by
For example, the probability density given
is
<$|<) is conserved
(4>\W*V\$.
(Z4-SO)
physical interpretation of the Dirac theory
of a four-component
wave
with a current density
function.
is still
based on the use
For the exact theory, the functions
by themselves cannot be regarded as different wave functions, one describing the positive-energy and the other the negative-energy particle. The only probability that is conserved refers to the state |<+)
and
!<-}
\4>)
\4>
+)
+
|4>->.
This means that the possibility of reducing the Dirac equation to two-component equations does not prevent transitions to negative energy states.
The
physical meaning of the functions [<+} and |0_)
of
+) and
|<
function
|0_) is
is
now
clear: (a) the
sum
a probability amplitude of the same kind as Dirac's wave
[^}; (b) in
the nonrelativistic limit the function |<_) refers to small
components and the function
|<
+)
is
a probability amplitude.
We may now carry out the required approximation for the low-energy limit. We shall retain only the terms proportional to l/m c We shall need 2
2
.
the following easily established relations
:
(X.4-8%)
[X.4.]
Physical Interpretation of the Dirac
Wave Equation
305
(X.4.S4)
\M
=
2mc(C7
-\
Hence equation (X.4-%6)
2mc
=
2 ,
2mc
yields
'
'
2mc 2
8m
2
c3
or rt*fi
,_
I
JL\
j
At
~LJ I JL
\
/
V
i
c)fy\
(J^.j^.O/J
JLL\
where
TJ
___
2m
+ 8^O v O//t
The term
(eh/2mc)cr-3C
is
-
'
2
^ O'/t-
ff
-CV
X
)
-
me*.
C/
an additional potential energy and
it
arises
from
the electron's magnetic moment,
<***>
*--^b--
2 2 For a static field we set -4 = 0. In this case the term X TT) (e7z,/4m c )a- ( 2 2 becomes (eft/4^ c )cr-(V^l4 X p), where the electric potential A is a func-
tion of r alone, so
_
N
6^
6
1
1 dAt r dr
,
.
f (*'V
fY (X440) .
the additional energy arising from "spin-orbit coupling." The remaining terms are small and need not be discussed in this connection. is
The Wave Equation of Particles of Spin fn with Mass
306
[Chap. X]
X.4.B. Problems 1. Find the position, momentum, and angular momentum operators in a representation where the energy of a free Dirac particle is given by *
Show
+mc 2
).
that the group velocity of a particle of spin J with mass
m The velocity velocity of 2.
2
in the original representation is
m is given
by
2 2
c )
Under what conditions
COL.
is
the
particle independent of a representation?
any Prove that the total angular
J= L of a Dirac particle
with mass
with the conservation of J'
The unitary operator
U is
is
momentum
+ 4ft[
01
a constant of the motion. Compare this result each term is conserved by itself.
= WJU, where defined
by
(X.4-15).
CHAPTER
XI
QUANTIZATION OF MASSLESS FIELDS
XI. 1.
The Basic Formalism
The explanation
of Quantization
of the duality properties of
matter as waves and particles
not completely achieved by the quantum theory we have discussed so far. We must ascribe particle features to fields; particles must be representable in is
terms of some kind of
fields.
amplitudes of the wave operators.
The
This
is
done by
"field quantization/
7
where the
ordinary space and time are replaced by quantization has first been discussed and applied
fields in
idea of field
by Jordan and Klein* and Jordan and Wigner.f We shall be concerned in this book only with the quantization of massless fields. The quantization of fields with mass is outside its scope. We shall not discuss the
phenomena that fall naturally within the territory of quantum electrodynamics. But quantization of the free electromagnetic field and also of the neutrino field is of basic
importance for further understanding of the
and philosophical foundations of quantum theory .J (See also II and IV.) Chapters with an outline of the Lagrangian formulation of classical fields. We begin We assume that the Lagrangian of a classical field is a function of field variables for example electromagnetic potentials or electric and magnetic fields and their first-order partial derivatives. Thus a Lagrangian is of the form statistical
where a and
/x
run through
"Langrangian density"
<
1
to
4.
The
action function
S corresponding to the
is
* P.
Jordan and 0. Klein, Z. Phys. 45 (1927), 751. fP. Jordan and E. Wigner, Z. Phys. 47 (1928), 631. t A good account of modern quantum electrodynamics can be found dynamics, Dover, New York, 1958; edited by J Schwinger.
in
Quantum
Electro-
307
308
Quantization of Massless Fields
where
is
a four-dimensional region of integration. ??<* of the form
We
[Chap. XI]
envisage variations
of field variables
such that the variations stationary values of
S
dr} Q
vanish on the surface bounding the region 0.
BS
The boundary
The
for such variations satisfy the equation
=
0.
(XI. Ug)
consists of physically indeare that that separated by spacelike points pendent space-time points; is, intervals. The totality of such points constitute a spacelike surface cr. may look upon as spanned by two spacelike surfaces 01 and 0-2 at times t\ and 2 . of the four-dimensional region
We
Thus
(XI.1.1) can be written as
*>-
(')**
(XI. US)
F entails two types of variations: each point by dfj a) and (b) altering the (a) region of integration by a displacement to of the points on the boundary The change 8F
of
changing the
an arbitrary function
field variables at
4
surfaces.
Hence 88
=
I
Zdxxd^
Jcri
~
f
J
(XI. 1.4)
where 5
e
=
8wa
+
orja
(XI.1.5)
According to Gauss' theorem we have d
d 4x
I
fa dx"
Hence the variation
8S=
3
=
of the action integral
-JL can be written as
d
f fa
(xi. ue)
[XI. I.]
The Basic Formalism of Quantization
From Hamilton's
principle (XI. 1.2) the equations of
309
motion follow as
and
+
da*.
J 02
(XI. 1.8)
Equations (XI. 1.7) are the field equations and (XI'.1.8) lead to the laws of conservation.* We shall be concerned only with the field equations. Since the
Lagrangian
is
a Lorentz-invariant quantity, the corresponding
field
equations
are, of course, Lorentz-covariant.
In order to establish a correspondence between field and particle we must try to make a transition from a continuously infinite degree of freedom of fields to a discrete infinite degree of freedom. This can be done, at a fixed
by decomposing the three dimensional coordinate space into small volume 8V M for s = 1, 2, The corresponding field variable in each volume 5F W will be designated by i?(rt t). At a given time t the time
t,
cells of
-
.
,
,
Lagrangian can be defined by
where j& w refers to the value of & in the sth ence with the Hamiltonian dynamics by (fa
i~&
where
rj a
=
=
*?a(r a ,
*\ -iS
-\
d&
""
/
j;\
/
-t\
d7] a (r 8, t)
dq a /dL defined
cells
7T
by
we can
p*a
=
write
= |^ drj a
as the canonically conjugate variable to
J.
*v
a7? a (r s , t)
In the limit of zero volume of the
*
We can set up a correspond-
0,
The Hamiltonian can now be
and we
cell.
7ra (r
also obtain
Schwinger, Phys. Rev., 82 (1951), 914.
7?
a , so
(XI.1.11)/
(XI. 1.10) yields
310
Quantization of Massless Fields
In the limit 67 (s)
[Chap. XI]
we have
*~
=
H(t)
where the Hamiltonian density H(r, H(r,
H(r,
f
t)
d*x,
(XI. US)
C.
(XI. US)
is
t)
=
f)
-
irti*
Quantization of a
field
consists of regarding the canonically conjugate
and wa as
field
operators and deriving, from analogy with particle rules. From the above definitions
variables
T?
dynamics, the corresponding commutation
and quantum postulates field
(III. 4.2) it follows that the
commutation
rules for
operators are
tfefo
0, w(ri,
01 =
fcr (r,,0,i<*(rj,01
0,
=
0, <>
fc7(r.,
t), ir0(rj,
01 =
~~'
ih8<#
Since the time development of the dynamical system is not specified, we can work with Schrodinger's picture, where the operators are time-independent.
In the limit BV W
so the
we have
*
commutation
relations are given
by*
=
'
'
=
0, *
The Hamiltonian (XI. US) can be employed motion for the field operators and ?ra
tions of
??
= ik^jf
,, H],
to write Heisenberg's equa-
:
i%
^=
IT,,
HJ.
(Z7.1J5)
using the commutation relations (XI. LI 4), the equations of motion (XI. 1.1 5) can be written as
By
where the functional derivatives at a fixed time are defined as
The
state of a quantized field
Schrodinger's equation,
is
described
(XI.U7)
*')
by a
state vector
|1P)
satisfying
The Photon
[XL2/
^
where
Field
describes the state of a system of
and so
photons, neutrinos,
The time-dependent
any number
of field
quanta
on.
field
operators are defined
*[';
where
311
: J5
by
(* j*>
% w*2w"'
H does not contain time explicitly. Hence (XI. 1.20)
In this way we have established a complete analogy between Hamiltonian particle
dynamics and
field
quantization.
XL2. The Photon Field The square
Hamiltonian operator
of the
of the free
Dirac particle
is
of the
same form as the classical expression for the square of the energy. For example, = ca -p + pmc* are c 2j> 2 and c 2p 2 + m 2c 2 the squares of H v = c
H
,
the Hamiltonian operator of the photon. Thus, in the expression
the second term on the right (the longitudinal part of the energy) can be eliminated without using the transversality condition. The first term, c 2^ 2 is ,
the square of the photon energy and
+
it
c*pp
can be expressed as
=
(cK-p
+
Hence we can choose
H ^ e
cK
p+
~
PP
(Xl.Jt.f)
as the "effective Hamiltonian operator" of a single-photon theory, discussed previously (see Chapter II). It operates on 77) in the same way as does cK-p
and
it
leads to the
Chapter
II) is
now
same
results.
replaced
det (H.
However, the determinantal equation
(see
by
-E) = -(E-
cpY(E
+ cp} =
0.
(XI. 2.2}
The root E = cp because of the transversality condition, cannot occur, The repeated root E = cp corresponds to the energy of the two states of y
polarization.
The Lagrangian density
of the electromagnetic field can
now be
given by
Quantization of Massless Fields
312
= summed
(17!
I
(ih
for repeated indices,
-
H^ j
i,
=
[r?)
=
ihfai
1, 2, 3.
The
-
[Chap. XI]
(XI .2.8)
v\(He}iW,
field is
quantized according
to the commutation rules D?.-(r,
0,
vy(r',
01 =
Dn(r
0,
;
0, *v(r',
03 =
0,
where
T< so
we may
=
|$ OTJi 7
field is
The
(XI. 2.5-)
also write
&<(r, 0, nKr
From
= iM,
definition
(XI.L1S)
given by
it
,
03 =
-
C^/Aff)
r').
follows that the Hamiltonian density of the
H
=
state [*} of the electromagnetic field
equation for the quantized
5,y 5(r
is
a solution of the Schrodinger
field,
where
The (XI.
equation V-iy This difficulty
field
.6).
by a subsidiary
V-iy
= is
is not consistent with the commutation rules removed by replacing the operator equation
condition,
(XI. 2.10)
0,
which
is
From
a further restriction on the state vector \&) of the
field.
the Hamiltonian (Xl&ff) and the commutation rules
we
obtain
.*J J) as the quantized photon wave equation. In order to appreciate the significance of the above quantization process
convenient to use spatial periodic boundary conditions, as in (11. 1.34), and expand the field operators as a superposition of states of the form it is
= n
x
(XI.ft.lSl)
n
X
The Photon
[XI.2.]
Field
The
and
1, corresponding to two states of polarizaare specified by the vectors e Af and they are polarization directions
where X assumes the values tion.
313
1
(
of the type jl)
and
1}
j
used in (11.1.45)- They satisfy the relations
=
cg><$>
The
8 ih
e$?e#*
=
8^.
(XI.S.1S)
functions
=
/&> are eigenfunetions of
H
e,
ttnefi?
given by
(#e )ufxf =
Enf,
(XL2.14)
and the functions u n (r) are normalized according to
fv ul(r)u n >(r) To a\ n
=
d*x
5 nn >.
commutation relations (XI. 2,4} must obey the commutation rules
satisfy the
and
a\ n
[&\n,
t
-
ttxvj
=
field is
the above definitions
it
or (XI.2.6) the operators
O nn /dxx',
where we used the completeness relation
From
(XI. 2.15)
of the eigenfunetions
follows that the Hamiltonian (XI\.9) of the
given by
H^^^EJf*, X
(XL2.18)
9
where the occupation number operator N\ s
is
defined
by
Nx8 = aUx,
(Z/JBJP)
and
E = 8
The
ho> 8 .
are integral numbers nx so that the total energy sum of the energies of two independent integers corresponding respectively to the numbers of light quanta with
eigenvalues of
N\8
of the field is given as a
and n__i s right and left Thus
n-is
,
circular polarizations (or
two
possibilities of spin orientation).
The
stationary states of the field satisfy the Schrodinger equation of the quantized field, as
MT
Jiu s
ns
|T&>
J
=
(XI. 9.21)
E\n),
where
n^
(XI. 2.2%)
314
Quantization of Massless Fields
We as a
have field
in this
[Chap. XI]
obtained a description of the electromagnetic field Each oscillator has an
way
of radiation oscillators [see (VII. 141)}-
energy which is an integral multiple of hus can be constructed in the form \n)
= mm
E
.
A
.
general solution of (XI. 2. 21)
-to)|n*>
'
'
K>
*
'
(XI.2.23)
-
where lCmm---n.---l 2 is the probability of finding n x photons of type photons of type 2, and so on. Each \n ) satisfies
H \n = 8
s}
H = N hv
njto.\n 9 ),
s
s
N
s,
The total momentum operator of the
PT =
i
[
C J
H
(
e \n)
=
s
N^ + N
field [see (II. 2.6)}
d*x
=-
C J[
l8 .
1,
n%
(XI. 2.2$
can be expressed as
v\K?ij, d*x.
(XI.2.25)
It yields
C
or, in
eJ"
X
XX'
S
vector notation, X
where we used the relations
K/ = 1
e^ X where
fe s is
=
eg>
-icrtV
(XI.2.27)
,
(Z/.*.S)
tfixx'fe,
the unit vector in the direction of propagation
kg =
ks and ,
i
(XI. 2.29)
co,fes .
c
The momentum operator P
operates
on the stationary
states according to
(XI.2.30)
},
where
ps = Hence the photon as a
^nA-
particle of zero
p}
mass
- iE = C
satisfies
0.
(XI. 2.81} the relation (XI.2.S2)
The electromagnetic field quantized according to the commutation rules (XI. 24) reduces, for h = 0, to the classical field theory and the particle The occupation number operator N\ s for fixed X and s has integral eigenvalues, so the number of quanta attached to each radiation t>scillator is not limited. Each radiation oscillator contains any number of features disappear.
The Photon
[XI.2.]
Field
same
identical particles in the
Einstein
The
statistics.
315
Hence we infer that photons obey Bosemakes it possible for the quantized field
state.
latter fact
to reduce to the classical field for
=
fi
0.
Only Bose-Einstein
will lead to Planck's distribution law. (See
quanta
problem
statistics
for light
6 of Chapter VII,
p. 207.)
Finally,
and
we
note the following important properties of the operators as
al,
~
a ....
al\n)
= v (n -f- l)JTii n / = vns \ni ns n ) ns /
*
s \ni
a<s\'ri)
s
*
-f-
s
1,
},
.
-
aijfti
-
-
8
1,
},
which follow from (VI I.I. 55) and (VI'1. 1.56). The operators a* and al are called "creation and annihilation operators," respectively. The operator as
number
increases the occupation
decreases
it
same
in the
state
in the sth state
by
by one and the operator al number states
one. All other occupation
are left unchanged.
XI.2.A.
The Propagator
of the Free
The quantization procedures used
in the
Photon
above have been carried out at a
fixed instant of time in terms of time-independent operators.
opment
of the system requires the use of time-dependent operators of the field variables, for
operators at two different instants.
field
dependent commutation
rules,
we
lli(r,
F#
is
use the expansions t), rt](r',
t>)1
=
devel-
and time-
which we need to know
dependent commutators the commutators of
where
The time
F,y
To
(XL 2.12)
obtain timein
,
the commutator, and calculate the "propagator of the free pho-
ton." Substituting from (XI. 2. 12)
nn f
we
get
XX'
(Xl&JSft
),
n
where, because of the invariance under translation, coordinate differences. If
we
we now set t =
f
t
in
-
13
depends only on the
(XL 2.34) and use the commutation relation (XI. 2.6),
find that
ZX(rK(r') = n
Hence
F
S(r
-
r').
316
Quantization of Massless Fields
From
(XI. 8.1 4)
we can
write (XI. 8. 84) in the form
F =
[e-(/ft)fr-')H.].j5 ( r
fj
The
operator exp
e~ W(t-t<)H
t
=
_ p?(^-O1 h 1
cos
_
r /).
(XI. 8.85)
t')H e ] can be cast as
(i/h)(t
[
[Chap. XI]
[_
^
e
sin
cp
p^-^)1 * J L
(XI .2.36)
where
Replacing the
5
function
by
its
Fourier representation
and using (X1.2. 80), we can write (XI. 2. 85)
in the
form
sn The integral in (XI .87) can be put into a manifestly covariant form. r 2 ) on the For that we introduce a function D(x) such that it equals 5(xl 2 r ) on the past part future part of the light cone (x > 0) and to 8(xl of the light cone (x*
<
Hence
0).
D(x)
The Fourier transform
=
=
has the form
-
[d(x,
of the
4 f D&ePv* d x
it
-
r)
8(X4
+ r)].
(XIJ6.88)
D function is
I j
r
[e*
e
-^] e -^r ^x
~ e-^ ]e~^ r
r
cos
r sin
fl
d8 d$ dr
Hence
D(x
-
x r)
=
g^
and
F = tf
The commutation
~ -^ (ih + H) D(x
x')
relations for time-dependent field variables
&K(r,
Hence the propagator
t), T,](r',
of the
')]
photon
= is
^ (t defined
I + H) D. by
(XIJtJSS)
now become
[XI.2.]
The Photon Sp (x
Field
-
x')
317
= j-k (ik
|+
He}
D(x
-
z'),
(XI.241}
r x represents the relative space-time coordinates.* From the definition of the D function we have
where x
so the photon propagator satisfies the equation
(ih
-
j
H*} Sp =
0.
(XIJBJS)
t
XL2.B. Problems Discuss the quantization of the four-dimensional form of Maxwell's equations [wave equation (X.8.1)] and point out its advantages and disad1.
vantages as compared to the quantization carried out in the text. 2. Find the total angular momentum of the quantized electromagnetic field. Then show that the photon is a particle of spin h that aligns itself parallel or antiparallel to the propagation direction k, where the eigenvalues of ATU and N-is represent the number of spins oriented parallel or antiparallel, respectively. s be the phase of an electromagnetic wave consisting of n9 light that Prove 9 and 8 are canonically conjugate and satisfy the comquanta. mutation relation
3.
Let
(j>
N
Hence show that the number of light quanta of a wave and its phase cannot be determined simultaneously with an arbitrary accuracy. The measure of spread in these quantities satisfies
A&AAT. 4.
reversal
and
symmetry
that total
6.
Show
that the operators T*
rules for angular
uses
D
From using the expression
space-reflection operations.
Show
The usual
text.
properties of the photon spin under time-
5.
*
I-
Prove the relations (XL8.88) used in the
(XI. 2.27} discuss the
tials,
^
number
of
photons in the
=
/
(ri\Ki\ii}
field
can be written as
d*x satisfy the commutation
momentum.
theory, in which one quantizes electromagnetic field in terms of alone in the corresponding commutation relations.
its
poten-
318
Quantization of Massless Fields
XI. 3.
The Quantization
[Chap. XI]
of the Neutrino Field
Neutrino as a particle of spin |A must be quantized according to the statistics. The Lagrangian of a neutrino field is given by
Fermi-Dirac
where
H = v
ca-p and
a,
ft
=
1, 2.
In accordance with the exclusion principle the occupation number operator and 1. Therefore the of the neutrino field must assume the eigenvalues canonical commutation rules used for the quantization of the electromagnetic are not suitable for the quantization of the neutrino field, since the rules
field
do not put an upper limit to the eigenvalues of the occupation number operand 1 is a projection operator and can ator. An operator with eigenvalues
and 1 [see be represented by a diagonal matrix with diagonal elements Section 1.3.8 and (VII. 1.45)]- For fields quantized according to the FermiDirac
no physical limit reducing the quantized
statistics there is
field into
a classical field. Therefore all particles of spin %h belong to nonclassical having no counterparts in the classical domain. The anticommutation rules of the neutrino field are Etf.(r,
l4>*(r,
*),
= = 03+
**(r', f)J +
t), 7r0(r',
[7ra (r, J), T0(r',
0,
tfttafiW
-
*)
J+
=
fields,
0,
(XLS.S)
r),
where (XI.3.8)
a
so
we may
also write
[
The Hamiltonian density
= M(r -
").
(Z/.S-4)
of the neutrino field is
H The
<)]+
= Uca-ji)^.
(XI. 3.5)
state vector 1^} of the neutrino field satisfies the Schrodinger equation
of the quantized field,
ih
where the total Hamiltonian
H We
shall use periodic
operators according to
H =
1*>
is
=
HI*),
defined
(XI.S.6)
by
<&(ca'p) a ^d*x.
(XI. 3. 7)
boundary conditions and expand the neutrino
field
The Quantization of the Neutrino
[XI.3.J
&(r,
=
f)
^^2 ai-aSV n
where
p,
=
I
respectively.
''''"
E
319
''\
(XI.S.9)
fA
=
1 now correspond to particle and antipartide states, functions are unctions of MJ$ //, spinor eigenf
and p
The
1
Field
= pEPug, E> = -\E n for
(XL 3.10)
(H,)a#i$
where E$>
=
!*!, for M
=
1,
and
\,
/z
= -1. The
eigenf unc-
tions are normalized according to
=
JuSJOO <**
The above
(XL3.11)
dnn>8w<.
eigenfunction and eigenvalue assignments are equivalent to We may, if we wish, regard the number ju as the "lepton
Dirac's hole theory.
1 corresponding to charge content" of the neutrinos, with & = 1 and /z = neutrino and antineutrino charge. We are assuming that the lepton charge is conserved. In this way negative energies are associated with particles having
negative lepton charge. However, the resultant energy of the
field,
as will
be shown below, is positive. To satisfy the anticommutatiion relations (XI. 8. if), the operators aM7l and aln must obey the anticommutation rules
[fl^n,
0,
E
==
[aMn
0,
,
]/*']+
=
$nn'&w',
where we used the completeness relation of the eigenfunctions
The
total
Hamiltonian of the
field
H
can
now be
(X/.S J S)
rO.
obtained as
=
where the occupation number operators JV^ are defined by
NV =
a^a
(XI. S.I 5)
s.
Using the anticommutation relations for n
=
= therefore A72
iV ^J
_A7" J-V
n'
and
jit
=
0;
/,
we
find that
(XLS.16)
/YT % 17\ \^L.U.li)
US'
Hence the occupation number operator N^ is a projection operator, as discussed in Section VII. 1. This is the only quantization procedure consistent with Pauli's exclusion principle. The total energy eigenvalue of the as follows
E=
^
\E s \(n ls
+ n_
ls ),
s
is
the
sum
field,
from (XL3.14),
of positive energies of particles
and
antiparticles.
(XL 8.18)
320
Quantization of Massless Fields
The vacuum
state, as in Section VII. 1, is defined
[Chap. XI]
by
0.
(XI.3.19)
Hence where the one-particle state
j^i) is defined
by (XI. SM)
Thus als and a^ are creation and annihilation operators of particles (M = 1) and antiparticles (M = 1), respectively. For example, the field operators a^ can act on a one-particle state according to
(XLSJBg)
We
thus see that a^,
to the
vacuum
which for
ja
states, respectively.
l^i),
it
=
1
corresponds to n-particle and n-antiparticle
The most
general n-particle state can be expressed as
and
1
it
reduces
A particular n-particle state is represented by
state.
=
when
acts on a one-particle state
jot
in (7II.1.6S).
For the spin component of the neutrino
If
we assume
where
H
field
a coordinate system where pi
we have
=
p%
=
0,
then
we can
write
v
Using the expansions (XI. 8.8) and (XI. 3.9}
we
obtain
Rr
L/l
where
jB^'
}
=
cps;
using the normalization conditions,
we
obtain
or
Hence we
see that spin projections of particles
to one another
and both
particles
and
and
antiparticles are opposite
antiparticles are completely polarized
[XI.S.]
The Quantization
of the Neutrino Field
S2I
For a one-particle state the projection operators assume the eigenvalue zero and A"_i, the eigenvalue 1.
In the longitudinal directions. A^i, for
a fixed
s will
For a one-antiparticle state the eigenvalue 1 for A^, and zero for 2V_i, occurs. Thus the particles have their spin antiparallel to their momenta and antiparticles occur with parallel spin
and momenta. Hence the neutrino and
left- and right-handed, respectively. We see that the neutrino and antineutrino are created in single spin states alone instead of two spin
antineutrino are
states, as in the case of particles
with mass. Moreover, a given state of polari-
zation (left- or right-handedness) cannot be transformed into an opposite state of polarization. Contrary to the case with photons, it is not possible
and change its direction of polarization. These results are in accord with the experimental facts on parity nonconservation in weak
to reflect the neutrino
interactions.
For example, the decay of a
ic*
at rest into a
ju
+ and a neutrino
(with spins as well as momenta antiparallel) is followed by the decay of a /*+ a neutrino and antineutrino plus positron, where the spins of v and v are
into
+
antiparallel; the
/z
is
and e+ is emitted in the + momentum. For the decay of a T~ into a fjr and p. the pr is in the same direction as its momentum and
therefore essentially polarized
direction opposite to the Vj
the polarization of
therefore the \r
must be right-handed.
XL3.A. The Propagator
We
of the Neutrino
same procedure as that used in deriving the photon propFor the neutrino propagator we obtain the expression
follow the
agator.
f
^ L -|
^
-
0~l J
(XI.S.25)
This can be written as F<#
=
[e-
(i/m -V H v] a rf(r
-
(XI. 3.26)
r').
Hence [^(r,
f),
Thus the propagator
4(r', *')]+
it satisfies
i (ih
of the neutrino
&(x and
=
-
=
is
I + #,) D(x
given
-
aO.
(XJ.S.S7)
by
^ (* | + B) D(x
-
x'),
(XI.3.28)
the equation
(iU
I
#,)
S,
=
0.
(XI.8.S9)
322
Quantization of Massless Fields
XI.4.
[Chap. XI]
Gauge Invariance
The conservation law
(1 1. 1.20) of
the electromagnetic
field
can be replaced
by
J;
<*!H|u>
+
V
.
((n\cp\-n))
=
(XI
0.
4^
This conservation law implies a more general interpretation of (??} than the one given in Chapter II. According to (XI 4.1) the photon energy density is conserved with its momentum flux density. In the latter sense the quantity (i?h)
should be regarded as the expected photon density so the integral
N=
I
(-n\n)
d?r is the
number
of photons in the field.
The
total
number
photons, as shown in the previous section, can further be split up into the of left
and
right circularly polarized
photon numbers,
The form of the conservation law (XI'4.1)
N
= N+ +
sum
N-, where
suggests the existence of a gauge
transformation group under which (XI4-1) remains unchanged. transformation is of the form
The gauge
W) = e
of
CX7.4-S)
an arbitrary real function of space and time. The transformations * (XI'4-8) are called "gauge transformations of the first kind" as compared to "gauge transformations of the second kind/ which are applied, is
of the type
7
for example, to the potentials of electromagnetic field
The transformation (XI 4$) corresponds to a rotation of the at a given space-time point. The conservation law (XI 4^) |T?)
state vector is
invariant
under the transformation (XI 4-8). All observables of a dynamical system are gauge-invariant. This is a law of nature. Gauge invariance will impose certain restrictions on the possible course of physical phenomena. For example, according to the principle of gauge invariance the potentials of the electromagnetic field and the wave functions of elementary particles are arbitrary up to a gauge transformation. Therefore potentials and wave functions are
not observable elements of a dynamical system. Gauge transformations are applied onto the unobservables of a dynamical system. Gauge invariance sets
up
definite relations
between the unobservables.
In order to study the gauge transformation properties of the photon wave equation let us operate on both sides of (XI 4-8) with the operator ihK^
and obtain *
W.
Pauli, Rev.
Mod.
Phys., 13 (1941), 203-232.
[XI.4.]
323
Gauge Invariance
For the wave equation to remain Invariant under the gauge transformation we must have I*>
and hence the gauge function S must be
^|||V)
=
0,
restricted according to
=
(XI 44}
0.
write the matrix form of this equation and apply the same method as used in calculating the eigenvalues of H, we get If
we
The transverse condition p-rj Thus the gauge transformation
does not allow the solution dS/dt of 2
(I)
a
free
-
w
photon 2
=
is
=
0.
restricted according to
(XI4 5} -
'
which resembles the classical eikonal equation of geometrical optics. The restriction (XI 4.5} on the gauge transformation can be removed completely the gauge transformaonly for a wave equation that allows the application of a wave equation For unobservables. of set another least at example, to tion of the
type
(XI4 6} -
is
invariant under the gauge transformations
and second kind, respectively. In this case the gauge function S, the field aw is restricted less depending on the type of equation satisfied by Infer that the requirement we Hence free a of function than the gauge photon.
of the first
of
an interaction of dynamical gauge invariance without restriction involves
systems.
In general an enlargement of a gauge transformation group
for
example
introduction of varmany-parameter gauge transformations will require the between interactions the and corresponding ious dynamical degrees of freedom
them. Thus a dynamical system with a restricted gauge group is, essentially, a free system and therefore not observed. In general we may now postulate the invariance of the Lagrangian with an d investigate its implications respect to gauge transformations (XI 4$}
324
Quantization of Massless Fields
with regard to conservation laws of the change under (XI.4-S), we then have
We shall show that
(XI. 4-8)
If
field.
[Chap. XI]
the Lagrangian does not
a conservation law. Since conserved quanti-
is
represented by Hermitian operators, it is necessary to symmetrize becomes a functhe Lagrangian with respect to 77 (or <j>) and rf. In this case
ties are
tion of
1
?;,
i
1?
",
and
their partial derivatives.
rd~
as
as
<%
\ru OTfJi\ OX*
/s /
Hence
aa
ae a
+
7/*
a??l
/n
t\ (dt]i \
OlJi
\dx)
{dx'J
which can be written as
where
For
particles
of the field
with charge,
and
its
AM
conservation
For photon and neutrino
interpreted as the charge-current density
is is
connected with gauge invariance.
fields,
symmetrized Lagrangians are given by
and
^ *""
ihc
2
I
I
The corresponding A are
These are energy and
momentum
densities of the
photon and neutrino
fields,
respectively.
XI.4.A. Problems 1.
Prove that under the simultaneous operation of charge conjugation and
parity a left-handed neutrino goes over into a right-handed antineutrino. >- /*+ + v and Hence show that the decays T+ by a (P3 C [see (X.^JS)] symmetry operation.
TT~~
>- JJT
+
v
are related
[XI-5.]
Spin and Statistics
325
Consider the reactions
2.
X + v*Y + er, X
and
Y
place unless v
=
where
are v.
some
Thus,
if
nuclei.
we p
Show
Y=
take
+
e-^X +
Y+
e~~
v,
that the two reactions cannot take
7 and
*
n
+v
>-
p
+ er,
X=
n,
we have
and
n
+
v
where the second reaction
is forbidden by lepton charge conservation. Thereabsorbable by protons but not by neutrons. In the same way v can be absorbed by neutrons but not by protons. Apply these arguments to T
fore
v is
and
ja*.
Consider the derivation of a four-component neutrino wave equation from the complex conjugate of the photon wave equation (X.S.I). Does the resulting wave equation describe the neutrino and antineutrino pairs discussed in the text or do these particles correspond to a different pair of neu3.
trinos?
Discuss the differences between a very high-energy Dirac particle with mass negligible compared to its kinetic energy and a massless Dirac particle. 4.
5.
Construct the wave equation for a Majorana neutrino [see (VIII.5.85)], v = v. What is the status of lepton charge conservation in a Majorana
where
representation? What are the differences between the Dirac, the Majorana, and the two-component neutrino? 6. Discuss discrete transformation properties of neutrino and photon fields. Prove that the two states of photon polarization have the same intrinsic
parities. 7.
Show
that the operators r
commutation
f
\
(v\
d*x satisfy angular
momentum
relations.
XI.5. Spin
The
=
and
Statistics
spin degree of freedom of elementary, atomic, and nuclear systems
entirely of relativistic origin.
The
concept of
statistics (Bose-Einstein
is
and
Fermi-Dirac distributions) belongs to quantum mechanics. The methods used and neutrino fields suggest a
for the quantization of the electromagnetic definite connection
between spin and statistics. The quanta of integral and have different statistical distributions. The require-
half-integral spin fields
ment
of Lorentz invariance of the physical laws, together with the
equations and corresponding representations
of the
wave
Lorentz group, implies
certain statistics for elementary systems. Therefore spin
and
statistics are
Quantization of Massless Fields
326
two basic concepts where
and quantum theory find a definite and was first observed by Pauli* and he has
relativity
indispensable unification. This fact given a rigorous proof for the case of free
been extended to interacting We shall give a summary
fields
[Chap. XI]
Pauli's theory
fields.
has recently
by Ltiders and Zumino.f
of the general proof of the connection between fields. In order to follow the basic spin and statistics for the case of massless two-dimensional complex linear arguments it will be necessary to use the the to representations of the Lorentz and their relations
transformations
VIII.8.C that the eigenfunctions of group. It has been shown in Section the representation have the general form
U = tT'W'-^Ct*)*! W are integers and satisfy the
where k and
k
For a given a and
a!
we
an
relations
g
a,
+
there are (a
(XI.5.1), so they can span
Pauli
^
(a
+
+
^
k'
+
l)(a'
l)(a'
(XI.5.1)
1)
a!.
eigenfunctions of the type
l)-dimensional space. Following
introduce the notation j
=
ia
f =
,
Ja',
and denote the corresponding representations of the Lorentz group by D 1 he numbers j and / assume integral or half-integral values. Thus the repirreducible. resentation is (2?+ 1)(2/+ l)-dimensional and is, of course, characterbe can Lorentz the of group All finite irreducible representations J/)
ized
has
by two numbers, j and /. (
a
+
i)( a
+
'
l)
=
(2j
A
+ 1)(2/ +
corresponds the scalar, to antisymmetric tensor (such as (0, 0)
iJfC),
(fi
to (1,
quantity F(jJ'), characterized 1)
(|, i)
by
.
(j,f),
independent components. Thus, to the vector, to (1,0) the self-dual
+ #C),
to
(0, 1)
its
complex conjugate a (-f, f )
to 1) the symmetrical tensor with vanishing trace,
so on. nonsymmetric tensor with sixteen components, and The photon and neutrino fields transform irreducibly under the groups characterized by (1, 0) and (i, 0), respectively. In both cases (where j ^ /)
the transformation matrices are complex.
The product
be decomposed into quantities, Fi(jij[)F^(j^\ can two angular momenta: according to the usual addition of
of
several F(j,f),
two
- J2 and/ = ji The spin value of the = j\ fa \ji 3\ j massless particles belonging to a given field must be determined by noting = J is not possible. For a neutrino s that for a photon s = 1; therefore ; =
+
-
-
.
and therefore j * t
\
\
=
is
+,
-
A
-
not possible. Hence in both cases the total angular
W. Pauli, Phys. Rev. 58 (1940), 716. G. Lxiders and B. Zumino, Phys. Rev. 110 (1958). See also N. Burgoyne, Nuovo
Cimento, 8 (1952), 607.
momentum values
327
Spin and Statistics
[XL5.]
s
s,
j begins with a certain
+
minimum
value*
s
and takes then the
.
I,
/ = an Integer and j / = In general (where m ^ 0) the statements j a half-Integer correspond to fields with integral and half-Integral spin. Hence there can at most be four types of quantities that can occur in the field-
+
+
theoretical description of particles.
= an
/=
an integer. The field variables and their adjoints of number an odd have components and belong to the representations D u 3 and D respectively. (a) j
integer,
'
(
'
}
'
}
,
(b)
j
number j
(c)
number (d) j
number
The
=
a half-integer,
of
components. a half-integer,
= of
=
j'
j'
=
a half-integer. The
=
field variables
have an even
The
field variables
have an even
a half-integer. The
field variables
have an even
an
integer.
components.
an
integer,
f =
of components.
classes (a)
and
(b)
belong to an integral spin
long to a half-integral spin
field,
and
(c)
and
(d) be-
field.
XI.5.A. Remarks on the Observables of Massless Fields The
total energy for the
but the probability density
<&L
s,
for
ju
=
1, 2, 3, 4, is
not positive definite as the fourth component of the current
unquantized neutrino
= AM-rtfi,
positive definite,
-^ The
total energy for the unquantized
possibility of a third solution
field is
(XI.B&)
where
= photon
(XI. 5. S) field is positive definite
with negative energy
is
but the
not excluded. The
current vector
is
conserved; that
is,
(XIJ.S)
The probability density s4 = -nl^iis positive definite. The photon field variables ^ transform irreducibly under the *
M.
Fierz, Helv. Phys* Acta, 13 (1940), 45.
representa-
Quantization of Massless Fields
328
[Chap. XI]
Both of the representations D (10) and to the covarianee group of the photon and the two states of
tions belonging to class (a) above. J)(OD "belong
D
D
(01) ao) or polarizations transform according to transform irreducibly under the representaThe neutrino field variables .
<
ization state.
The
of the neutrino
equation
D
(1/2 0) There is only one polarcorresponding to (0 1/2> the transform will complex conjugate representation
tions belonging to class
(c),
.
D
wave
function.
The complex conjugate
of the neutrino
wave
is
ih
<& = (c^-pWft | at
(XI.S.S)
Hence the representation D (a l/2) does not belong to the transformation group of the neutrino. However, if we were to assume the existence of another pair of neutrinos, then the representation
so ccr^-p
jT)(o
1/2)
is
not the same as
C ould
c
correspond to a group of physical transformations. may be involved in reactions of the type
The
latter
type of neutrinos ff
+ ?' _>. p + e+
n
>-
p
+ e+ +
(XI. 5.7)
v\
and v are the new neutrino and antineutrino, respectively. Thus v can be absorbed only by the antineutron and v' only by the antiproton. r
where
r
v
XI.5.B. Field Quantization According to
Commutators and Anticommutators The negative-energy
eigenstate for a neutrino field does not arise
if it is
quantized in accordance with the exclusion principle. In this case, as has been shown before, all negative energy states are filled and positive energy states are empty, so no transitions to negative energy states can occur. This
a consequence of using anticommutators for the quantization of the neutrino field. The anticommutator or the propagator
is
S = r
(XI.5.8*)
^-j^\D(x-x'}] where j and
/
assume half-integral transformation property for $ could be obtained
of the neutrino field belongs to the class
The same type of we used the commutators,
instead of the anticommutators, for the
quantization of the neutrino field.
However, the energy would not then be
values.
even
if
positive definite.
Hence
it
is
necessary, in connection with. Dirac's hole
theory, to apply the exclusion principle
Dirac
and quantize according to Fermi-
statistics.
For the photon
field
the propagator (or commutator)
-xV]
SP
is
of the
form (XI. 5. 9)
[XI.5.]
and
It
Spin and Statistics
329
belongs to a class where both j and
/
assume
integral values. If
we
V(8T#Y)ij, then the commutator for the variables of an number of derivatives of the function D. The D funceven consists X> from 2. tion, as seen (XI. 38), is even in space coordinates and odd in the time
by x
replace (1 1. 1.29)
coordinate. Thus, the expression
*
=
xiVoi
tx.-(z'),
+ Cx<(*"), xira,
f
symmetrical in x and x", is equal to an even number of spacelike derivatives times an odd number of timelike derivatives of D. Therefore, falls into
X
the same class as the propagator of the neutrino field. This result is in contradiction with the property of the commutator of x and % f The consistency = to vanish. In particular, for x' = z", the requirement requires -
X
cannot be
X
satisfied
with an anticonrmutator, since XtX*
+ x*x*
is
a positive
quantity. Hence we infer that quantization of the photon field according to the exclusion principle is not possible. In general, all integral spin fields must
be quantized according to Bose-Einstein
statistics.
XL5.C. Problems Prove that Bose-Einstein quantization of the neutrino energy without a lower bound. 1.
field implies
an
Give a proof different than the one in the text that Fenni-Dirac quantization of the photon field leads to an algebraic contradiction with the commutativity of physical events separated by a spacelike interval. 2.
3.
Discuss the transformation of the quantities
and
where the d2^ are four-dimensional surface elements, under reflection of time. Hence show that time reflection interchanges the states of polarization. 4.
Consider the spin matrices
where subscripts
A
and
B
= ~(jA7B run through
JBJA),
1 to 5.
Show
(XI. 5. 10}
that time-reflection
transformation of a Dirac spinor can be obtained with a rotation through the angle ir in (45) -plane; that is, 3
Hence show that
=
exp
(iir$ff&)
=
io-45-
w> = -m
where
W)
Quantization of Massless Fields
330
A
n contains
spinor of rank
spinor
we have
fields of
spin
J/zfi,
l)h
(fn
[Chap. XI] .
For such a
the expression
as the basic invariant.
Show
that the time-reversal operator 3
-
is
given
by
H **%
s=l
and that Prove that a Dirae spinor with four components can be decomposed into two irreducible parts with two components each, characterized by of the matrix y 5 with (J, 0) and (0, $), corresponding to the diagonalization* 1. Hence show that when applied to Dirac spinors the its eigenvalues 1 and 5.
transformation
for the irreducible quantities
vi& 0) =
w(i,
tfed, 0),
0)
and v(0,
J),
ti(0, i)
yields the forms
= -(0,
4).
Discuss the significance of the above for the neutrino field. 6. Show that the results of the previous problem can be generalized by
Since for a vector /
=/=
|,
then the above transformation changes the
every vector. Construct the expectation value of the energy
sign of 7.
the quantized electromagnetic field in the
and
momentum
vectors of
form
discuss its transformation properties
under space- and time-reflection
transformations. 8.
Compare the
n
reactions >-
p
-f-
e~
+
?,
^
>
P
+
e
+
+ /,
and show their relations to the representations (|, 0) and (0, \) group. Are these reactions charge conjugates of one another? 9. Are there any relations between invariance under time connection between spin and statistics? *
W.
Pauli, Niels
Bohr and
the
of the
Lorentz
reflection
Development of Physics, Pergamon, London, 1955.
and
Spin and Statistics
[XI. S.] 10.
331.
Derive the photon and neutrino propagators by solving the equations (ih
~ -
H J
-
S p (x
xO =
2io(x
-
x')
and
where
H
e
and
.HV are
the photon and neutrino Hamiltonians, respectively. function there is just one more function satis-
Prove that apart from the
D
fying the equation
QDcwCs and that
it is
-
*')
= 2(s -
re'),
an even function of space and time coordinates. What would
be the consequence of using the DCD function in place of the D function in the quantization of the photon and neutrino fields? Can one maintain that quantization of integral spin not possible? 11.
fields
according to the exclusion principle
is
Prove that Z>(r, 0)
=
0,
12. How would you interpret the massless spin \Ji wave equation obtained through stereographic projection of the complex conjugate of the photon wave equation? (Use a complex orthogonal representation of the
Lorentz group.) 13.
Can one
regard the two possible states of polarization of the photon observables? Does a single state of polarization of the
field as its indefinite
neutrino imply a definite observable for the field? 14. Is there any way to differentiate between the two reactions
where 15.
v
1
and
v
f
are the second pair of neutrinos discussed in the text?
connection of spin and statistics, prove that it is not poscombine two neutrinos to obtain a photon, and vice versa. Using the Fermi-Dirac distribution, discuss the differences between
By using the
sible to 16.
neutrino gases obeying the equations v^Puli')
=
and ffft
Pn\^
0.
17. Show that the expectation values of the creation and annihilation operators of the photons are given by
(d)
=
^
where N\ s represents the average number of photons
of
mode X and
s.
332
Quantization of Massless Fields
[Chap. XI]
18. Assuming that the radiation field is built up by emission processes which are statistically independent events, show that the actual number N\ of photons in different modes are distributed around the averages (Nx.)
s
according to the Poisson formula
known
19. It is its
path
is
that in a strong gravitational field deflection of light from fact. Using the equation (XI. 4-6} for a photon in a
an observable
Newtonian
= GM/r, a* = 0, and i = 1, 2, 3, and wave function (as in Chapter IX), show that photon
gravitational
also the expansion of a
the resulting deflection of relativity,
is
field,
where a4
one-half of the value obtained in the general theory
CHAPTER
XII
PROBABILITY
IN
CLASSICAL AND
QUANTUM PHYSICS XII.l.
If for
The J.W.K.B. Method and Dependence of the Wave Function on h the solutions ^ of a
wave equation where
the integral
taken over the whole of space is finite or equal to unity, then we interpret 2 as a probability density. The wave functions that do not satisfy the |^| normalizability condition are also possible solutions of a describe a possible physical process. In experiments with
wave equation and beams of particles
extended over a long time interval, where the physical situation does not 2 is to be interpreted as the average number change with time, the quantity
M
of particles per unit
volume (assuming that the interactions between the
particles are negligible)
.
In this case the current vector
J in
the conservation
equation (obtained from Schrodinger's equation),
= |p + V-J
0,
(XII. 1.1}
be interpreted as the average number of particles crossing a unit area perpendicular to / per unit time, where
is
to
We
can, for example, observe electrons either as particles causing a flash on a screen, a kick of a counter, or so on, or as waves producing a diffraction pattern.
We
use a
wave function
to get the probability of finding
some
elec-
333
Probability in Classical and
334
Quantum
Physics
[Chap. XII]
trons in a given volume or the probable number of electrons crossing a unit area per unit time. If dV is a volume element at a point r, then the probability of finding an electron in
the average
number
dV
at time
t
of electrons in
is |^j
2
dV.
We may also speak of finding V by calculating the
a certain volume
quantity
*M*dV.
(XII. 1.S)
Let us consider a stationary state of a beam of particles. In a stationary we have a definite energy of the system, satisfying the statement E = T V = constant total energy, where T = kinetic energy, 7 = poten-
state
+
tial
energy. In the stationary state
the phase factor e~~ <*W^ depends on the product Et of energy and time and therefore it cannot be measured experimentally. Furthermore, the potential T at a point r can not be determined experimentally. This energy V = TQ
because the kinetic energy jp 2 /2m will be uncertain if we try to measure the position. But a measurement of potential energy in a sufficiently large is
region
of course, possible.
is,
The quantity
TQ = T
E
+ V can,
however,
be measured at a given point.
A beam of particles of infinite breadth in the z direction can be represented (i/h)Et], where by a plane wave ^ = A exp [ikz h If the
wave function
is
normalized in a volume V, then
we have
=
2 1/7. Thus |A| can be regarded as the average number of particles per unit volume. The average number of particles crossing a unit area perpendicular to the z direction, per unit time, is given by |A| 2 z;, where v is obtained from E = ($mv*. The probability of finding one electron in a
Hence
2
|A|
volume element dV at time
t
M The number
2
is
dV =
2
|A|
dV =
(XII. 14}
^~-
of particles crossing a given area per unit
time
is
given
by
the
normal component of the current vector with respect to the area. If / is the current vector and dS is the surface area, then the probability that a particle will cross
an element dS
of area in the time dt is
J-dSdt,
where
/
is
given by (XI7. 1.2}. For a plane wave of the form
(XI 1. 1.5}
J.W.K.B. Method and Dependence of Wave Function
[XII.l.]
335
the probability (XI 1. 1.5) is given by ]A ~vdS dt. For a beam of electrons or protons, J can be measured ]
by measuring the charge falling on a collector. If we have rnairy streams of electrons, sav, thev can be represented by a superposition of plane waves ^ = ]C
A n e< M r 'p*e-< 'V iE i
i
(XII. 1.6)
'>.
n
The the
waves would not affect the number of electrons. from different sources the wave function is originate
interference of the
beams
^ =
^A
ne
i
**eW'"p*e- (i
ME
'*
If
(XII. 1.7)
9
n
where
are arbitrary phases.
4> n
The average number of state
beam,
equal to the
electrons moving into a volume V is, in a stationarynumber moving out. This fact is expressed by
V-J =
0.
(XII. 1.8)
For a nonstationary state we use the conservation law (XI 1. 1.1).
XII.l. A.
The Connection Between Classical and Quantum Mechanical Description
The time development of a wave packet is fixed by Schrodinger's equaThus a wave packet approximation may be regarded as the nearest analogue to a classical description. A wave packet must, on the average, have the properties of a classical system with long lifetime and must move tion.
according to classical laws. In accordance with our discussion of Schrodinger's equation, let us assume that a time-dependent wave function in Schrodinger's picture
is
given by
lKr,0
where a and S are
real functions of r
= aeW s and
t.
(XII. 1.9)
,
An example
of the
form
is
provided
by the transformation functions discussed in Chapter V. Let us assume that a and S are slowly varying functions of their arguments. This assumption is equivalent to saying that the change of potential energy within a de Broglie wavelength X = h/p is small compared to kinetic energy, so the quantum
from the wave character of particles will not show up; therethe corresponding description will be a good approximation to a classical
features arising fore
picture.
Now,
in analogy to a
harmonic
oscillator state [see (VII.1.17)~\, let us
define the state of a dynamical system
by (XII. 1.10}
Probability in Classical and
336
where
|0) will
be
Quantum Physics
called the constant state of zero energy
ator satisfying Schrodinger's equation. Thus,
then
it
follows
where now S
is
if
we
[Chap. XII]
and &()
is
an oper-
write
from (KILLS) and (XILL10) that an operator function of the position operator q and time a
=
t,
and
(XILI .18)
(rt\0).
Hence
_dS dt
or
(XILLU)
,.
ot
Using the relation e
we
*
(i/rt)Si-fc>(t/7i)<S JL/O
""""""
J>
In I
.
..
)
dq
obtain
= H\ and a.0\
/
a.O
(ZIJJJ5)
0^
If
we
neglect the terms involving h
which
is
Let us
obtain
Hamilton-Jacobi equation. now assume that 8 can be expanded into a power
S The Hamiltonian
It can,
we
up
for
-&+& +
&+
series of
-.
an electron in an external electromagnetic
to the first order in
7t,
the form
(XII.1.16) field is
be written as
+ ^-(^-:^)
(^-^)
J.W.K.B. Method and Dependence of Wave Function
[XII. 1.]
337
Using the commutation relations
Ca p*
we can
write (XII. 1.18)
m
m
where
Substituting (XII. 1. IS) and (XII. 1.16) in (XII. 1. IS)
we
obtain the equations
-t - s + where, in obtaining (XILI .22),
we used j>|0>
That
is, |0) is
the equation
=
0.
(XII.13S)
momentum so the term involving p in
a state of zero
(XII. 1.19) can be dropped. Equation (XII. 1.21) leads to the- classical Hamilton-Jacobi equation. Equation (XII. 1.22) can be written as
= V9 The wave
where first
>S
function to the
order in h
and Si are now functions
order in h
is
e 2Sl
of r
is
i.
The
probability density to the
just the conservation equation for the probability
and the current
density,
J = From
and
is
therefore
so equation (XII. 1.24)
density
first
m
251 . Troe
Heisenberg's equations of motion
(XII. 1.26)
we have
and hence
J = proves that (XII. 1.24)
is
in h, the expression eSi
is
pv
(XII. 1.28)
the conservation law. Therefore, to the first order the amplitude of the wave function ^. The motion
Probability in Classical and
338
Quantum Physics
[Chap. XII]
2Sl at any point and time, density e moving in the space of variables r with a velocity v = (l/m)^. Here equation (XII. 1.^4), written in the form
can be pictured
In
terms of a
fluid, of
expresses the conservation of current density for such a fluid. From the above argument it follows that
is
the operator statement of the fluid density, where
LS,
Writing
we
=
1
S = Sa -
we assume
that
0-
ihSi,
obtain
&(f)*(t)
=
e
2
*
(XII.L29)
where &*&(() now corresponds to the operator form of the amplitude a in (XII. 1.9).
XII.2. Probability The
basic difference
and Waves
between
classical
and quantum mechanical probabil-
because in quantum mechanics probabilities are calculated in terms of wave functions satisfying Schrodinger s equation. The wave function is a ities arises
probability amplitude. In classical mechanics
we
chanics
For an
calculate directly the probability
for example, statistical
me-
itself.
quantum probability consider the penetration of a a hi a hole particle through diaphragm placed at some distance from a photoillustration of the
graphic plate.
the particle is recorded at one point A of the plate then it to observe an effect of the particle at another point B] there could be no correlation between the two events at A and at B.
Classically
if
would be impossible that
is,
lines represent plane waves constituting a hk. moving toward the diaphragm with a momentum p Quantum mechanically the plane waves representing the particle will be diffracted at the hole and will emerge to the right in the form of spherical waves having a limited radial extension. The aperture 6 of the waves is given by 6 A/a. The spread Aq in position of the particle is equal to the radius a = h/Aq or ApAq = h. of the diaphragm. Hence Ap 6p = (\/a)p = h/a
In Fig. 12.1 the vertical parallel
one-particle state
=
^
we imagine a shutter on the hole, then during the time At when the Also, shutter is open, the number of harmonic components present in the waves after they emerge from the hole is Av ~ I /At = AE/fi or AtAE = h. if
Probability
[XII.2.]
and Waves
339
Diaphragm
FIG XI 1 .
distance
.
1
.
Penetration of a particle through a hole in a diaphragm placed at some
from a photographic
plate.
According to quantum description the spreads AE, Aq, Ap result from interthat is, exchange of energy and momentum with the diaphragm and also with the shutter. These interactions take place in accordance with actions
the conservation of energy and momentum. The shutter, which leaves the hole opened for a time At moves with a velocity v a/ At and therefore a momentum transfer Ap between the particle and the diaphragm will cause
^
}
an energy exchange with the
which
is
of the
same order
balance of energy and
particle,
of
amounting to
magnitude
momentum
is
as the energy spread AE.
Hence a
possible.*
Here we have assumed that the diaphragm and shutter are sufficiently heavy. We might be interested to know the momentum and energy of shutter *
This problem has formed the basis of interesting and detailed discussions between
Einstein and Bohr. Einstein was not convinced of the limitation on a control of momentum and energy transfer involved in the passage of the particle through the hole and subsequent specification of the state of a particle. See the article by N. Bohr in P. A. Schilpp (Ed.), Albert Einstein: Philosopher -Scientist, Harper, New York, 1959, Vol. 1.
Probability in Classical and
340
Quantum Physics
[Chap. XII]
knowledge should allow us to control the momentum and energy exchange between the particle and shutter or since diaphragm. But here we would expect a conflict with physical reality,
and diaphragm.
Sufficiently accurate
in accordance with uncertainty relations the location of the shutter in space
and
This uncertainty was the time, for example, becomes indeterminate. of the particle after it state the that means It raised by Einstein.
question
the argument emerges from the hole is not specified correctly. However, momentum balance is based on the given in the above for the energy and have fixed posiassumption that the diaphragm and the photographic plate mechanical description, it is not within the quantum and tions in
space
that,
possible to predict exactly
where the
particle will
be recorded.
good examples of Compton a be can particle. In both cases replaced by situations where the diaphragm
The
there it is
effect furnish
photoelectric effect or the
a large latitude in the knowledge of the position of the particle and then possible to predict, by means of an experiment, the direction in is
which an electron or a photon
mentum
is,
is
scattered.
The balance
of energy
and mo-
further illustration let us consider then, clearly understood. For with two parallel slits, as shown in Fig. 12.2, where S is a source
a diaphragm
with the same energy. The arrangement can be used for two different experiments. We can try one of the slits (or to discover the electron (a) as a particle going through as a wave going through both holes at the same time. The elecor of electrons
holes)
(b)
on tron will behave either as a particle or as a wave, depending interested to find out about
what we
are
it.
Under the usual conditions the beam
produce on the sufficiently intense, then
of electrons will
detector an interference pattern. If the beam is the interference pattern will consist of a large number of small spots
on the
function of the detector. Before the arrival of the electron at A, the wave beam covers both slits 1 and 2. After the interaction with the slits 1 and 2, the plane waves must interfere destructively and form wave packets of widths AI and A 2 the sizes of the slits. We can compute the probability of finding the electrons near 1 and 2. ,
The corresponding wave
functions,
being wave packets, represent single
and analogy exists between this situation and classical optics, a of Huygen the propagation of electron waves can be described by means If we know the value of the wave function on a given wave front,
electrons.
An
principle.
then we can express its value elsewhere as "the sum of contributions from different elements of the wave front." It implies that the probability amplitude as a function of time arises from the entire motion of the particle. of the partiIf ^ 1; t} and |^8, t) are the states corresponding to the passage
Probability
[XII.2.J
and Waves
341
Screen with two
slits
Detector screen
FIG. XI 1. 2. placed cles
at
some distance from a photographic
through
to the
Penetration of a particle through two parallel
two
and
slits 1
slits
2,
a diaphragm
plate.
respectively, then the overall state corresponding
taken together I*,
The wave
in
slits
is
t)
the
=
sum
|*i,
t}
of the
+
|%,
two
states:
(XIL2.1)
t).
function at an arbitrary point r to the right of the *(r,
t)
=
fc(r,
slits is
+ t*W,
(XIL2.2]
where \l/i(r, f) represents that part of the wave reaching the point r that has come from slit 1, and V^C**, t} represents that part which has come from slit 2.
The
calculation of probabilities proceeds as follows:
(a)
If
only
the point r
1
slit
is
open, then the probability that an electron reaches
is
P!(r, (b) If
only
slit
2
is
i)
= Mr,
2
i)|
-
open, then the probability in question P,(r,
f)
=
is
Probability in Classical and
342 (c)
If
both
slits
Quantum
Physics
[Chap. XII]
are open, then
=
P(r,
Thus the Laplacian
l*i
+ fa\*
=
+ P + iflfc
Pi
2
by the appearance from the wave prop-
rule for the probabilities is modified
of the interference terms
^+ 2
*i$. These terms
arise
erties of the electrons.
When
the position of an electron is observed, the interaction process inwill not change the probability but the wave func-
volved in an observation
an unpredictable and uncontrollable way. This kind of change in the wave function can be fitted into the mathematical scheme of quantum mechanics in the form of phase factors. We know that a state |^, t)
tion will change in
completely determined, except for a phase factor, when its scalar product with every vector of a quantum frame of reference is known. Thus in an act of observation each part of the wave function corresponding to a definite is
position of the electron is changed during the course of interaction between the electron and the observing apparatus in such a way that the state is undetermined by an unpredictable and uncontrollable phase factor, e*X In this case the total probability amplitude
^(r,
where
71
and
f)
is
+ ^V
= ^fc(r,
2 (r,
72 are functions of position.
p=p +p + 2
l
We see that the interaction the interference terms.
At is
is
from the act
of observation
has changed
before the waves actually do interfere, the
phases 71 and 72 are equal. Outside the distribution of particles
probability
$^(72-71)
arising
slits,
The
(XII. 8. 8)
),
affected
slits,
by
71 9^ 72.
Therefore the statistical
the phase through the interference
terms.
To
continue with the second double-slit experiment, let us suppose that S is very low, so the detector at B (such as a will record pulses representing the arrival of a particle, sepGeiger counter)
the intensity of the source
N
arated
by gaps
in time during
which nothing
to disturb the above state of affairs, then
observation to find out
if
arrives. If
we say
nothing
else is
done
we are carrying out an The result is that in this
that
the electrons are particles.
case the electrons are actually particles.*
whole of screen B is covered with detectors, with a very weak source one detector should respond; then after a small time interval another detector or the same one should record the arrival of an electron. For various If the
S, only
*
and
R. P. Feynman, Proceedings of the Second Berkley Symposium on Mathematical Statistics Probability, edited by J. Neyman, Univ. of California Press, Berkeley and Los Angeles,
1951.
Probability
[XII.2.]
and Waves
343
we shall measure the number of pulses per second, positions r of the detector that an electron This is equivalent to determining the relative probability
P
passes from
S
to r as a function of
r.
Qualitatively, this probability
is illus-
trated in Figs. 12.3 to 12.6.
FIG. XIL3.
The probability 2 dosed.
distribu-
FIG. XII. 5.
The
classical
that if both slits are
open
probability
the particle goes
through one hole or the other. The probability
P of arrival of the particles at a point G
on
the detector is the
sum
PI of those coming from ability
Pz of
those
FIG. XII. 4.
of the probability
and prob-
slit 1
coming from
FIG. XI 1. 6. that
it is
from
The quantum
probability
or
from one
pendent of
The waves
its
2.
slit
The passage
of the
of the slits is not inde-
passage from the other
will
interfere
and hence
slit.
the
Laplacian rule of addition of probabilities
slit 2.
+P
rule
is
distribu-
probability
slit 1 closed.
not true that the electron passes
slit 1
particle
Pi
For each electron there
The
tion for the case of
tion for the case of slit
a probability
2
will be replaced by the
P=
Pi
of going
+
Pa
quantum
+ interference
through
slit
terms.
1 or 2, so
the chance of arrival at r, when no interference is possible through their wave nature, should be the sum PI, the chance of arrival through slit 1, and P2 the chance of arrival through slit 2. The probabilities PI and P can be 2
,
determined separately by closing one of the
slits
The sum
corresponding to the arrival
two probabilities PI and P2 is the classical distribution of the electrons and is represented in what one observes experimentally. The result Fig. 12.5. However, this is not because the electrons coming through slits 1 and 2 p = p 4. p 2 i s
of electrons
l
from that particular
slit.
of the
obtained,
were particles alone, without wave properties. Therefore, this kind of statisinterference tical interpretation does not take into account the destructive
Probability in Classical and
344
Quantum
[Chap. XII]
Physics
brought about by the interaction between the electron and the observing apparatus. The Interference arises from the wave nature of the electrons.
The wave and
particle properties of the electron
and consequently the
destructive interference effect can only be included within the quantum mechanical description. The chance of arrival at r with both slits open is not the sum of the chance with slit 1 alone open plus the chance with slit 2 alone open. The intensity of distribution in the interference pattern must be derived
from the probability amplitudes and not from the probabilities themselves. The resulting probability P must be derived from a probability amplitude of arrival at r as a square of its
to be defined as the
P^
Since slit 1
or
Pi
slit 2.
sum
+ P^
But
if
of
modulus The .
amplitude
is
two probability amplitudes.
then
not true that the electron passes through the electrons to see through which slit they
it is
we "watch"
pass, then they will pass either through
slit 1
or through
from an incoherent superposition
at r results
total probability
slit 2.
The amplitude and fa]
of the amplitudes fa
we interact
the light used in locating the position of the electron, for example, with the electron, the total amplitude at r is $ = faeiyi faei72 Because of the unknown amount of disturbance of the electrons caused by
that
is,
after
+
.
"watching" them, the functions 71 and 72 can change in such a way that the interference between fa and fa will be destroyed. Consequently the probability P2 If we do that a particle can be found at the point r becomes P = PI
+
.
not try to see through which slit the electrons pass, then the probability that 2 an electron can be found at the point r is P = \fa fa\
+
It is to
passes,
be understood that
we change
if
we
.
try to see through which
the chance of the electron's arrival at
r.
slit
an electron
The use
of a
weak
intensity or low energy for the light to locate the electron will not help to reduce the disturbance of the electron. The photon's nature of light implies
an exchange of long its
of a
whole energy
wavelength
X,
the
limitations also, since
scattered from behind
One might think that with the use of light small. But this has tell whether it was slit 2. In accordance with photon theory and
hv.
momentum ft/A can be made now we shall not be able to
slit 1
or
the uncertainty principle, a source of light of wavelength X cannot be located in space with a precision greater than that of the order of X.
Thus the act
of observation
must produce enough disturbance to
alter
the distribution from Fig. 12.5 to Fig. 12.6. According to the uncertainty principle no apparatus exists or can be designed to determine through which
Whenever we try to separate an observer and obmore objectively, when we do not attempt to determine
hole the electron passes.
served
or,
to put it
through which hole the electron passes we are not in a position to say that it will pass through one hole or the other.
[XII.2.1
Probability
and Waves
S45
For the occurrence of successive events, the
calculation of probability
Is
based on the knowledge of a probability amplitude
* where
\frsi
is
= fefe,
the amplitude to go from
S
(XII.S.S)
to the hole at
1,
and
fe
is
the ampli-
tude to go from the hole at 1 to R. For a hole that is large compared to the wavelength of the electron we have to consider each differential of the area
from S to this area times the and sum these amplitudes over the
of the hole, calculate the amplitude of going
from
amplitude of going
this area to R,
Each
total area of the hole.
differential area constitutes
an alternative.
Despite the indeterministic interpretation, the probability amplitude Is obtained from a completely deterministic equation (Schrodinger's equation). A
knowledge of ^ at
t
=
implies
XII.2.A.
all
arising
subsequent times. The from the finite value
not an entirely settled question.*
is
The Quantum Mechanical Path of
A cloud
h
knowledge at
quantum mechanics
indeterministic trend in of Planck's constant
its
an Electron
an experimentally determined patl The track has quite large dimensions compared to the pos-
chamber track
of the particle.
sible size of the electron.
of a particle
The track
is
of the particle
is
made visible through the
around every positive and negative ion by condensation of a vapor admixed to the gas of the chamber. In order to form a quantum mechanical path of a particle for every position
formation of drops
measurement one has
to cause further disturbance of the system.
A quantum
mechanical path is, therefore, built up by a series of experiments. If the electron at time tij starts from, say, position /% and later is found at position r 3 it need not have had a definite position r2 at time t z in order to have a ,
position r s at a later time fe. The fact is that ri and r 3 exist as a result of measurements carried out at times ti and k- But what were the positions of the
and r 3 corresponding to times ti < t% < 3? The answer depends on the position measurements carried out between the times U and fe. electron between
ri
* A. Einstein, B. Podolski, and N. Rosen, Phys. Rev., 47 (1935), 777. N. Bonr, Phys. Rev., 47 (1935), 676. D. Bohm and Y. Aharonov, Phys. Rev., 108 (1957), 1070. A. Peres and P. Singer, Nuow Cimento, 15 (1960), 902. D. Bohm and Y. Aharonov, Nuovo Cimento, 17 (1960), 964. D. Bohm, Phys. Rev., 85 (1952), 166, 180. L. de Broglie, C. R. Acad. Sci. Paris, 233 (1951), 641. L. de Broglie and J. P. Vigier, C. R. Acad. Sri. Paris, 233 (1951), 1031, 234
(1952), 265.
Probability in Classical and
346
Quantum
[Chap. XII]
Physics
measurement between ti and 3 could be made and a number r% could be found such that the electron's position at time 3 would still be rs This statement in quantum mechanics is not true, since the number r 3 found
Classically, a
.
at time
fe
is
r3 because of not measuring r2 at time
electron need not have a specified value r2 at time the measured value r3 at time 3
fe
fe.
The
position of the
in order for it to
have
.
In the classical case the probabilities are not subject to Heisenberg's uncertainty principle, while a quantum probability has to comply with its require-
ments. If
we
Pa
let
i
be the probability that
if
A measurement B
the measurement
for
example
will give the gave the result a, then result b. Similarly we can define P& Pabc, e- If the events between Pabc a and b are independent of those between b and c, then
the position at time
ta
-
,
Pabc
(XII. 2.6}
PabPbc*
a complete specification of the state, then also true in quantum mechanics. In general, we have the prob-
If the statement that
(XII. 2.6} is law
B
=
is
b is
ability
V~^
Pac
/
y
/ TT r*\ \j\.JLJ- .&./ )
p
j JLabc'
^
b
The summation over
b
means that the quantity B must have had some value A and C. We can say that it had the value 6;
at the time intermediate to since
we
are dealing directly with the probabilities themselves, no other
needed to calculate the probability. However, in quantum mechanics we first calculate the probability amplitudes. Accordingly we replace the law quantity
is
P ac =
2P
Pbc
(XII. 2.8}
*****
(XII. 2.9}
ab
b
by 4>ac
=
] b
where the $ are probability amplitudes, Pab
2
== l^aftj
Pbc
;
==
2
l^&cj
?
Pac
=
2
|$ac|
,
SO Pic 5* Pac.
(XII. $.10)
Statement (XII. 2.10}, as discussed before, contains the fundamental difference between the calculation of probabilities in classical and quantum physics. it
had
Why P ^ Pac is
to go
c
clear.
For the particle to go from a to c classically, B had to have some definite value 6.
through a condition such that
The law of composition of probabilities as given by (XII. 2. 8} holds only where an attempt is made to measure B between the experiments A and C and where this measurement did not effect the result c found later by measuring
Probability
[XII.2.]
The statement
C.
measure
and Waves
347
B had some
that
value, even
if
one made no attempt to
a meaningless statement in quantum mechanics. The laws it, (XII.2.8) and (XII. 2.9) are to be used in accordance with our decision is
whether we do or do not attempt to measure B. Of course, a measurement of B cannot escape the limitations imposed by the principle of uncertainty. The attempt to measure B must disturb the system, at least enough to replace the law (XII. 2. 8} by the law (XII. 2. 9).
Now let ta
and
fe,
ta)
and
let \qi,
tO
K,
be the position operators corresponding to the times \q", t b ) be the eigenstates of q(t a ) and q(t b ), respec-
q(t b )
and
ta)
The amplitude
tively. |ffi,
q(t a )
and
<
ffl
an element of the unitary operator transforming
is
&
fc>,
\qi,
where
q'a
=
qi.
= Z7()|g, O,
k)
The transformation function <*
=
(qat a \qi%)
<
a& is
= <#! Z7(fe,
ta
(XII.2.11)
given
)\qX,
ta
by
).
(XII. 2, 12)
a transformation from a representation in which g(i a) is diagonal to one in which q(h) is diagonal. If we choose a time tb in the interval (ta t c )
This
is
,
we can
write the usual relation
=
SO
=
/ Mbc dqi".
(XII. 2.13)
Since the probability calculated from (XII.2.13) leads to interference terms, it is a typical representation of the wave nature of matter.
we can say that
When measurements
disturb the system the effect of this disturbance can
be taken into account by means of phase factors. Therefore, if (XII.2.13) is the transformation function before a measurement, then, after the measurement, the system being disturbed by an unknown and unpredictable amount, the transformation function is (XIL2.14) If
#(&), the phases T(g&) must remain unknown, so the resulting the square of the modulus of ac averaged over all phases. This
we measure
probability
is
<
gives
= Thus the
electron acts as a
particle. If
that
it
we attempt
behaves as a
wave
f if
(XII.2.15)
we do not attempt
to find its path, as
particle,
dqi
if it
were a
to verify that
particle,
it is
a
then we find
and (XII.2.15) holds.*
* R. P. Feynman, Rev. Mod. Phys., 20 (1948), 267. This paper contains Feynman's formulation of nonrelativistic quantum mechanics.
Probability in Classical and
348
[Chap. XII]
Quantum Physics
2 In order to construct a path we shall need more general forms of (XII. .13)
and (ZIIJ8J5), given by
The probability of the ured,
results a,
c,
fc,
for example, to occur
-
if 6, d,
are meas-
the classical formula
is
M a and c no attempt is made to measure the sequence of events between k will occur is and between c and k, then the probability that sequence a, c, If
*!
I
or
PL*
=
-
-
dqldq'a
\f
dqi 4>rtM>*
<M
'
2
(XII
-
.19)
XIL3. Definition of a Probability Amplitude for a Space-Time Path be an observable referring to the position of a particle, and let = 1, 2, be the operators corresponding to position measureg(fc), for z =e is a small positive time interval. The *_i ments at times U e, where can be used to define a quantum mechan&), operators gft), (fe), Let
q(t)
,
+
*
'
,
ical path.
The quantum frames
of references
corresponding to g(k) are con-
structed in terms of the eigenstates \q[, fe), (gi*, fe>, _ i 2 each defining a representation at time ^
,
|?i,
O
'
"
"
of
ffW? for
ti.
,
From gi, ft,
a classical point of view the successive values of the observables >the above path is define a path g'(0- In the limit e
",,
a continuous one. The probability that such a path will occur 0, kt us say. If fl*+i. of gi, qi, P('
g{
'
'
q'n,
'
,
,
'
'
,
a function
is
we
integrate
a particular region R, we obtain the probability ) ov ^r giH-i, g{, P(that of a path going through the q which lie in the region R. The probability so 6 and and a on, is -+i, lies between a* and 6< and gf+i lies between '
*
*
,
w
gj
4
integration is to be taken over those ranges of were actually within the region R. If all the #1, g2
The symbol B means that the the variables which
lie
,
R were taken, then the law measured, and only those lying in the region correct be quantum mechanically. If no such detailed (XII. 3.1) would also
[XII.3.]
A
Probability Amplitude for a Space-Time Path
measurements were
available, then the
amplitude for the region
=
lim 6
-0 J
>-
e
number
of Interest
the probability
is
by
given
$(,
U
The complex number In the limit
R
S49
q't,
.)
ql+ lt
-
dqtfqi +l
-
(XII. 3.2}
a function of the variables defining the path. $ will depend on the entire path q'(f) rather
the function
than just on the values of
at the particular times
g<
i*.
In this case we
call
r $[#'(0] the probability amplitude functional of the paths q (t).
We may now
state
Feynman's
first
postulate.
"(7) If an ideal measurement is performed to determine whether a particle has a path lying in a region of space-time, then the probability that the result will be affirmative is the absolute square of
from each path in
the
a sum of complex
contributions, one
region"
Feynman's second postulate gives a prescription for the computation of the probability amplitude function of the paths q'(t). The contribute paths equally in magnitude, but the phase of their con"(77) <
tribution is the classical action
S
(in units K),
i.e.
the time integral of the
Lagran-
gian taken along the path."
The
contribution $[q'(t)] from a given path exp {(i/K)S[q'(t)~\}j where the action S is given by
q'(f)
L[q'(t), q'(t)] dt
proportional to
(XILS.S)
the classical Lagrangian. The action S[q'(f)] must be computed for possible paths q'(t) and not just for one point q'(t). The action integral
and all
L
is
is
(XI1. 8. 8)
is
a
minimum
for a classical path (see Section III.2.A). This sugS as the sum of the actions corresponding
gests that we can write the action to the time intervals (k, t i+^) :
(XII. 84)
J), i
where SCft'+i,
t)
= min P* L[q'(t), Jti
q'(t)] dt.
(XIL8.fi
In Feynman's formulation of quantum mechanics the use of the correspondence principle in taking over the classical Lagrangian differs from Schwinger's formulation in that it does not make use of a Lagrangian operator corresponding to a given classical Lagrangian. Feynman's theory deals directly with the transformation functions and not with their differential characterizations. The latter procedure is the main starting point in Schwinger's action principle,
detail (see Section V.5.B).
which we have already discussed in some
Probability in Classical and
350
Because of the
infinite extent of
Quantum
time the
sum
[Chap. XII]
Physics
in (XII. 3. 4} is infinite
and
hence meaningless. This, however, is not an important obstacle and will be clarified as we proceed. For the present let us restrict ourselves to a finite,
but
and detailed discussions of
arbitrarily long, time interval. Interesting
various problems raised
by Feynman's formulation have been given
in the
literature.*
Feynman's postulates
(I)
and
with (XII. 34) can be com-
(II) together
bined into the statement that
= Km
exp
S(qUi, go
(XII. 3.6}
,
e-*0
where the IfA are normalization factors responding to the time intervals
for the transformation functions cor-
(t t +i
ti
=
e),
so each transformation is
weighed equally. The constant A will be determined later on. The integration in (XII. 3. 6) is taken over all values lying in the region R. $, gf+i, with the physical in3. and definition coupled (XII. 5], Equation (XII. 8.6} ,
the probability that the particle will be found in R, constitute Feynman's formulation of quantum mechanics. terpretation that
\(t>(R}\* is
XII. 3. A. Feynman's Definition of the Wave Function
A quantum version
with "fixed end points" used in classical mechanics. How-
of the principle of least action
cannot be understood in the manner
it is
ever, following Dirac, for a finite time interval
L(f) <#
exp
"U
(ta , fe)
etfAWw.)
=
J
so B(fe, ta) corresponds to (q'tb\q"t a ) in
quantum
we can
B(tb
theory.
q" correspond to the eigenvalues of the operators q(t a )
and
tb
,
intervals t i+i
ti
=
e.
(t a , fe)
Then we may write
B(t,
The numbers and
into a large tQ }
Y. Nambu, Prog. Theor. Phys., Japan, 7 (1952), 131.
W. Tobacman, Nuovo
Cimento, 10 (1956),
3,
1213.
K. Symanzik, Z. Naturforsch, 9a (1954), 809. K. Goto, Nuovo Cimento, 10 (1956), 3, 533. F. Coester, Phys, Rev., 95 (1954), 1318. M. Jauch, Helv. Phys. Acta, 29 (1954), 287.
J.
G. Wick, Phys. Rev., 80 (1950), 268. Van Hove, Physica, 18 (1952), 145. W. Thirring, Ann. Phys., 3 (1958), 91. L.
,
q(tb)
q'
and
at times
ta
respectively.
Let us now divide the time interval
*
write
as
number
of time
A
[XII.3.]
Probability Amplitude for a Space-Time Path B(t,
f
=
)
B(t,
t
m )B(t m
,
The corresponding quantum equation
f
m -0
is
-
-
B(t* tJBfa,
351
k).
(XII.3.8)
in terms of transformation functions:
(XII.3.9)
Each
#
in (XII.3.8)
must be regarded as a function of the g at the two ends which it refers. This makes the right side of (XII.3.8)
of the time interval to
a function of B(t,
all
the intermediate
stationary, the
tQ)
quantum
Since the small variations of the q leave analogue of the action principle is contained q.
law of the transformation function given by (XII.3.9). This relationship has first been pointed out by Dirac. Feynman has extended the above procedure to a time interval of infinite extent. in the composition
The quantity
can be written as an
<
from successive sections
arising
infinite
of the path.
By
product of contributions it tend making e = t i+i
we
obtain a continuously infinite product of transformation functions a "quantum path." This method can be used to define a quantity generating the properties of a wave function. having to zero,
A more
abstract definition of a
transformations of
Feynman path can be based on
the unitary
quantum mechanics. The composition property f
unitary operator U(tt
),
referring to a time interval
=
U(tt')
Z7(tfOC7()
(t, t')
}
is
expressed
U(t^tf),
of the
by
(XII. 3.10)
which can be generalized into U(tt')
= j[ = n
(XIL8.11)
C7(O,
l
where Cn
The transformation (q'\U(tt')\q')
=
=
tn
tn-1,
Hm n
tn
=
if'.
>co
function can be expressed, therefore, as
lim [ >oo J
n
f <'|C7(0|gi> di<3i|
J
W)l> (XII.3.12}
This method is suitable for further formal developments of Feynman's ideas. To continue with our discussion of Feynman' s postulates (I) and (II),
we choose a
particular time
three parts: (a) a region
region
R"
K
with respect to which the region R consists of lying earlier in time than some t'(t < ); (b) a
t
r
lying later in time than some t"(t" > )] (c) the region between t' 1 all the values of q coordinates are unrestricted (Fig. 12.7).
and ^'in which
We
have
f
\
,
R")\* as the probability that the path occupies R' and R".
352
Probability in Classical and
*"
Quantum Physics
The probability can be interpreted mean that if the path has been
to
be found in R".
in R', then it will
Thus that
if
\4>(R',
malized, if
is
properly nor-
the relative probability
the system
was
in region
be found later in R".
it will
FIG. XII.7.
[Chap. XII]
If
R
r ,
we
assume that the particular time t corresponds to one particular point Schematic
diagram
of
k of the subdivision of time into
region R.
steps
that
e
is,
we take
if
t
=
fe,
where the index k depends on the subdivision t then the exponential in the integrand of (XII. 3. 6) can be split up into a product of two factors, .
exp
The
first
*
r*-
-i
IV
exp
A-i
ft
factor contains coordinates with index k or higher, while the second
contains only coordinates with index k or lower. The integration in (XII..8.6) all g*, for i > k can be performed on the first factor, and it results in a
over
}
function of qi (times the second factor). Next we integrate the second factor over all variables #*, for i < k. This results in a function of qi. The final form of (XII. 3.6)
is
',
R")
=
(XII.3.14)
where
2
-
(XII.3.15)
,
dqL
(XII.3.16)
.
The
asterisk (on %*) denotes the
complex conjugate
of
% The function ^
depends only upon the region R' previous to t, and completely defined that region is known. It is independent of the future of the system with respect to the time t. The information for the future of the system is contained it is
if
in %.
The
\f/
and
x, defined in
the above manner, serve to separate past and Thus the function $ completely determines
future experiences of the system.
the past state of the system, and the description is a causal one. The function ^ can be used to predict future probabilities. This the following sense:
we could choose a time
have corresponding past and future regions
different r'
(XII.3.14), the probability of the path ending in
and
from
r".
t,
is done in and we could
Then, according to
any region R" would be the
A
[XII.3.]
Probability Amplitude for a Space-Time Path
353
same for R as for r', because all that we have done is to shift the strip between t" and t'. This means that future measurements will not distinguish whether the system had occupied R or r'. In this sense the wave function ^ is sufficient to define those attributes that are left from past history which determine future behavior. We must remember that the wave function determines future probabilities only, and it does not contain deterministic information f
r
about future events. Regardless of the preparation of state ^, equation (XII.8.14) asserts that the chance of finding the system in R" is always the
same
as finding
in r".
it
Finally, the probability that a transition
place
from state # to
state
x
will take
is '
*ZT-
(XIL3.17)
XII. 3. B. Feynman's Derivation of Schrodinger's Equation In the following
it will
be assumed that the Lagrangian
system is time development of the wave function, it finite time difference e, equation (XII. 3. 15) a simple recursive relation. At the time (XII. 3. 15) 1
,
t
+
t
+
is
L
of the
wave equation
quadratic in the velocities. Since the
dynamical
describes the
natural to expect that for a permit the development of
will e
the equation corresponding to
is
e)
=
exp
S(gj +1> i
ass
-
gO
(XII. 3.18)
,
00
which can be written as
+
)
=
+
6)
=
exp
q +l g ,
exp
Hence ^( SJ +1
,
t
exp
This gives the time development of the wave function
i(g', Q.
By
a suitable
equation (XII. 8. 19) must lead to Schrodinger's equation. *- 0. We is not exact, but is only true in the limit e
choice of A,
Equation (XII. 8. 19) can, therefore,
(XI1. 3 .19) to equation.
make
the
first
e
as small as
order in
e is
we
please.
Assumption
of the validity of
sufficient for the derivation of Schrodinger's
Probability in Classical and
354
We
Quantum
Physics
problem in which a
will consider a one-dimensional
[Chap. XII]
single particle is
acted upon by a potential V(q ). In Cartesian coordinates the path of a free particle is a "straight line." The integral (XII.3.5) can be taken along a f
and
straight line, classical
This
is
to the first order in
e
we may use
the procedure used in the
theory (111.2.14} and write
the same as
In the absence of a vector potential and other terms write S(qi + i,
= ie
80
~
^-
(^
we
can, as in (1 11. 8
.
^Crf-n).
Hence (XII. 8.19) becomes +lf
Let g+i
^, +
+ = e)
e)
q'
=
-
/
exp
and gi+i exp
/
-
-
gi
=
j^f ) exp
The appropriate boundary
,
so g*
=
g'
-
|.
Then (XII. SS)
[^] ^ -
condition on $(q
f ,
f)
f
7(^0]} *
{f [2 (Ski.^)'
,
f
'
leads to
(ZJI^)
'
can be imposed by requir;
ing that the integral (XII. 3 $4) converge. This means that ^(g t] must fall since e is very small, off sufficiently for large q'. In the integration over ,
,
exp (im^/2h)
Thus
it is
oscillates
only a small
the order (&e/m) 1/2
.
We
extremely rapidly, except in the region about that contributes effectively to the integral, for
can expand $(q
We shall now use the relations
f
f,
as a Taylor series.
Hence
=
0.
is
of
A Probability
[XII.3.]
Expanding the
Amplitude
left side of
for a
Space-Time Path
(XII. 3.25) to the
order in
first
e,
we
355 get
L
To have both
sides agree to zero order in
we must
Hence, expanding the exponential containing V(q
which
f
),
set
we
obtain
Schrodinger's equation for a one-dimensional system. Equation (XI 1. 3. 19) gives the time development of the wave function is
e. We can compare it with Huygen's principle and regard it as Huygen's principle for matter waves: "I/ the amplitude of the wave is known on a given surface the amplitude at a nearby point can be considered as a sum of contributions from all points of the surface. Each contribution is delayed in phase by an amount proportional to the time it would
during a small time interval
in optics,
take the light to get
from
the surface to the point along the ray of least time in
geometrical optics."
For a time-independent Hamiltonian the Schrodinger equation
can be solved by
fat) Putting
t
tQ
=
e,
=
e
we can write it fa
t
+
as
e)
= e-MWfa
(XII. 3.32)
t).
Hence
(jfa
*
+
e>
=
/ tfle-WWlq") dq"
(q"fa
t)
or
j
A
f
(xn.s.ss)
.
with (XI 1. 8. 19) shows that it expresses the an approximate integral operator for small e. quantity (q \Q" \q ) by But exp [( i/h)tH] is a unitary operator and therefore we can construct an e-dependent position operator:
comparison of f
this equation
(i/h)H
ff
q( e )
=
e (tVA)ff0 oe -(tVft)ff.
Equation (XII. S. 88) asserts that the wave function
&(q' 9
1
+
e)
at time
356
Probability in Classical and
Quantum
Physics
[Chap. XII]
+
e representing a state in which g(c) is diagonal, is related to a wave function representing a state in which q corresponding to time t is diagonal, through
t
a transformation function, {g'le-CH/awrig")
= I e vnsw
For further discussion and applications
of
Feynman's formulation the by Mathews and
reader should consult the papers by Polkinghorne* and
Salam.f *
J.
G. Polkmghorae, Proc. Roy. Soc. London, A, 230 (1955), 272. Mathews and A. Salam, Nuovo Cimento, 10 (1955), 2, 120.
t P. T.
(XII. 3.34)
CHAPTER
XIII
ENERGY LEVELS AND SYMMETRIES OF SIMPLE SYSTEMS
XIII.l.
The Hydrogen Atom
The hydrogen atom, like a harmonic oscillator, is another physical system that can be treated exactly both in nonrelativistic and relativistic quantum it
theory. Historically, in physics,
tions
and interesting
It is therefore
discussion.
has always been a "test atom" for the new theories
and every new attempt
for its
study gave
rise to further
discoveries about the interaction of matter
worthwhile to devote to the hydrogen atom a
The hydrogen atom
observa-
and radiation.
fairly
complete
has, essentially, a simple spectrum, but its
observation requires very high experimental precision. The basic characthe atomic hydrogen spectrum are found to be in the visible region,
teristics of
A strong red line
called
Ha has a wavelength 6563 A and appears in the solar
absorption spectrum as a Fraunhofer line (usually called Fraunhofer Besides the red a line there are strong blue Hp lines and violet
H
H
their respective
Fraunhofer
Hs
line,
stars
A
wavelengths are 4861
F and
C
lines.
A further
and 4340 A, and they correspond
to
line falling into the visible region is the
with wavelength 4101 A. These
and they can be produced
line).
y lines;
lines are
in the laboratory
observed in the spectra of in a by electric discharges
hydrogen gas.
The most recent experiments on the hydrogen spectrum have used the method of electron bombardment. In this method the hydrogen gas is excited by a controlled beam of electrons, which produce excited atomic states in hydrogen. In this
way
the fine structures of various atomic spectra have been
studied exhaustively. 357
358
Energy Levels and Symmetries of Simple Systems
[Chap. XIII]
XIII. 1. A. Helativistic Treatment The
limiting case of the relativist ic
the effects arising from the is
hydrogen atom can be deduced as the theory. The Hamiltonian, for the case where
nonrelativistic theory of the
given by
finite electron
H
=
ca-p
and proton mass
+ /Me + 2
ratio are negligible,
(XIII. LI)
7(r),
the potential energy of the electron. Separation of the total can proceed as in the neutrino Hamiltonian angular momentum part of The (P is now to be replaced by the corresponding 8. (IX. 30). parity operator
where V(r)
is
H
parity operator & of the four-dimensional representation [see (VIII. 5.4)]Hence the constant of the motion corresponding to (IX. 8. 87) of the neutrino is
in the present case given as
f
which
of course
in (IX.,8.88)
is
=
+
jff(a
(XIII. LS)
o-JB),
commutes with the Hamiltonian (XIII. 1.1). The operator
by
replaced
= --r.
OT
Thus the Hamiltonian
of the
(XIII. LS)
hydrogen atom in polar coordinates takes the
form
H=
f)
cvjfa
+
Pmc*
+
^K +
(XIII.14)
V(r),
<x r and {$ anticommute with the rest of the operators in H. The eigenvalues f of Dirac's f operator replace the quantum numbers Z and j. The eigenfunctions of H are of the form (IX.8.84) except that the spin vectors |1) and 1} are now four-dimensional spinors. However, the spherical sym-
where the operators '
J
|
metry allows the reduction into two sets the same structure. For example, because Xi
=
xz
=
and x3
=
r
and obtain a r
and
=
of
two-component equations with symmetry we can set
of spherical
as In this case we may replace the the anticommuting operators o-2 and J3 .
/3 by anticommuting operators respectively. The Hamiltonian becomes
<*3
H
=
c(p r
-
^]
Hence the eigenvalue equation
me 2
+V-E he
,
(XIILLff)
V(r).
E\$) can be written as
H\\l/)
he --
--
r
^cp "r
-me + V - E 2
=
0.
The Hydrogen Atom
[XIII.l.]
Dropping the indices j and
-If,
we
359 finally
have the equations
where
=
1 (^
1 = 1 On* + E),
-15),
U\q\
= f = ay
+
(XIII. 1.8)
1).
For the attractive coulomb force the potential energy
is
V(r)
=
e 2 /r
and
equations (XIII. 1.6) become
ri
eel -
La_
rj
~ + al -
ra
w
T[_dr
rj
i ~
T-
t;
L^
rj
,
r.
r,
La+
--i + -b = a =
* sl w
rj
0,'
0,'
where we have used the substitutions /
=
- 6~ r / a w,
g
=
- e~ T /a v,
a
<
= V(a+ a_)
(XIII. 1.10)
that are suitable for the discussion of a bound-state problem.
a is
=
The constant (X1II.1.11}
the Sommerfeld fine structure constant.
The number
q
assumes the values
Equations (XIII. LIST) correspond to a selection rule imposed by the conservation of parity.
The most general solutions of equations (XIII. 1.6) have been discussed by Darwin.* Dirac has solved equations (XIII. 1.9) for the hydrogen atom.j
To (l/c)f
(XIII. 1.6) we put G, and obtain the exact equations
find the nonrelativistic
=
F,
and
g
= ib
ilG
limit
I
of
I
OT
T
E=
me 2
+
T,
I
(XIII. 1. IS)
*
f
C. G. Darwin, Proc. JKoy. Soc. London, A, 118 (1928), 654. P. A.
M.
Dirac, Principles of
Quantum Mechanics, Oxford Univ.
Press, Oxford, 1960.
360
Energy Levels and Symmetries of Simple Systems
The second term on the left side of the first equation to 2m/h, and so we get the approximate equations
is
[Chap. XIII]
negligible
compared
(xm.i.14)
2mldr
(XIII.1.15)
we
Eliminating G,
where we use the
To
obtain
relation q(q
solve equations
We first
=
1)
(XII1. 1.9)
cast the equations in the
a
-u r
+
1(1
as follows
1),
we apply a power
series
from (XII1.1.12}. expansion method.
form dv
u = -h
v -,
-
u
.
T-
a_
dr
v
a
2 where we observe that the definition of the constant a by a same as the vanishing of the determinant of
"JL
!'
a_
a
1
J_
a
&+.
(XIII. 1.17}
=
a+a_
is
the
(XIII. 1.18} _.
For the power
series solution let us
u = ]T
assume that v
Csr*,
= ]T df,
where the exponent values
s 0j s
+
1,
o
s
+
(XIII. 1.19}
s
a
begins with an initial value s and runs through the Substitution of (XIII. 1.19} in equations 2, .
(XIII. 1.17} yields the recursion formulas ds-l
q)d 8
,
Ca-4^
a (XIII.1.20}
-ad,
+
(8-
q)c,
=
=3 tfr-f
From
the vanishing of the determinant of (XIII. 1 .18}, it follows that the determinant of the coefficients d s and c s on the left of (XIII. 1.20} must also vanish.
Thus the
initial
value of the exponent
-
a 2 ),
s is
given by
(XII1. 1.21}
The Hydrogen Atom
[XIII.l.j
where the negative value
361
discarded; otherwise
Is
it
would lead to an
Infinite
probability of finding the particle at the origin. If
we now multiply the first of (XIII. 1.20} by a and we obtain
the second by a+ and
subtract the results,
[aa
-
-
(s
+
q)a+]c,
+
[aa+
+
(s
=
q)a]d s
0.
(XIII. 132)
The
probability of finding the particle at the origin must vanish. This boundcan be incorporated by requiring that rf and rg must tend to condition ary *- 0. The existence of discrete zero with r energy eigenvalues depends on the
convergence of the series (XIII. 1.19). For large
the second of equations
s
(XI II, 1.20) and equation (XIII. 1.8$) yield sc s
ds-i
c s _i
=
=
a+c s
[-
a
a+
,
ads
.
Hence
=
Jii_
A^
d* ds-i
as
c s-i
7
so the series converges:
If
E > me
a bound
2 ,
then a
state. If
pure imaginary and therefore it does not correspond to < me*, then for positive a the series must break off on the
is
E
we would get an infinite probability of finding the at distances. Suppose that the series (XIII. 1.19) terminates large particle with the terms whose coefficients are c s and ds In this case we must have side of large s; otherwise
.
=
and d 8+ i
=
0; then equations
,
a_
a
(XII'I.I. 20) yield a+
a
Using (XIII .1.22) to eliminate d we get a+ (s
a_[aa
g)]
=
+
a[a+ a
a(s
+
q)].
Hence the energy eigenvalues are
E = we where
s differs
from
s
/ 2
f 1
by an integer n s
=
$o
+n=
The corresponding wave function 2_ a2) e -r/a( r } V( a If
we
neglect
depends on n
a2
+
=
(1/137)
|g|.
The
2 ,
^
/v 2 \-l/2
+ yj 2
^
0.
Thus a2
2
q is
+ n.
form
of the
p iy nom i a ] Of degree
compared to
states with
g
(XIII.1.
,
2 ,
n.
then the energy
quantum numbers
J?,
(n, q)
effectively,
and
(n,
g)
Energy Levels and Symmetries of Simple Systems
362
same energy eigenvalues
ha.ve the for
which
=
I
or the
[Chap. XIII]
spectral lines with the
two
same
j,
j db J exactly coincide.
derived by Sommerfeld by the use of Bohr's orbit theory. Its expansion in powers of or leads to
Formula (XIII. 1.23) was
E-
where
The
= -
me*
E
first
+ -5- fi -
and from the
of
to .V
1, is
+
2
|g[)
.
limit,
The
nonrelativistic theory.
=
N
the nonrelativistic theory,
In the nonrelativistic (n
(XIIL1.24)
"I
.
+
n
principal
=n+
\q\
replaced in the relativistic theory
+ =
+
1
2 the so-called "Rydberg constant/' defined by R = %e*m/h term in (XII LI .24) corresponds to what we obtain both from Bohr's
ranging from
AT 2
*
is
orbit theory
number
-T; ~1
first
6
2
)0
where a
+
and
>-
The corresponding energy
+
V(a 2 level
I
+
s
with
I
by
s 2 )]. >-
quantum 1,
(XIII. 1.25)
+
n
|
we
obtain
formula and eigenfunctions
are
EN = ~
WW
(XIII.1.26)
and
Fn lm=
Rnl(r)Y lm (6,),
where the radial part of the wave function functions*
by
Hn-l"!
~L[(
+
PIT/' pZT/s 6 2nJ L n J
3 !]
where the Laguerre function L?(p)
U(p)
They
=
^
is
L,(p),
is
defined in terms of Laguerre
^ L^ + r2Zr-| "
'
L n J
denned by L,(P)
=
ep
^
(-"P
F
).
are normalized according to
XIII.2. Accidental
Degeneracy and the
Momentum-Space
Description of
Bound
States
Because of the spherical symmetry of the potential energy there exist many hydrogen atom with different magnetic quantum number
states for the
*
A
In this book we shall not be concerned with the details of the eigenfunctions. plete discussion of the relativistic and nonrelativistic eigenfunctions of the hydrogen is contained in H. A. Bethe and E. E. Salpeter, Quantum Mechanics of One- and Electron Atoms, Academic, New York, 1957.
com-
atom Two-
[XIII.2-]
L3 =
Degeneracy and Momentum-Space Description
mh, which have the same total angular
momentum
Z
363
and the same
energy. This corresponds to a degeneracy in the hydrogen atomic levels and it is a consequence of the invariance of potential energy with respect to the
Thus
invariance under the operation of the elements of a symmetry group, together with certain conservation laws, leads to some kind of forced degeneracy of the system. In the following we shall show that a symmetry group in a given representation for example con-
three-dimensional rotation group.
figuration space or q representation need not be the only one pertaining to a given system in another representation momentum-space representation or p representation. In some quantum mechanical systems a certain symmetry that does not occur in a representation may reveal itself by going to another
representation. This
only arise
is
a "hidden symmetry" of a dynamical system and can of the system under a different group of
from the invariance
transformations. Historically, a
spectrum of the
symmetry
of the
hydrogen atom.
It
above type was observed by Pock* in the was found that for a given principal quan-
and the z component of the angular tum number n, angular momentum momentum, m, there were n* degenerate states instead of (21 + 1) with 1 and m values from to n Z to Z ranging from This "accidental deI,
Z.
generacy" was due
to a higher
symmetry
in the system.
Such accidental
degeneracies are connected with the existence of further constants of the
motion than can be recognized in a particular representation with a definite symmetry group. The latter was observed in the early stages of quantum mechanics by Pauli.f Such symmetries were
later investigated for various
systems by others. J An elementary discussion of the accidental degeneracy is summarized by Mclntosh. A general formulation for Kepler-type problems has recently been discussed by Alliluev.|| We shall try to give a fairly comprehensive account of accidental degeneracy for the hydrogen atom from a group-theoretical point of view. But for this
purpose we must
*
t
first
give a
momentum-space representation
of
a bound
V. Fock, Z. Phys., 98 (1935), 145. Pauli, Z. Phys., 36 (1926), 336.
W.
JO. Laporte, Phys. Rev., 50 (1936), 400 (A). V. Bargmann, Z. Phys., 99 (1936), 576. 0. Laporte and G. Y. Rainich, Trans. Amer. 'Math. Soc., 39 (1936), 154. A. W. Saenz, unpublished Ph.D. thesis, University of Michigan, 1949. J. M. Jauch, Phys. Rev., 55 (1940), 1132 (A). J.
II
M. Jauch and E. L. Hill, Phys. Rev., 57 (1940), 641. H. V. Mclntosh, Am. J. Phys., 27 (1959), 620. S. P. Alliluev, Soviet
Phys.
JETP,
6 (1958), 156.
state.
364
Energy Levels and Symmetries of Simple Systems
XIII. 2. A.
The Momentum-Space of Bound States
For the stationary state with energy <j>(p, f)
in the
wave equation
[Chap. XIII]
Description
E the momentum-space wave function
(V.2.80) has the
form
In this case the wave equation (V.2.30) becomes an integral equation,
E~ (
where V(p
p') for
Coulomb
potential energy F(r)
=
Ze 2 /r
is
given by
[see (VJtJSf)]
The integral on the right can easily be performed. To we introduce a "screening" factor e~ r where e is a ,
simplify the calculation positive small
number.
Introducing polar coordinates with the polar axis in the direction of the fc-axis, we obtain
7(p where k
The
- pO =
= (l/A)(p-pO-
integration leads to
Hence the Fourier transform
The
r^ " ^n e dO dr, 00
~|~-3 /
of F(r) is
integral equation (XIII.2.1)
becomes
For bound-state solutions we write
E=
W, where
W
>
0,
and impose
- oo
the function
|p|
the same as the vanishing of the probability of finding the particle at infinity. Thus the integral equation for bound states is
2m
Degeneracy and Momentum-Space Description
[XIIL2.]
The wave function
satisfies
the normalization condition 2
j If
we put p' = p
q and
(p, 6,
i4>(p):
d*p
=
1.
(XIII. 2.5)
+
q and assume that (q, 9, $) are the polar coordinates of <) those of p, with respect to a fixed direction, we can write
= fiMY^O,
where /(p)
365
0),
(XIII. 2.6)
[or /(
given by cos a
=
+
cos e cos
sin
sin
The angle between p and q is
6
cos ($
<).
(XIIL2.7)
Hence, the integral equation (XIII. $4} becomes
=
Ze* rr-^
I
-
r /
sm ^ 6 .
i
+ 2m By
using the addition theorem and the orthogonality property of spherical can be reduced to a one-
harmonics the integral equation (XI1I.2.8) dimensional integral equation of the form 1
f
*dqqQi
where the Legendre function function of the
first
kind, PI,
of the
second kind,
Qi, is related to
the Legendre
by
'(*)
=
*
du f\A 7^7, M/
J
-
(XIII.2.10)
JU
The integral equation (XIII. 2.9) can be solved exactly. The discrete eigenvalues correspond to the energy levels of the hydrogen atom as obtained from solving the Schrodinger equation in the configuration space. The bound-state radial momentum space eigenfunctions for hydrogen are
UZ where C^
is
expansion of
4
-
a Gegenbauer function, defined as the 2ax + a 2 )~x (1 :
coefficient of a" in the
366
Energy Levels and Symmetries of Simple Systems
The Gegenbauer function* (1
[Compare
-
x*)
is
g-
[Chap. XIII]
a solution of the hypergeometric equation
(2X
+ 1)*^ + n(n +
2X)(?
=
0.
this with (VIII.7.1S).]
2mE in (XI 1 1. 2. 8), we can Now, putting pi = (XIII. 2.4) for bound states in the form
rewrite
equation
The HamUtonian in
configuration space is invariant under the transformations of the three-dimensional rotation group. These transformations are irreducible.
But, as written in the form (XIII. 2.11), the bound-state problem in momentum space exhibits a symmetry of the four-dimensional orthogonal group.
This higher symmetry of the system can be represented by
all
possible motions
of a hypersphere of unit radius in four-dimensional Euclidean space. The latter can in turn be represented by orthogonal transformations in the equatorial
plane consisting of the stereographic projections of the points of the hypersphere. This means that the corresponding symmetry group is reducible and
one speaks of "accidental degeneracy." By analogy with the three-dimensional stereographic projections discussed in Section (VIIL4, we may regard the vector pi/p*, for i = 1, 2, 3, as the stereographic projections on the equatorial plane of a point
qi, q%,
(ft,
q*
on the
hypersphere flf
+ ql +
ql
+ ql
=
1.
(XIII. 2.12)
Thus we can write 2i
=
2pip 4
p = 2
2p 2p4
_
p?
+ pi +
pi.
Hence, solving for the p, we get
14,
p
=
Pi
=
r
where
P + 2
From
(XIII.8.14)
we
^
g
2
=
<&
+^+
flS-
obtain
* W. Magnus and F. Oberhettinger, Formulas and Theories for the Special Functions of Mathematical Physics, Chelsea, London, 1949.
[XIIL2.]
Degeneracy and Momentum-Space Description 2 !*-p'! =
is
^jr^-r^f
the "angle" between two points = cos w, with given by
where
co
Q and Q
Is
f
of the hypersphere
and
q^
= =
cos w
cos 7
The polar coordinates
=
#3
7
cos X cos X
cos 6 cos
0'
+ sin X sin X' cos 7, + sin 6 sin cos 0'
for the q are defined
sin 6 cos
qi
By
367
cos
ski X,
<
#
as a function of
by
= =
#2
sin X,
4
sin
cos
#1, # 2 ,
sin
sin X,
X.
the Jacobiaii for the transformation
regarding p to the q can easily be calculated. Thus q4
<')
(<
from the
-- -
we have
where in polar coordinates we have sin 2 X $4
d
(sin -
\)- sin
dd d6 -
=
COS X
sin 2 X sin
*
Hence the projected wave equation (Kill. 2.11) becomes
where we use the substitution <j>
= A(l
+g
2
4)
(XIIL 2. 17)
^,
A
being a constant. Equation (Kill. 2.16), as can be seen from the manner of its derivation, is invariant with respect to the group of four-dimensional orthogonal transformations. The latter is the hidden symmetry group of the bound-state problem for the hydrogen atom. The eigenvalue of the integral equation (XIII.2.16) is Zme*/hp4) expressed 2 by Zme /hp4
=n+
1
or
[Compare this with (XII 1. 1.26).] As an interesting exercise the reader should show that the
integral equation (XIII. 2. 16) can be replaced by the differential equation (VIII. 7. 8) for the four-dimensional orthogonal group. In this way we observe that the solutions of (XIII.2.16) are just hyperspherical functions defined by (VIII.7.16).
These functions provide an irreducible representation
of the group.
368
Energy Levels and Symmetries of Simple Systems
[Chap. XIII]
XIII .2.B. Accidental Degeneracy and Motion in a Constant
Magnetic Field
The nonrelativistic theory has been discussed by Kennard, Darwin, Landau, Page, and Uhlenbeck and Young.* A more detailed discussion, involving various symmetries of the problem, has been given by Johnson and Lippnaann.f Here we shall begin by establishing a complete analogy with the one-dimensional harmonic oscillator discussed in detail in Chapter VII. We shall assume a constant magnetic field in the z direction in space. The Hamiltonian for an electron in a magnetic
H where
it
=p+
=
(XIII. H .19)
~^ **,
(e/c)A satisfies the commutation relations !>*, *vl
From
given by
field is
= -
fu
fa,
=
Heisenberg's equations of motion, ih(dir/dt) }
dt
where
aj c
is
'
dt
the Larmor frequency
co c
[see
(XIII.S.80)
e*i3ez.
=
[ff, -ff],
dt
(X.4-S)]:
we
obtain
'
jloe
= /
The ir s
=
(XI11. 2 M)
third equation of
T03.
potential
Using
A
so the constant of the
of the
|(3C
X
motion
7r 3
is
two equations TT+
in
=
=
by
just
pos
=
constant.
(XI11. 2. 21) can 6^7r +
motion,
easily infer that the vector
r) 55 (-iofcSe, i&iSC, 0),
pz first
A, we
for a constant magnetic field in the z direction is given
A =
The
yields the constant
= VX
the equation 3C
x_
,
readily be solved
=
e-*^irl,
(XIII.2.22)
by (XIII.S.SS)
where TT-jL
=
TTi
dh
2-7T2.
These solutions are quite similar to those of the one-dimensional oscillator represented by (111. 1.35).
classical
harmonic
* E. H. Kennard, Z. Phys., 44 (1927), 326. G. C. Darwin, Proc. Roy. Soc. London, 117 (1928), 258. L. Landau, Z. Phys., 64 (1930), 629.
L. Page, Phys. Rev., 36 (1930), 444. Uhlenbeck and Young, Phys. Rev., 36 (1930), 1721. fM. H. Johnson and B. A. Lippmann, Phys. Rev., 76 (1949), 828; 77 (1950), 702; 78 (1950), 329 (A).
[XIII.2.]
By
Degeneracy and Momentum-Space Description
369
introducing the operators a + and a_, 1
v +>
a-
1
=
the Hamiltonian (XIII.8.19) can be rewritten as
H
=
(N +
Ra> e
+
I)
^
(XIII. 2^4)
TT!,
N
is defined by JV" = a+a_ and has where the occupation number operator the same properties as the one introduced by (VI I.I. IS). The operators a+ and a_ satisfy the commutation relation [a_, a+J = 1.
From
(XIII.2.22) and the form (XII 1.8.84) of the Hamiltonian we see that the transverse and longitudinal motions of the electron are independent as in classical theory. The motion along the magnetic field that of a free particle. The eigenvalues of the motion, as follows from the form of the Hamiltonian, are given by of one another is
E=
o
+ I) + ~^ pi
(n
(XIII. 2 M)
where the first term refers to the eigenvalues of the transverse motion and the second term to the continuous eigenvalues of the free motion along the magnetic field.
=
Heisenberg's equations of motion, ih(dxt/dt) *E! dt
_ ""
JL
m
^2_! "
"
71
1'
m
dt
for the position of the electron hence
Z3
we
=
lp +
=
%
3
[#i, H"]^ } yield
fe ~
"
71
2'
dt
1 "
m
71
3
easily obtain
o a;
and
where
rc
=
Xi
ix%.
a
i
The commutation
,
relation
[a;?-,
#+] =
27i/mco c
gives the result [T-? 3/1,
Hence we
i /-v^i ??ZCU cX2j|
-VM /
(imT py} \j&.JLJLJL.a.&i) $>
-?"fi
^/t'*
infer that the canonically conjugate observables
a??
and
mu xl c
obey
the uncertainty relation
ArfAxi
^
(XIII.2.28)
3 772-COc
Therefore
it is
not possible
particle exactly.
initially to locate
However, at a
later time
t
the center of the "orbit" of the
it is
possible to measure the center
Energy Levels and Symmetries of Simple Systems
370
of the orbit,
of the transverse coordinates
by a simultaneous determination
and momenta, which are subject
to the uncertainty relations
Ax2 Ap2 S P-
^ p,
AxiApi
[Chap. XIII]
we
note that because of the commutation relations (XIII. 2.27) the eigenvalues of Xi and x will coincide with the continuum of real numbers. Since or? and ar commute with the Hamiltonian, then to each energy will correspond an infinite number of eigenf unctions. Hence it follows that the
Finally,
is
energy
infinitely degenerate.
The hidden symmetry
of the
motion can be seen by writing the Hamil-
tonian (XIII. 2.19) in the form
~-
H= =
HL
&wi
(pi
+
+
pi)
=
~ &
+ xl) + HL
nu**(xl
(XIII. 2.29)
,
=
+
3 o^pi- The first Xip 2 iuJ/3, and (l/2m)pl a two-dimensional Hamiltonian. of the refers to part of the Hamiltonian and L 3 isotropic harmonic oscillator. Furthermore, HL commutes with
where
o>
ico c
,
H
H
commutes with ps The transverse part
of the
.
in the
form fftr
=
hu(Ni
+ N* +
1)
a
=
where the operators a
,
a satisfy the
commutation
for
=
=
+ 0^2") +
defined
(XIII.2.80)
ft,
by '
?
v
ma^)>
relations
c(i
5
af
^a ^ V//^ (2mw/i)
eigenvalues of the motion are
E=
+ 7ko(ai
1, 2,
[a.-,
The
Hamiltonian can be written
a/] = ft*. now given by
+ n* +
1)
+
r^- pi 11%
t
(XIIL2.S1)
+
5
^m^
(XIIL2.S2)
c,
N
where ni and n^ are eigenvalues of the occupation number operators a = 1,2. The degeneracy arising in the transverse part of the a-aday for a motion
is
oscillator.
of the
type discussed for the three-dimensional isotropic harmonic
Hence there
is
(n
+
l)-fold degeneracy,
where n
==
n\
+ n^.
we regard the
operators of, a as the components of a "spinor" in a two-dimensional space, then the Hamiltonian Htr as given by (XIII.2.80) If
is
invariant with respect to the group of unitary transformations
(XIII. 2. S3)
where the infinitesimal generators of trices
a-f.
The group
U
transformations are Pauli spin ma(XIII. 2 .33) is isomorphic
of unitary transformations
to the three-dimensional rotation group.
The
latter is the
hidden symmetry
The Zeeman
[XIII.3.]
of the oscillator of
Effect
and the Lamb Shift
371
and is the cause of the accidental degeneracy of the eigenvalues
H.
Accidental degeneracy for simple relativistic systems has been discussed by Johnson and Lippmann. However, investigations on the hidden symmetries of
bound
have not been pushed far enough to let us know if the degeneracy can find some applications in elementary
states
concept of accidental particle physics.
XIIL2.C. Problems 1.
any relation between the symmetry group of the two-dimenharmonic oscillator and the theory of angular momentum?
Is there
sional
2. The non-Hermitian operator afa^" is, for the Hamiltonian (XIII.%.25\ a constant of the motion. Show that it represents the annihilation of one
from one coordinate, leaving it in an eigenstate, and the creation of the same energy in the other coordinate, which is also left in an eigenstate. What changes have taken place in the oscillator in its
quantum
energy or
of energy
its
(foco c )
eigenstate? Is this related to accidental degeneracy of the oscil-
lator?
XIII.3.
The in
The Zeeman
Effect
and the Lamh
splitting of the energy levels of
an external magnetic
field is called
electron spin
We know,
and
an atom (with a one-valence electron) effect. These displacements
the Zeeman
from the interaction
of the energy levels arise
orbital angular
Shift
of the external field with
momentum.
with the hypothesis of spin, that an electron in the has a magnetic moment of magnitude /z = an field of external presence in the electron spin is ^Ffft. The correin the direction which =Fe/i/2mc in accordance
= (eh/2mc)
that
is,
as follows from (X.4.7),
H where
=
regarding the electron as a Klein-Gordon particle
is
eA*
(XIII. 8.1)
the electron's charge. We shall be interested in the nonrelativistic 2 H] we shall neglect also the term (e*/2mc )A*. The latter is responsible
diamagnetism of helium, whose ground state is characterized by and L = and therefore there can be no Zeeman splitting in this case.
for the s
+ c ^ [(p - ~ A\ + mVJ,
e is
limit of
=
given by
Thus the Hamiltonian
is
Energy Levels and Symmetries of Simple Systems
372
l
#=
[Chap. XIII]
A.^. P + eA + me
2m
(XIII. 3. 2)
where we use A-p = p-A, which follows from the commutation relation 3 L^'J PJ! = ^(dAi/dx ) and the Lorentz condition V*A = 0. In a uniform magnetic field we have
p-A = A-p = Taking the
X jp)
$3C-(r
z-axis in the direction of JC,
we
==
(XIII. 3.3)
i5Fe-.
obtain
p-A = and
or
l
#=
dm
+ v + wL
F is the potential energy of the electron and co = e3/2mc = |co c is the angular frequency of the Larmor precession. Hence we see that the external
where
magnetic
field effects
total effect of
the orbital motion of the electron
X is
=
H'
+
coL 3
juo- 3
by
X.
o>L 3
Thus the
.
(XIII. 3.6)
commutes with the Hamiltonian and is, therefore, a constant of the motion. The eigenstates of p 2 /2m + V are also eigenstates of L 3 hence the energy eigenvalues of H as defined in (XIII. 8. 5) are given by
The
interaction energy
;
E= where EQ
term
in
orbital
is
H
f
#0
+
(XIII. 8.7)
Ticom,
the energy eigenvalue in the absence of a magnetic field. The first can be regarded as a magnetic moment energy arising from the
motion
of the electron. Therefore the corresponding energy eigenvalues
hum do
not depend on the quantum numbers n and I. Result (XIII.3.7) implies that the eigenvalues in a magnetic field are split up. The corresponding
splitting of spectral lines is given
iw>
= =
(E
v,
- BO =
by
-E'o)+-~(m- m ) r
(EQ
Cm - m'). + 'ZTT
(XIII. S. 8)
the emitted light is linearly polarized parallel to the magnetic field, then the magnetic quantum number does not change and the selection rule If
(IV. 4-1 4) holds. In this case the frequency of line does not change: v mm
However,
if
the emitted light
is
=
?
.
(XIII.S.9)
linearly polarized perpendicular to the direc-
The Zeeman
[XIII.3.]
tion of the magnetic
and the Lamb Shift
Effect
field,
then the selection rule (IV 4.13) holds. The corre-
sponding line frequencies are given vm
where
VQ
light is
373
,
m
by
i
=
"o =*=
(XIII. 8. 10)
2r
corresponds to the frequency of the unperturbed line. viewed in a direction perpendicular to the magnetic
of every unperturbed line of the
atom
If
the emitted
field, in place there will appear a triplet of three
+
U/^TT and V Q equidistant lines, with the two components V G u/Zir polarized to the field and the in a direction perpendicular component polarized parallel to the field. When a particular line is observed in a direction parallel i/
to the
only the components
field,
+ oj/27r
j>
cular polarization about the axis of triplet are separated 1
-
f (
and
VQ
u/2-jr
appear with
The two outer components
3C.
cir-
of the
by
-
m m+l
vm
.
=
\
,
m -,}
-u =
e
nn ^
3C
=
Cm
where 3C is measured in gauss units. A more general theory must treat the electrons as Dirac particles and hence the effect of the external field on the energy must be included. In this
H
case the perturbing energy
f
=
momentum we
write
it
S
(L
+ g S),
gs
=
= iM- [J
+
==
M *e-
r
interaction energy
Thus the complete energy calculation
m
is
1
+
fc
-
(XIIL8.18)
1)5].
term
spin-orbit coupling
W+H
(XIII.S.1S)
80 .
by Bethe* gives the perturbed level formula
the eigenvalue of
-
(g s
is
Enljm
where
In terms of the total angular
must include the
Hi =
A detailed
(XIII. 8.11)
s
2.
as
as
H The complete
by (XIII.3.6) can be written
%ha and
H' where, for Dirac theory,
as given
= (TM)ny +
Jz,
3CMJWI,
(XIII. 8.14)
and
-
-
(xm s 1S} .
.
" the "Lande splitting f actor. The quantity (H 80 ) n ij represents the eigenvalue of 80 The splitting of the spectral lines in a magnetic field in this case is
H
*
.
H. A. Bethe and E. E. Salpeter, Quantum Mechanics
Academic,
New
York, 1957.
of One-
and Two-Electron Systems,
374
Energy Levels and Symmetries of Simple Systems
[Chap. XIII]
more complicated; it is called the "anomalous Zeeman effect. " The Lande g factor above comes from the expectation value of the spin S with respect to the simultaneous eigenstates of 2 S 2 and J 2 From (IV. 4-8) an(i (1^-4-4),
is
.
,
,
with K replaced by S, we obtain
p (/ s + s/
itJ2 [J2 sjj =
The average value
2
,
,
2
of the left side
is
s(s
)
(xm.8.18)
r-j(j-s).
and therefore the right side yields
zero
+
-
2
1)
-
1(1
H
is
f
In the derivation of the energy formula (Kill. 8.14) it was assumed that a small perturbation to the spin-orbit coupling energy so The second term
H
-
term obtained from calculating the average with respect to the eigenstates of so The case where so is small value of compared to H' (strong magnetic field) is called the "complete Pashen-Back effect." In this case one calculates the average value of ao with respect in (XIII.3.14) is a first-order
H
H
f
H
.
H
H
to the eigenstates of
f ,
corresponding to its eigenvalues /iOe(my
with
s
=
1
and my
=m
+
s.
+
(XI 1 1. 3. 18)
*),
The average value
(Hso) = ms jI
of
H
80 is
(XIII. 3. 19)
>
-f-
^
where
*2
i
\ a
if
i
Hence the
total energy of the
atom
E = ^(mj + The
i
<
i,
if
is
s)
+ms -jj + EQ.
(XIII. 3.20)
recent microwave techniques used in the measurements of the
Zeeman
anom-
have provided greater precision and have led to observable discrepancies between Dirac theory and experimental observation when gs differs slightly from 2 (attributed to an anomalous magnetic moment alous
effect
of the electron).
The same experimental techniques have been used for the measurements magnetic field. The experiments of Lamb and Retherford*
in the absence of a *
W.
E.
Lamb and R. C. Retherford, Pkya. Rev. W. E. Lamb, Rep. Pn g. Phys., 14
1014. See also
79 (1950), 549; 81 (1951), 222; 86 (1951), (1951), 19.
The Zeeman
[XIII.3.]
have shown that a shift in the
S
Effect
any n there
for
state of about
and the Lamb is
q
Shift
375
=^
2o g2
"~"~
10%
-|
of the fine structure separation be-
tween the
=
levels j
f
and
2P
=
^ 3/2
2),
much
out without
difficulty.
> g
The
is,
o*
the 2Si/ 2 state
lies
higher than
The experimental separations 2Si/ 2
2P
3 /2
_..-.---""
~
by which
the energy
2P 1/2
l
FIG. XIII. 1.
.
results for the
2Pi/ 2
and
The
states with n =
& 3 in
hydrogen.
2$i/2
give also a very accurate experimental value for the fine structure
separation 2Pi/2
2Ps/2.
According to Dirac's theory the levels having the
number n and total angular momentum n = 2 in Fig. 13.2.
A
=
2s
experiments prove that there is a Lamb shift in the hydrogen energy levels; that
jg
""~~~-
the metastability of 2S
Lamb's experiments on these states have been carried states (n
"o
Be-
shift is zero in Dirac's theory.
cause of
g
This
f.
same
radial
j are degenerate. This
is
quantum shown for
further splitting of levels comes from the electron's interaction with the The nuclear magnetic moment is, roughly, of the
nuclear magnetic moment.
M
is the nucleon mass, being smaller than the electron's eh/Mc, where magnetic moment by a factor of one thousand. Therefore, the electron's interaction with the nuclear moment will be smaller than the ordinary fine structure
order of
splitting levels of
factor of the
by a
atomic states
order. The most closely adjacent energy the relative orientation of orbital and electron
same
differ in
spin angular fl
=2
examination of the
I
0.091 crrr 1
\
t
-
2p 3/2
f
0.365 cm- 1 s
1
1/2
FIG. XIII. 2.
=
momenta. In a
2
levels in
The fine
^ /2
structure of
n
fine
closer
structure
it is found that each spectral line of f the fine structure can in turn be re-
solved into further lines or "hyperfine structure/' This arises from the interaction of the magnetic
moment mo-
of the nucleus with the orbital
Dirac's theory.
tion of the electron.
The Lamb shift, 2Si/$ 2Pi/ 2 is completely accounted for by an improved formulation of quantum electrodynamics.* ,
J. M. Jauch and F. Rohrlich, Theory of Photons and Electrons, Ad dison- Wesley, Reading, Mass., 1955. This book discusses the subject matter in detail and contains also a complete list of references to the papers by R. P. Feynman, J. Schwmger, and F. J. Dyson. *
Energy Levels and Symmetries of Simple Systems
376
The
[Chap. XIII]
some of the Zeeman components of the states 2Si- 2 2Pi/ 2 and 2P3/2 were the main results of the Lamb-Retherford experiment. See Figs. XIIL3, XIIL4, and XIII.5. The experimental results of Lamb and Retherford are displaced by about separations of
,
,
-10 /(units of 5214 gauss)
FIG. XIII. 3. from Dirac's
Zeeman component energy
theory.
levels
for
n
=
2 in hydrogen,
calculated
[XIII.3-]
The Zeeman
Effect
and the Lami> Shift
377
2.5
42
"E
3
1.0
1.5
-2.0
X (units FIG. XIII. 4. 1000 Me/sec.
of
5214 gauss)
Enlargement of Fig. XIII. 3.
The degeneracy in the 2&/2, and 2Pi/ 2 levels is removed conThe relativistic theory does not predict any upward
trary to Dirac's theory.
displacement of
2Si/2
compared to 2Pi/ 2
in the zero
magnetic
field,
where these
levels coincide.
The
selection rules for electric dipole radiation in the
anomalous Zeeman
Energy Levels and Symmetries of Simple Systems
378
[Chap. XIII]
-1.5
-2.0
X (units FIG. XIII. 5.
of
Zeeman component energy from quantum electrodynamics.
region are
m" =
m'h
5214 gauss) levels
for
n = 2
for the electric vector of radio
in hydrogen calculated
waves polarized
parallel
magnetic field. The radio waves are used to induce transitions among the states n = 2 corresponding to the doublet separation k = 0.365 cm.- 1 or a wavelength X = 2.74 cm and frequency 10,950 Me/sec. For perpendicular to the
The Zeeman
[XIII.3.]
Effect
polarization the selection rule the allowed transitions are
(2S 1/2
,
(2Si, s> j,
s'3
J,
s
s^
s
is
and the Lamb Shift
m"
m'j
= P) = Jft)
= -|70
=
1.
-
(2P3
-
(2P 1;2
-
2, ,
(2P3;2)
=-#)- (2P
1/2 ,
For
= = mj = my = mi m,
379 parallel polarization
*), i),
-i), -i),
while for perpendicular polarizations the allowed transitions are
(2S1/2
,
*
(2Si/2, ss
= Jft) = P) = P)
^ (2P 3/2 m, = /s,
(2P1/2
>-
- Jft) -P) -P)
XIII.3.A. Mesic
|),
,
+ (2P 8
,
>.
(2P3/2;
>-
(2P 3/2
,
* (2P 1/2?
m, = my = m, = mj = my =
-i), -i), i),
-|), |).
Atoms Muonium, Positronium ?
Negative mesons (such as ir~ and K~) and the negative muon ^~ can come and can form bound states by capture processes. Such states
to rest in matter
are called "mesic atoms." Since orbit size
mass
of the particle, the mesic
particles will quickly cascade
and cause various nuclear
is
inversely proportional to the
atoms have smaller
orbits.
through successive orbits
These reactions
reactions.
in
In general, these to the nucleus
down
most cases (by way
of
radiation of the charged fragments) produce radiating tracks in emulsions referred to as a star.
by and
7T"
and
K~
K~
is
that
The fundamental fjr
difference
between captures by
ju~
and
does not interact with the nucleus strongly, while TT
are involved in strong capture reactions. Typical examples are capture
processes in deuteron and hydrogen,
+D TT- + H
7T"
The second
n
+ 7.
+H
In the case of the negative kaon
+p +p K- + n K- + n KK-
~
+ n, + TT.
reaction competes with TT"
and
n ^n
>-
>-
(K~~~)
we can have
+ IT + TT+ *- A + TT,
+ TT, + TT-
^ 2r >
A
1. The strange meson K~ also and A are baryons with strangeness 1. Hence the above reactions are strangeness-conserving has strangeness
where
380
Energy Levels and Symmetries of Simple Systems
[Chap. XIII]
reactions.
When
muon
a
interaction
stopped the formation of a star takes place via weak
is
:
\r
+p
*
n
+
v.
Being a slow capture process, the inuon tends to spend most of in the
its lifetime
ground state before undergoing a decay or nuclear capture. The energy states for a muon is given by the Dirac energy level formula
bound
of
(XI 11.1
8),
Muonium from the
where is
m
is
a bound
effect of larger
to be replaced by 206.77 ra, the mass of the muon. + state of an electron and a positive muon, y. Apart .
magnetic
moment and
to the proton, the energy level scheme
there
is
no experimental evidence for
is
it,
smaller mass of
similar to a
+ p,
compared hydrogen atom. So far
but theoretically
it is
considered as a
possibility. is a bound state of an electron and a positron. It is a hydrogenatom and can be formed when positron is slowed down in a hydrogen gas. Its formation is essentially through a radiative capture process. The reduced mass of the atom is mR = (mim 2)/(mi + m 2) = |m, where m is the mass
Positronium
like
of the electron or positron.
In the nonrelativistic theory, therefore, the energy
levels of the positroniuin are exactly half of the
two spin states S In the singlet
S
=
I
and S
=
hydrogen energy levels. The and singlet S states.
correspond to triplet
state two-y-ray annihilation of the positronium is possible,
since zero total angular momentum is a possible two-7-ray state. The fact that a two-photon system can never be in a state with unit total angular momentum (see Section IX.2.A) implies that a triplet positronium state cannot decay into
two photons. However, a three-photon annihilation of the triplet S state of positronium is an allowed transition. According to quantum electrodynamics the mean lifetimes* for annihilation of a singlet and a triplet S state of positronium are
Wet = triplet
where n
is
the radial
the triplet and singlet
=
X 10- n sec, X 10"" n sec, 10
1.25 1.4
7
3
3
quantum number. The total energy separation between S states with the same value of n is (in frequency units)
given by
A#o
=
~i 2.044
X
in
Corrections to *
A. Ore and
AE
arising
from the
Lamb
10 5 Me/sec. shift
and other
effects originating
J. Powell, Phys. Rej., 75 (1953), 1696, 1963. DeBenedetti and H. C. Corben, Ann, Rev. Nuc. Sd., 4 (1954), 191. J. Pirenne, Arch. Sci. Phys. Nat., 29 (1947), 121, 207, 265. V. Berestetski and L: Landau, J. Exp. Theor. Phys. USSR., 19 (1949), 673, 1130. R. A. Ferrell, Phys. Rev., 84 (1951), 858.
S.
The Zeeman
[XIII.3.]
and the Lamb Shift
Effect
381
from quantum electrodynamics have been calculated by Karplus and Klein.* The total theoretical value with ft = 1 is
A# = The experimental
X
2.0337
10 5 Me/sec.
value for the splitting between singlet and triplet com-
ponents of the positronium ground statef AEexp
=
(2.0338
=fc
is
0.0004)
X
10 5 Me/sec,
good agreement with the theoretical result. Further discussions of the Lamb and hydrogenlike atoms will be given in Chapter XV, Section XV.4.
in
shift
XIII. 3. B. Parity Selection Rules and Parity of the Levels Consider the
electric dipole
moment
of
a system of
D= -e]g
electric charges,
(XIII.SJK1)
co .
i
D
The operator [(P,
D]+ =
0,
x
(P|r )
j
taking representatives,
J Using
=
8(r
anticommutes with the parity operator
'\r'),
-
r')(P'
we
dV
dV
(P'
=
d=l,
r
(r\D\r
we
}
+ j d(r' -
+
($>")(r\D\r")
Hence (
so the only nonvanishing representative of (P"
states of opposite parity.
for systems in If in
we take
radiation,
states involving orbital
(IV. 4-9)
where AZ *
is
=
replaced I"
D
r")
=
&*f
(r|D|r'>
=
0.
(XIILSJtS)
0,
comes from (XIII. 3. 23)
~-(P', 7
is
A parity
which no angular
(IV. 4-4)
on the dipole
0.
a "parity selection rule/ which means that all due to electric dipole radiation arise from transitions between
or opposite parity. This spectral lines
=
dV
obtain
|D|r">
7
Thus from
(P.
obtain
-
D
selection rule
momentum
for the operator
is
of particular importance
selection rule exists. ic,
we
obtain another restriction
arising from taking representatives with respect to angular momenta. In this case the selection rule
by |AZ|
=
1
(XIIL8.S4)
V.
R. Karplus and A. Klein, Phya. Rev., 87 (1952), 848. See also T. Fulton and P. Martin, Phys. Rev., 95 (1954), 811, for similar calculation on the excited states of positronium. t R. Weinstein, M. Deutsch, and S. Brown, Phys. Rev., 98 (1955), 223.
$82
Energy Levels and Symmetries of Simple Systems
The
[Chap. XIII]
may be expected to be valid for all quantum momentum type. For example, it is observed that for
selection rule (IV. 4.9}
numbers
of angular
strongly interacting particles, such as baryons and mesons, the weak interactions can induce decays that violate conservation of strangeness and, therefore, + K~~ of isotopic spin. The following decays of the strange particles H, 3=*=,
K
can provide possible
>A + T,
H If
selection rules.*
S
we assume
>-
that there
is
*
(XIU.S. 25)
27r(or STT).
a selection rule
AS = quantum number
for strangeness
K
+ pion,
nucleon
>
Thus
dbl
(XIII.8.26)
S, then, as follows
from (VI. 8.16), one has
the selection rule
AT3 =
(XIII.8.27)
J.
+^+v
if However, the recent observation of the reactions KQ *with v not 7T+ do Ko (XIII.8.27). M~ + agree By analogy with (IV. 4.9), it has been suggested! that there -
and
+
may
a
exist
further I spin selection rule for the total I spin, |Arj
= i
(XIII.8.28)
Because of the nonconservation of the total I spin by electromagnetic interactions, the rule (XIII. 8. 28)
The concept spectra.
energy
is
of parity plays
We shall give a
subject to electromagnetic corrections. role in the classification of atomic
an important
brief discussion of the application of parity to
atomic
levels.
For an atom with
where
N electrons the Hamiltonian
of the
is
form
H
= HQ
+ H',
N
The term H'
represents mutual repulsion of the electrons. If
then the electrons
will
move
we
in the field of the nucleus alone.
7
neglect fl
,
In the latter
case the Schrodinger equation
-
*d,
2,
,
tf)
can be solved exactly, where the numbers
wave function
= m(l, 2,---,N) 1, 2,
,
(XHLSJM)
N in the argument of the
refer to the coordinates of the electrons.
The wave
function
^
can be expressed as a product of the individual electron wave functions * M. G ell-Mann and A. H. Eosenfeld, Annual Review of Nuclear Science, Vol. 7, Annual Reviews, Palo Alto, 1957, p. 413. t M. Gell-Mann and A. Pais, Proceedings of the International Conference on High-Energy
Physics,
Pergamon
Press,
London, 1955.
The Zeeman
[XIII.3-]
and the Lamb Shift
Effect
383
A*
* =
H *W*),
(XIILS.30)
s=l
N are principal, orbital, and magnetic m* for s = 1, 2, Z., of the sth numbers electron. Each tn m satisfies the wave equation quantum where
-
n,,
-
-
,
ai s
where
ls
=
1, 2,
,
N
1
and
m = s
lt
+
ls
,
s
-
-
-
1,
ls .
,
Thus for a given
n f or for a given energy eigenvalue Ent there can be various quantum numbers and therefore various states. Furthermore, for a given set of n8j the interchange of the electrons
among themselves does not
affect the corresponding energy
eigenvalue. There is thus a degeneracy of the levels. The inclusion of H s in the Hamiltonian will remove the degeneracy only partially. The complete removal of degeneracy and hence the splitting of the levels requires a
to the
treatment of the N-electron atom. In principle such an approach problem under certain approximations is possible. However, this aspect
of the
problem
relativistic
If
L
netic
not be considered here.
the total angular
is
values!/
will
=
quantum
so a level
momentum
of the atom, then
corresponding to S, number runs through 2L
0, 1, 2,
,
M
P,D,F,
+
L
with quantum number
has 2L
1
+
values 1
it
can assume the
levels.
L,
(
The total mag-
L
+
I,
-,!/),
eigenfunctions. These eigen-
are distinguished
by their magnetic quantum numbers. In this section quantum numbers and quantum states, to distinguish them from the corresponding quantum numbers and quantum states for which we used small letters before. From the discussion of the repref unctions
we are using
capital letters to represent
sentation of the rotation group
it
follows that the eigenvalues
belong to the representations
D
a)
sponding to
L =
three, five, seven,
0, 1, 2, -
-
symmetry
F
-
*
0, 1, 2,
levels corre-
levels,
with one,
eigenfunctions, respectively. Besides the rotational sym-
metry (characterized by the quantum number L) we of a level.
The
of the rotation group.
are designated as S, P, D,
L =
The
also
have a
reflection
latter refers to the "parity of a level." Therefore,
must be based on the representation of the reflection group which consists of the identity 7 3 and the 7 3 Every orthogonal matrix can be obtained from a rotation reflection from a parity point
of
view the
classification of a level
.
7 3 Thus two representations of the matrix by multiplication with 7 3 or rotation-reflection group can be obtained from every representation D (l} of the .
D
(l) 7 3 The two irrewith 73 and with pure rotation group by combining ducible representations of the three-dimensional rotation-reflection group are
(21
A
+
The matrix
D a)
.
corresponds both to a rotation matrix A in the "identical representation" and to a rotation-reflection matrix l)-dimensional.
384
Energy Levels and Symmetries of Simple Systems
of the reflection group.
The matrix
-D
a)
corresponds to
representation" of the reflection group.* The even parity levels of the A7-electron sentation of the reflection
atom belong
group. The odd parity
-A
[Chap. XIII] in the
"negative
to the identical repre-
levels belong to negative
the parity of the level representation of the reflection group. We incorporate and -, corresponding to its angular momentum state with the subscripts
+
S+J
P+, P-,
*
and odd parity states, by writing are the most common analysis shows that S+, P_, D-H F-,
to even
,SL,
-
electrostatic interactions
=
Spectroscopic
of the Levels
XIIL3.C. Multiplet Structure In an TV-electron atom the
.
levels.
can couple the orbital
orbital angular AT, into the resultant
for i momenta L 1, 2, momentum L and spin angular momentum S, respectively. The magnetic = L + S. A given L and S interaction between L and S leads to a quantized J From the addition rule of two angular correspond to a certain multiplet.
angular
(i)
,
,
momenta,
J = L
+ S,L + S-l,-",\L-S\+l,\L-S\, relative orientations of L and S and follows that for L > S there are 2S + L > S there are for L < S there are 2L + 1 relative orientations. Where number 28 + I the the to multiplet; 2S+1 different J values corresponding 1
it
and S = \ has a multiplicity of 2. 2 2 In spectroscopic notation they are indicated as P 1/2 and P 3 /2, where the sub2 2 and P3 /2 are (2J + I = 2) | andf correspond to J values. The levels Pi/ 2
is
called the multiplicity.
A level with L =
1
scripts
+I=
an(j (2J
M
(2S +
(2L +
1)
corresponding to the magnetic quantum &. Hence in the absence of magnetic interactions all sublevels of a multiplet are degenerate and each multiplet
4) -fold degenerate,
numbers
=
LE
+
1)
corresponding to different value.
ent
J
The magnetic values.
A
J
coupling
values differs in the corresponding energy eigenremoves the degeneracy in the sublevels of differ-
complete removal of degeneracy
is
produced by an external
M
values. magnetic field which splits different For a two-electron atom we have parallel or antiparallel spin states, repre= the two levels are 3S and 1 S sented by the values 5 = 1 and S = 0. For L (triplet
number
and
singlet spin states). Since
of levels is
2L
+
1
=
S
is
greater than L, the corresponding
1.
S
0,
In general, for an A^-electron atom the quantity if is even, and the values f |, 1, 2,
is
even we obtain
For odd
,
N
,
,
states.
singlet, triplet, quintet,
,
can assume the values if
N
ding spin states are doublet, quartet, sextet, *
E. P. Wigner, Group Theory, Academic,
New
York, 1960, p. 176.
is
odd.
Thus
if
AT
AT the correspon-
CHAPTER XIV
APPROXIMATION
METHODS XIV.l. General
Remarks on Perturbation Theory
quantum mechanics the number
of problems that can be solved exactly have to use approximations, and for this purpose various methods are available. The most important of these is the perturbation theory. This method is based on the fundamental assumption that the total energy H
In
are rather few.
We
(Hamiltonian) of the system can be separated into two parts. (a) A part ffo, which is simple. The corresponding Schrodinger equation can be solved exactly. (This is not a necessity but in most problems an exact solution for HQ does exist.)
(b)
A part H', which is small compared to H
original
A
system and gives
rise to
Q.
This
a small correction to
is
a perturbation on the
it.
separation of the Hamiltonian into two parts is motivated either by difficulties or by the nature of a particular physical process
mathematical
where no new information
obtained by an exact treatment, as with Coulomb does not involve any new physiscattering. sight the separation of cal assumption about the system, but a reasonable physical interpretation is always provided by the nature of the problem.
At
is
H
first
In principle the knowledge of the Hamiltonian of a dynamical system
is
by means of Schrodinger's equation, to predict all of its most probable behavior. Thus any assumptions on total H, such as separation into two, will automatically affect the state of the system. There is no a sufficient,
priori justification for the separability of
ever, a solution in the is
form
of a
power
an
H into two or more parts.
series in
powers
of a small
How-
parameter
the result of the perturbation theory. The proof of convergence of this a formidable task. Despite this, the perturbation theory is considered
series is
385
Approximation Methods
336 quite satisfactory even
the
first
The
if
[Chap. XIV]
the series does not converge. In most cases of interest
approximation leads to acceptable results. the Hamiltonian is taken over from the
split of
classical analogy.
Here
one assumes a possible separation into free and interacting parts. The separability of the total energy into two parts is inherent in both classical
also
and quantum theories. We seem to know, in advance, how much of the energy is free and how much remains for interaction. For a particular problem the method of perturbation theory to be applied will depend on the time behavior of the Hamiltonian. For a time-independent "
Hamiltonian, for example, one applies stationary-state perturbation" theory, where the effect of a small perturbation consists of changing the original state
motion of the dynamical system. The "Stark effect" is an example of stationary-state perturbation where an external electric field causes changes in the initial state of the system. For a time-dependent Hamiltonian, where the unperturbed part is time-independent, one is interested in the time varia-
of
tion of
a stationary
but
third,
conditions,
which
by a time-dependent perturbing
state caused
method
part.
A
the so-called perturbation of boundary include either a change in the boundary conditions
less practical
may
is
or a change in the shape of the boundary surface or both. This is especially useful when one has no detailed knowledge of the perturbation or the interaction of the system.
XIV. 1. A. Stationary-State Perturbation Theory
We
shall
assume that
(a)
the perturbed part of the Hamiltonian has dis-
and (b) the separations of energy levels are large comthe changes in them caused by the perturbation. Let the total
crete eigenvalues only,
pared to Hamiltonian be
H
where
r
eigenvalue
Thus
is
E
H
H + H',
of
H
lies
(XIV. 1.1}
Q
the small perturbing energy, and where
for a state \&)
we assume that each
very close to one and only one eigenvalue
we
E
Q
of
H
Q.
have, approximately,
(E
The
=
-
JB
general eigenvalue equation
)
^ (\H'\&).
(XIV. 1.2}
is
H\E)
=
E\E)
(XIV. 1.3}
H'\E).
(XIV. 14}
or
(E
The
-
H}\E} =
stationary-state perturbation theory deals with the solutions of the
We
proceed by assuming that the eigenstate \E) and corresponding energy eigenvalue E can be expanded according to
eigenvalue equation (XIV. 14}*
General Remarks on Perturbation Theory
[XTV.l.]
= E=
\E)
where
and EI and so
\Ei)
quantities, If
we
E,
on.
(EQ (E. (E.
sides,
-
(XIV. 1.5)
,
-
l
The nth-order
same order on both
-
Z)
are first-order quantities,
substitute (XIV. 1.5) and
of the
+ \Ei) + \E + +E +E +
[EQ )
387
2)
|j
and E%
are second-order
quantities are of the order of
(XIV. 1.6)
we
(XIV. 1.6)
,
in
n
f
)
.
(XIV. 14) and equate the terms
get the set of equations
- H,)\E = 0, - H )|#i> + Wo> = H'\E*}, - H )\E + E.IE,) + E,\E } = H'\E&
(XIV. 1.7)
Q)
Q
(H
Z)
(XIV. 1.8)
(XIV. 1.9)
Q
H
We
set up a representation. If can use the eigenstates \EQ) of Q to is a degeneracy we introduce further observables f, say. The
there
new
representation would then be simultaneous eigenstates of the complete commuting set of observables HQ and ". If \EQ, f ') is a simultaneous eigenstate
of the
of
complete commuting set of observables HQ and f then the scalar product
\E'o}
,
and
\E'Q ',
"}
is
of the ;
<j%
where ^(f'O
is
,
form
r ^>
some function
=
fe-oW^r"),
(xiv. 1.10)
of the variables f".
using these results and
taking the representative of (XIV. 1.8) in the (Ei
By
new
we obtain
representation
r
There are two cases to be considered, this case
(a)
Let us suppose that
E'o
(XIV. 1.11) yields the result
which constitutes a set of linear homogeneous equations for F(f'); they are to be used to determine the energy EI. The possible values of EI are just eigenvalues of the matrix
whose rows and columns of
H'mn
for
EI
same energy E'Q Each of the eigenvalues an energy level of the perturbed system the unperturbed system. As an interesting
refer to the
.
gives, to the first order,
lying close to the energy level
EQ
of
H
Qj are just example, let us suppose that the extra variables f in addition to the total angular momenta; then we can label the states by the eigenvalues of /3. Thus the matrix equation (XIV. 1 .12) becomes ,
.
(XIV. 1.14)
Approximation Methods
388
Hence the equation determining E\
is
+
a (2j
7 [Chap. XIV ]
1) -dimensional
eigenvalue
equation,
det \H'm m ,,
where
H^ =
In this
-
>
Ei8 M fm >] ,
=
(XIV. 1.15)
0,
<#o, m"\H'\E'*, m').
way we
obtain 2j
of the unperturbed system.
+
energy eigenvalues EI that are close to EQ 1 does not exceed the number The number 2j 1
+
independent states of the unperturbed system belonging to the level EQ. Hence we see that the energy levels EQ and EQ that coincided for the unperturbed system has been separated by the perturbation H'.
of
H
Now, knowing
from (XIV. 1.1$), we can determine
F(f')
to the zerotb.
order the representative (XIV. 1.10) of the stationary states of the perturbed
system belonging to energy levels lying close to EQ. (b) Let us now suppose that there is only one stationary state of the unperturbed system belonging to each energy level. In this case no extra variables f are required;
H by itself can fix a representation.
Thus
for
EQ
=
EQ
we have Ei
There to
is
=
only one energy level EQ
+
EI of the perturbed
system lying close
unperturbed system, and the change in energy is order to the corresponding diagonal element of the perturbing
level of the
any energy
equal in the
(XIV.1.16)
(E{>\H'\ES>.
first
energy H' in the representation for the unperturbed system. Equations (XIV. 1.10) and (XIV. 1.11) in this case read respectively, (E'o'\ES)
= fiw,
(XIV.1.17)
and r
(EQ For EQ 7
EQ
rr
- EQ
f f
f r
)(E Q
\Ei)
+ E$w, =
rf
(EQ \H
f
\Eo).
(XIV. 1.18}
we have '
fTTT\r
1
1ti\
(XI V.I. 19)
,
jG/0
A similar treatment
of
(XIV. 1 .19} leads to the energy change (XIV.1.2ff)
XIV.l.B. The Brillouin-Wigner Perturbation Expansion
An improved form of the stationary-state perturbation theory can be based on a solution of Schrodinger s equation (XIV. 1. 4) where the energy denominators do not depend on the differences of unperturbed energy eigenvalues. It is a definite improvement over the former method. 7
t
"
[XIV. 1.]
General Remarks on Perturbation Theory
389
Let us write Schrodinger's equation (XIV. 1.4) in the form
-
(E
=
ffoX*)
#'!*>,
(XIV. 141)
where the eigenstate j*) corresponds to the eigenvalue vector ]&) can be split up as
=
j*>
where assume that
j^o) is
Q)
(XIV. LSS)
!*>,
H
Q
<^o|^o) it is
orthogonal to #); that
Hence
it
=
1,
=
0.
follows that the normalization of \&)
<*b|*>
have seen
to (XI V.I .23)
in Section 1.5.
we can form
(XIV. 1
8)
(XIV. 1
4)
is,
<^o!#>
We
H. The state
normalized as
[iPo) is
and
of
Q corresponding to the eigenvalue EQ. We shall does not coincide with the total energy E. The state vector
an eigenstate of
E
& +
E
B
=
is
given
by
1.
(XIV. 1.25)
that for every vector normalized according
the corresponding projection operators
M=
by
(XIV. 1 8)
|*bX*b|
and
p= Subtracting the equations obtaui
_
M
H \&)
=
=
-
Q
H \&) Q
i
(E
i
(E
- |*b)<#b - H )\&) and -E
H')\V)
Q
Operating by projection operator M", equation
By
substituting in (XIV. 1
Hence, since
E
9
E
,
we
8) for .Eol^b},
(E
(XIV. 1.9V)
.
f
- # )|#> =
HQ\&O) =
\V ).
E & Q
)
we
(XIV.1J68)
(XI'V.I. 28) yields
obtain
PH'\V).
we obtain (XIV. 1.29)
'
Thus the Schrodinger equation (XIV. 1.21)
is
now transformed
into
an
integral equation, >
(XIV.1.30)
.
where
E= So
far,
everything
is
exact. If
Eo
+ (V \H' *>. Q
we assume
turbed and unperturbed energy (E
(XIV. LSI}
that the difference between perEQ) is greater than (tf'blH'l^b), then the
Approximation Methods
390
[Chap. XIV]
perturbation expansion consists of the iteration of the integral equation
(XIV. 1. SO). Iteration
of
(XIV. 1. SO) leads
to
= |#r ) -f 1^} + j#- ) + E = EQ + Ei + Ei + 2
|tfr)
(XIV. 1.38)
,
(XIV. 1.33)
,
where v. 1.34}
(XIY.1.35) o
7
and
E = l
E,
= <* |ff
The perturbing energy EI
(XIV.1.S6)
<*b|ff'l*b>,
PR' * >,
'
(XIV. 1 .87)
agrees with previous result (XIV. 1 .16) but higher-
order perturbations do not agree.
XIV.2. Examples of Stationary-State
Perturbation Theory XIV.2.A. The Linear Stark Effect
An
application of the second-order calculation of the energy can best be
illustrated in the Stark effect of the atoms other than hydrogen. (Hydrogen must be given a special treatment because of the degeneracy of levels of the same n and different L) It consists of a shift of energy levels of an atom in an
external electric for
field.
Let this field be in the
an arbitrary atom
z direction.
The total Hamiltonian
is
H = Jp + V + H',
H'
=
eS
]T) Zi
(XIV. 2.1)
,
i
where the sum
is
taken over the electrons. In general,
=
(E,\H'\E,)
=
r
I
(E,\r)(r\H'\r')(r \E,) d*x
dV,
the average value of interaction energy with respect to field-free states, vanishes. In terms of the eigenfunctions ^o = (r\Eo) of the field-free atoms, the
average of H'
is
written as (H')
=
I
W
2
ff' d*x.
(XIV. 2.2)
Examples of Stationary-State Perturbation Theory
[XIV.2.]
H
391
r
on the coordinates z,, the integrand Because of the linear dependence of a of reflection coordinates and therefore {#'} vanishes. changes sign under
However, for a transition between two states ^/ and $p
of opposite parity
the matrix element
(E$\H'\E> of H',
where
and
\^F
T/T
=
f
&H
r
fr d*x,
(XIV. 2.3}
refer to eigenstates of opposite parity, does not vanish.
For hydrogenlike atoms, for example, the wave function, being of the form. fi(r)Yim (Q, <), implies that the integral (XIV. 2.3) does not vanish provided that for a fixed
Hence we I
m the angular momentum quantum number
see that for hydrogenlike
values for a fixed principal
atoms the degeneracy,
I
changes by
L
arising from several
quantum number n, can be removed. In this weak external field 8 are superI values and the effect linear in
case the eigenfunctions in the presence of the postions of field-free functions with different 8;
that
The
the first-order perturbation energy does not vanish. linear Stark effect can most conveniently be treated is,
by
expressing
the Schrodinger equation in parabolic coordinates, which are defined *i
=
V(fo) cos f
,
x,
= V(^" sin f,
=
r
x^
=
|(
-
by
(XIV Jt 4)
,,).
Hence we have r
=
The
*(*
line
+
17),
+x
n
s,
= r-xs,
f
=
tan- 1
element and volume element in parabolic coordinates are
Hence the Laplacian operator takes the form Tf2*
y
ZZ:
^^
I I
^~~
I
^^^-^ ^^____~
I
I
I I
^_ fn
1 I
I
"
-
Thus the Schrodinger equation for hydrogen corresponding tonian (XIV. 2.1 ) can be written as
= By assuming a solution of the form KI
+
tions
K2j
.
j
j
\[/
=
/i(?)/2(^)e"
w/r
0.
to the Hamil-
(XIV. 2. 7)
and writing me 2Z/h2
=
the differential equation (XIV. 2.7) can be separated into two equa-
Approximation Methods
392
m
/2
1^"
The two equations the
differ
1
+4
only in the sign of the last terms. For 8 behaves like e (1 /2) ** and for small like
=
0, '
equation for large bound-state solutions are of the form
first
so its
[Chap. XIV]
= V(2mE'/h^. A similar solution holds for the second equation. and substituting (XIV .9) in the first equation of (XIV. 8. 8) Putting x =
where i
fc
we
obtain
This equation
is
satisfied
functions
by Laguerre tn
=
m
=
iJSWC*),
where
is
f
- Km' +
1)
a nonnegative integer. In the same way, for w 2 (fo?)
we have
with
Hence
n1 where like
E = r
atoms.
+n +m + I = f
2
+ K^=Z
m& * }
l(
E, leads to the usual formula for the energy levels of hydrogeneigenfunctions for field-free case are given by
The normalized
X It will be
~(Kl
e
found convenient to treat
/ci
and
/c 2
as the eigenvalues in equations
(XIV. 2.8} and record them in the form Ki
Kz
Hence *
it
= =
KIQ /C 2 Q
+ KI + +4+
,
*
'
* ,
fl /2
= foi+fl = /02 + /2
+ +
"
' ,
'
*
follows* from the first-order perturbation theory that
H. A. Bethe and E. E, Salpeter, Quantum Mechanics of One- and Two-Electron Systems, New York, 1957.
Academic,
Examples of Stationary-State Perturbation Theory
[XIV.2.]
= where /i
is
if (6n + 6%m + m/2 + "
4
+
'
6?l1
the eigenfunction for field-free atoms.
3m/
+ 2)
393
>
From (me z /ff)Z =
KI
+
*2
we
obtain
f/2E'\
h_
\V
e2 If
we
x
=
write
(Z/a)
m
3
E(UI
2 .
)
Z=
ax
+ 6/#
+ y,
we
obtain, for small y, the result
2
and
=
re
=
=
0) is
of HI
+n
and E(n 2
0)
The maximum value
2
and assume a solution of the form x = (Z/a) (ab/Z*) or
V.E',
as the energy of the atoms in the field S. levels
8ft
The Stark
separation of the extreme
given by
is
2(n
Thus the separation of spectral terms the principal quantum number.
1),
so
n 2 where n
in the Stark effect varies as
,
is
XIV.2.B. The Quadratic Stark Effect This
is
a second-order
effect.
H can be calculated from
Hence
= e*&NN'8mm cos 6 Pi mPi> m fj = \(n'l'm' z\rilm)\*, ,
where
N and
7
^V"
are normalization factors and the
The only components I'
_
i
-j-i
an(j
m
f
where the summation
differing
>
sin
dO
f
RmRi>n>r* dr
R are radial wave functions.
from zero come from the
selection rules
m> Hence
is
taken over V
== Z
d=
m = f
1,
m, and
n',
the energies EQ and EQ correspond to Imn and Z'mV, respectively.
and where
Approximation Methods
394
We
[Chap. XIV]
not discuss the details of the quadratic Stark effect but it must be mentioned that the quadratic effect refers to a displacement of the fieldshall
free lines
where the red
above a
critical field strength 8 C
lines are displaced
more than the
violet lines. Further,
the Stark lines disappear; this
is
called
"quenching of the lines."
V= important to remark that from the potential energy e&zZe^/rof the electron we can infer that the electric field is capable of ionizing It
also
is
The potential energy has two minima, one at the atomic center and another at distances which are sufficiently far from the atom. The second
the atom.
minimum than
where the potential is lower In between the two minima (potential
occurs in the direction of negative
value at the atomic center.
its
troughs) there electron will
is
a potential barrier. There
make
transitions
z,
a
is
that the
finite probability
between the two troughs and hence a probabil-
The
penetration of the potential barrier leads to an acceleration of the electron away from the atom; therefore ity of penetration of the potential barrier.
an ionization takes
place.
XIV.3. Application of Unitary Transformations to Stationary-State Perturbation Theory
We
U by
define a unitary operator
U = X) E n \
where \E n) and
\En) are the
U
En
respectively.
U\E%)
=
H\En )
= En \E n) we
H U\E n The boundary ing energy
1
)
and
The unitary
H
,
oper-
U
)
\En)
(XIV.8.2)
\E n ). }
obtain
= En U\E$
condition that for H'
En must tend to
lim
>-
and En
U = Y!
0,
}
(XIV. 3.3)
the state \E n ) and the correspond-
respectively, leads to
|J
can be expanded according to
U = 1 + Ui + U + U in the representation -
2
The matrix elements S) of
Q
by (XIV. 8.1) has the property of transforming the state system into the state \En ) of the perturbed system. Thus
using this in the equations
Thus,
and E%,
H (= H + H
as defined
\En) of the free
By
(XIV.S.1)
i
normalized eigenstates of
corresponding to the eigenvalues ator
M
of
-
-
defined
the unperturbed system HQ, or Urs
=
(E\U\E<>)
=
Q
(XIV. 84)
.
(E r \E s),
by the
state vectors
Application of Unitary Transformations
[XTV.3.]
395
are stationary-state transformation functions. Furthermore, from
\^
\
77*
1
/
'
*'
i
77^X7
follows that the
Now,
U
rs
j
T
T
'
/~v TT^ o
T7I 0\
Lrs^r)
(XI
\
*-\
,5.t>)
n
r
it
^
= ~\ /
\
JT'O! "IT1
\j r)'(& r [&3/
j
are also the expansion coefficients of j7 8 ).
writing (XIV. 3.3} in the form
- HdU\E
(E n
and using the expansion
of
TTI t
we obtain the
n
En
= H'U\E
(XIV. 8.6)
,
= &n
TT'O
representatives of
j
"T
IT1 !
^n
1JT2
I
^n
"T
i
H~
*
" *
,
form
in the
(XIV. 8. 6}
or
Hence, equating terms of the same order on both
sides,
- Em Umn + Eti mn = l
(El
(En
From (XIV. 8. 7}
- Em
for
n
)
9*
H'mn
)
Uln
+ En Umn + E%S nm l
l
=
we
obtain
(XIV. 3.7}
,
J^ H'm U'sn
,
(XIV. 3. 8)
m we get the first-order part of the unitary operator: TJf
For n
=
(YTV 3 Q} \-A~J. V .O.&J
J2H
TJf u mn
-rro
TyTO
m we have En = l
Equation (XIV. 8. 8}
forn^m gives TT/
/TTTO ^ZI/ W
(XIV. 8.10}
H'nn.
irrO \
rr2
-C'm^ ty
TJ/
TJ/
i
mw "T
7770
^
""
/
^w TTTO
j
H-n
TjrO -
fif
IT'
" ^T?O s
or
Hence rr/ jcjr/
(XIV.8.11}
For n
=
m, (XIV. 3.8} yields the equation i
Em =
/
j
ttmsUsm
===
H-mmUmm
\
/^
Approximation Methods
396
[Chap. XIV]
Hence
which agrees with the usual method.
XTV.4. Time-Dependent Perturbation Theory concerned with the time variation
is
Time-dependent perturbation theory
of the stationary state of the unperturbed system under the action of a time-
dependent perturbation. In other words, time-dependent perturbation theory is designed for the formulation of transitions between the stationary states of the unperturbed system. For the sake of a first orientation
spectrum and nondegenerate
we
assume that HQ has a The perturbing energy H'
discrete
shall
eigenstates.
is
time-
dependent. We shall use the eigenvalues of a general set of observables f to label the states at time L At the initial time t = to the state with energy E s is
E
s,
U)
t = t Q the system is unperturbed. The apparent with the uncertainty principle arising from the choice of E8 at time t Q
and we assume that at
conflict
be clarified when we discuss the boundary conditions. The Schrodinger equation to be solved is
will
where
H
Since the \E n state
|f', t)
,
to)
(XIV. 4-2)
form a complete representation, we can express a general
t)
= ]T
\E n
,
t
)(E n
,
(XIV4.3)
iolr', t}.
E
The state ,
+ H'(t).
as \f,
\En
= #
n at time t with respect to the probability that Ho has the eigenvalue or the that the t) |f', probability system is in the unperturbed state
to)
at time
t
is
t)
=
=
^
t\S
HA \E n
,,
U)(En ,,
t*\?, t)
(XIV44) \(E n ulf, ,
t)\*.
This gives us the physical meaning for the expansion coefficients (E n h\', t). It will be more convenient to work with the "interaction picture.' We ,
3
introduce the unitary transformation \?,t)
= e-W s*\I,t)
and transform Schrodinger's equation
into
(XIV 4.8)
Time-Dependent Perturbation Theory
[XIV A.}
=
where the new interaction energy Hi(t) is given as = eW s'H (()e-WV H T (t)
H
before,
(XIV 4-6)
ffr(0!7,*>,
j!7,f>
As
397
f
we may expand
the state vector
(XIV 4.7)
.
[/, t)
of the interaction pic-
ture according to 1
7, *)
=
Under a unitary transformation so the expansion
coefficients
\En U)(E n
]
(E n
,
,
(XIV 4.8}
fe|7, t}.
the algebraic relations remain invariant, fo|7, f) have the same physical meaning as
all ,
(En ,t\t',t):
Let us put an (t)
=
=
t\d H9 B.\f, t)
<7, t\d
HA\I,
(XIV. 43)
t).
and from (XIV 4.6) we can obtain the equa-
(E,i, fol7, i),
tions
where
=
HT
(En t,\Hj\Em ,
,
(XIV4.H)
t,).
Equations (XIV4-10) are a set of infinite number of differential equations defining each a n in terms of am The time variation of a n depends on the perturbing energy '(t) and on the .
H
type of
unperturbed system with which
it
started. This
definition of the probability amplitudes a n
.
is
also seen
The boundary
from the
condition
is
con-
tained in the statement that the system starts out at a definite unperturbed state \E n to) at time t Q The state \E n U) niay be taken as one of the stationary .
,
,
states of
an atom. The
time prior to the application of the possible to determine the original energy
infinite extent of
perturbing energy at time makes it of the system with arbitrarily great precision, so no conflict with the uncertainty principle arises. The time available prior to the disturbance of the tQ
En
<=o to t tQ stationary state extends from t Let us assume that the particular state we start with .
is
an excited state
of
the atom, \E 8J to). This means that at time t = U all the probability amplitudes am are zero except a s say. If the system is actually in the stationary then we can take \a8 2 = 1. However, if the state \E 8 U) with energy 8 ,
E
,
,
perturbing energy is switched on slowly (or adiabatically) then the probability 5^ 5, for transitions to other states, begin gradually amplitudes a w where to form. Just after t = U the amplitudes a m where s, are quite small; ,
m
,
m^
accordingly, the probability amplitude for finding the system in the state 2 will remain close to unity. This state of \E 9 t Q ) will start decreasing, but |a ,
Approximation Methods
398
[Chap. XIY]
perturbation will continue for some period of time, depending on Hi(t). see the situation more clearly, let us expand a n (f) in powers of HI as
a n (t)
=
<&>
To
(XIV 4-12}
T
where
cff(t)
are of the rth order in HI.
We use (XIV 4.7)
and write equation (XIV 4
Using the expansion (XIV 4-^)
| o> = ^
ih
Equating
coefficients of the
w
*0) as
obtain
HLaS (Oe TO
same order we get
(XIV4.15)
*>, Ulf
where
r
=
.
0, 1, 2,
Thus the
0)
n
They specify the initial conditions of 0) the problem. Initially we assumed that all the a except c$\ where n s, are zero, so the system is in a definite stationary state when H' starts perturb= dsQ \). Therefore, in accordance with ing. From (XIV 4-1&) w have a s (t) o4
are constants in time.
^
,
the initial conditions, (XIV. 2.15} is then
we can put
0)
a^
=
5 ns .
The
first-order
equation from
or
We have thus made an estimation of the order of We may now work in terms of the a n rather than the a proportional to H'.
By
magnitude } .
If
m^
separating out the term referring to a s
5,
of the a n
then am
we can
.
is
rewrite
(XIV 4.13):
in~a n (t) =
H
r
ns (f)a s
(t)e^^-^ +
HL(t)am (t)e^^-^.
(XIV 4^6)
WIT^S
Hence, to the
first
order in H',
we
obtain
^a
s
= FU,
(XIV. 4-17)
Time-Dependent Perturbation Theory
[XIV.4.]
which
solved
is
by
=
a.(0
If Z?' is
399
independent of time a.(0
~
exp
H
'
r
ss (t)
fi.
(XIV 4-18)
we have
=
-
exp
I
(*
a real function the probability |a*(Z)( 2 does not change, so no transition can result from it. But, to a first-order approximation, the term This is the same H'ss changes the angular frequency by an amount (1/K)H' SS
HsS
Since
is
.
as changing the unperturbed energy
E=
H that #o + H'n r
by
ss ]
is.
(XIV'4.20)
.
XIV.4.A. Sudden Switch-On of a Time-
Dependent Perturbing Energy From the previous discussion, obtain the equations ihj-t
This
is
solved
if
n
=
a n (t)
9^
=
past and put
t
=
a n (f)
<#.
r
=
before
and as
I
and
s,
= _1
[' J
(XIV4-21)
its application,
we choose
In this case
fi
and hence the
the initial time from the
(XIV 4^1)
fl^>c*/c.-JW
leads to
(XIV 4.22)
now assume that the perturbing energy is independent of time except
it is
switched on and
off at
some small time
the first-order probability amplitude at time
where we use the boundary condition E n probability that the system is at time t state s at
The
=
-%f* HUtW'W**-*^.
H
that
take am
Hi. 6 (Ww.-A>.
In order to ensure the vanishing of vanishing of all the a n except a s for n
Let us
we
by <*(*)
infinite
s,
t
=
=
t
interval
(0, f).
Therefore,
is
13 *
=
>.
n when
it
at time
in the state
t
Hence, the was in the
is
largest contribution to the probability
comes from those states
foi
Approximation Methods
400
En = E
which.
that
9
Thus the conservation
[Chap. XIV]
for those states for which the energy
is,
of energy,
when
is
t
very large
(in
is
conserved.
accordance with the
is an automatic consequence of the time-dependent In general, a large probability results from perturbation theory.
principle of uncertainty),
or
AE ^
(XIV 4-25)
7. t
see that for large t the spread AE in energy is very small and therefore conservation of energy comes in accordance with the principle of uncertainty. can be regarded as a final-state energy. If we The energy E n in (XIV
We
4^)
confine ourselves to the case of discrete energies the probability will not be as large as
it will
energies. This
made
be in the case of continuous or nearly continuous final-state because the energy spread AE in relation (XIV 4.25) can be
is
smaller than the discrete energy spread.
In order to calculate the transition probability per unit time, we must first count the number of possible final states. Let us suppose that the system is contained in a large cube of dimensions L and that the eigenstates \E n t} are normalized to unity in the cube. If a group of final states n have nearly ,
E
the same energy as the initial state E s we can find a density function for these final states. Let k be the wave number vector. If we assume periodic ,
3 boundary conditions on the walls of the cube L the k values must satisfy the relations ki = (2-jr/L)n^ where n are positive and negative integers. 4
The number
of possible
k
values
is
given by
L
AT-
3
,
Hence
AN =
n 2 dn dQ, =
L
3
dk dO.
k*
We may now define the number of final states per unit volume as AN = dk JO = Wdkdti /T7 TTr ,
where
7
,
pk
jj
pk is the density of the final states.
.
,
,
.
(XIV 4.26)
>
.
3
E = hV/2m we :
Using
)
2mE dEdti dE dO = -jr TT-rp
can write
,
PE
where PE dE
E+
dE.
We
is
the
shall
number
of final states
assume that
independent of the energy E.
whose energies
(XIV 4.87) lie
in the range
E
}
this density pE of the final states is nearly
Time-Dependent Perturbation Theory
[XIV.4.]
401
With these premises we can
calculate a transition probability from a state E with energy 9 to a state IE, t) at time t with energy E. Since \E 9 IQ) at fo we want to measure the energy E we shall not be interested in times t, which
=
,
are short or of the order of one period h/AE, say. In order to be consistent with our assumptions that t must be small enough that the amplitude a s does
not vary much, we must state that the times under consideration are small compared to the lifetime of the initial state, but large enough for relation
(XIV. 4-^) to we can write
hold, so a sharp
measurement
WO! = 2
\
where we use the
5
Y
H
\
'**\**(
of energy is possible.
E* - E
For large
t
(XIV 4^8)
*^>
function property fr
x
,
S(x)
=
-1 v Inn IT
cos x\
1 1
X-co A
X*
For a nearly continuous final-state distribution the transition probability per unit time to one of the final states is obtained by multiplying (XIV. 4.88) by the number of final states and integrating over the energy of the final states.
Thus
W= A
\
J
|a n
2 |
p(n)
dEn
.
most favorable when p(ri) is indeand when the interaction energy does not vary appreciably
definite transition to final states is
pendent of
En
over the extent of the final energies. In this case the transition probability per unit time is
WFO = jwhere the
letters
and
F
2
(XIV 4*8)
|fffcl p(n),
refer to initial
and
final states.
E ) in (XIV. 4^8) does not designate a degeneracy; it The factor 8(En only means that the energy is conserved in the transition and that transitions s
take place between states of equal unperturbed energy. Expression (XIV. 4-27) for the number of final states, using can also be written as dN_
V where we use the relation
mentum particle
dE
_ ~
4T??
2
dp
_ ~
2
47rff
(2irK)* v dp,
with
2
p /2m,
dE
(27rK)*v' v
and p being the velocity and mo-
mass system of the final-state particles (emitted must be multiplied by the spin multiplicity in the This nucleus). The relevant factor is (2/SV + l)(2Jj? + 1), where Sp = the spin
in the center of
+
final state.
emitted particle and IF = the spin of the formula for the number of final states is of the
E
final nucleus.
Thus the
correct
402
Approximation Methods
[Chap. XIV]
In the case of the photon emission the multiplicity of states must be ascribed to the two possible states of polarization, so the factor (28F 1) must be
+
replaced by the number two. Result (XIV. 4.29) is the fundamental formula of the time-dependent
perturbation theory. In
many
HFO vanishes and r
cases
higher approximations. For this we shall apply the
we have to proceed to method of an integral
equation formulation of the time-dependent perturbation theory.
XIV.5.
The
Integral Equation Formulation of the
Time-Dependent Perturbation Theory The formalism bound
to be developed in the following
and related problems. In
states,
all
cases
is
applicable to scattering,
we have a
of the system, including the interaction energy.
total
Hamiltonian
For simplicity we can take
a system consisting of two interacting parts. The Hamiltonian of the system has the following characteristic properties. (a)
The
total
H
'
Hamiltonian
is
H
= HQ
+ H',
where
H
Q
is
such that in the
some similar internal structures and would suffer neither a scattering nor form a bound state. (b) In a scattering process we would be interested in a transition from one absence of
eigenstate of (c)
the two parts of the system would have
H
Q
to another.
This transition would take place under the action of the perturbing
energy H'. (d)
For bound
states
we would be interested in
finding the stationary states
corresponding to a time-independent total Hamiltonian. All these problems can, in principle, be formulated in a rather simple
way
by using the unitary operator
where the state vector
|f ', t)
satisfies
in the interaction picture, as in U(tfa)
shows that
if
t
=
Schrodinger's equation,
(XIV 4.5). The
definition (XIV. 5.1) of that and the fa, /, eigenvalues f of the f are the same at times fa and t. If f is the energy
then
U=
'
'
complete commuting set itself, then the definition (XIV. 5.1) contains the law of conservation of
[XTV.5.]
Formulation of Time-Dependent Perturbation Theory
energy. The integration in (XIV. 5.1) can also be taken as a discrete tion or both, depending on the nature of the problem.
403
summa-
We
can use the eigenstates If', fo) of f and set up a representation. In this representation the matrix elements of the unitary operator are ?',
*o|f",
t)df"#",t\U(t,
A comparison with the previous formulation of the time-dependent theory shows that the transformation functions (XIV. 5. 2) are the probability amplitudes of a transition taking place from the state |f ', i ) to a state If", Z) during In the case of discrete states, the total probability that a transition will take place is the time interval
(fo, t).
p=
We
/
Kr,
=
oif",
i.
thus see that the whole problem of calculating transition probabilities
can be reduced to the problem of finding the representatives of the unitary operator U.
we
we
operate on both sides of (XIV. 6.1) by obtain the integral equation
If
=
Ufa)
The
1
-
H
I
HI and
z (?)U(1fto)
di
integrate
from
to
I.7.B).
the
first
(XIV. 6. 8)
.
and
order in Hr(t') the iteration method yields
Ufa) = In general,
t,
f
solutions of this integral equation can be found (see Sections I.7.A
To
to
1-|
rHi(t')
dt
f
=
+
1
n> Jto
Ui(t).
(XIV.54)
we have
U=
1
+
Ui
+U + Z
-
+ Un +
,
where
u*
(t)
=
'
n\
(T)"
/*
"
K p H*> H*v)
-
i
*
H'(] **
*M (ZI7.5.5)
(See Section 1.7.8.)
time h (= 0), Ho is the only observable forming a complete commuting and set up a representation, where set by itself, then we can use it for f at If at
H\E n The
,
to)
= En E n
transition probability to the first order
is
,
ft,).
Approximation Methods
404
~
EH
~ \E,e~~^E EM dt
r
r
[Chap. XTV]
t
f
or ffk,(f )
JT'
where we use the
and
letters
F
and 0,
initial states, respectively. If
during
that t,
#'
(XIV. 6. 6)
,
index n, to signify final
in place of discrete
we assume
action from initial time to final time
its
2
(i/w-f H'FO then
independent of time
is
we
get
Further discussion of this transition probability proceeds as in the previous section. The above formalism is also an illustration of the physical meaning of a transformation function
which
a representative of the unitary operator of f at the initial
is
in a certain representation obtained
from the eigenstates
time.
H
f
the representative FO vanishes or is small compared to other representatives of H', then it is necessary to calculate higher-order transition probabilIf
The
ities.
second-order transition probability amplitude
as follows
is,
from
(XIV. 5.5), given by Z7$ where the
= -~
IH^Hioe^^'-^^e^^''^
dkdh X)
/ implies summation over intermediate
letter
assuming that
f*
H
f
of
is
time
independent the integrals can be carried out and
(o, t)
_
#0)
1
EF - Eo The transition Ep = EI. The since of a
E
first
final states.
EQ = EF The
we assume
initial
o.
before,
we
obtain
e (i/h)t(EF -Ei)
EF -
_
1
El
will increase occurs for
EF =
E
and
a statement of conservation of energy state while EF is the energy of any one
transition
Ep = EI need not
occur.
which a summation
is
The
states
taken.
that only energy-conserving transitions can occur, then the
term in the square brackets
HFQ =
As
at least during the time interval
is
called "intermediate states" over
order theory. If
I
which the probability
transition
the energy of the
group of
$i are If
is
for
-
states.
^],
H
in
(XIV. 5. 8) must be treated as
in the first-
1
does not produce any transitions in the first order, then In this case the summation in (XIV, 5. 8) need not include the terms
= F and
1
=
0.
The second term
Note that the terms I
= F and
1
=
are not singular. in the square brackets has a large energy denominator
Resonance Transitions and the Compound Nucleus
[XIV.6.]
EF
EI (nonconservation of energy in the intermediate negligible compared to the first term. The result is then
The second Hence the
(XIV. 5.9) can be treated
factor in
transition probability per unit time
or
"
states)
405
and
is
as in the first-order theory.
is
"
~~*
F-
(XIV.5.10)
In the second-order perturbation theory the transition from the initial to the final states F takes place through the intermediate state I.
state
These states
exist for a short time,
and
their energy, in accordance with the
uncertainty principle, is not measurable. In connection with the density of final states pr occurring in (XIV.5.10), it is important to note that if the particles have spins the density of states increased
the multiplicity of spins. This is essentially the case for a from a nucleus. This multiplicity depends on the spin orientation. Thus the correct density of final states is obtained by multiplying is
by
particle scattered
PF
by
(2/S
+
1)(2Z
+
1),
where 8
is
the spin of the emitted particle and / the
spin of the nucleus.
XIV.6. Resonance Transitions and the
Compound Nucleus A
particle scattered
ferring
it
from a nucleus can loose some
of its
energy by trans-
to the nucleus particles. This results in a redistribution of energy
among all the particles of the nucleus and the incident particle forming a "compound nucleus with the original nucleus. According to Bohr's theory, each of the particles of the compound nucleus will have some energy but not enough energy to escape from the compound nucleus. There is, however, a 77
probability that after a certain time the incident particle may regain the lost energy and escape from the nucleus, leaving it in an excited state. If the
escaping particle of the nucleus is
is
same kind as the incident one and the initial state unaltered, then the process is an "elastic scattering.
of the
left
7'
77
for only one special case; the probability of "inelastic collisions a different an or in excited state left a nucleus particle being example emitted is larger than the probability of elastic collisions. There is also a
This
is
high probability of emission of radiation.
Approximation Methods
406
[Chap. XIV]
Nucleai collisions can be described in terms of the compound nucleus as transitions
+
from compound
Incident particle emitted particle.
+
and initial states: > compound nucleus
states to the final
initial nucleus
>-
final nucleus
The compound nucleus can conveniently be represented by the intermediate states of the time-dependent perturbation theory. In this way the quasistationary states of the compound nucleus may have some interesting consequences. One of these is the so-called "resonance phenomenon." If the
+
energy of the incident particle is such that the total energy of the particle nucleus is nearly equal to one of the energy levels of the compound nucleus, then the probability of the formation of a compound or an intermediate state
between two resonance
increases. If the total energy falls
of
compound
nucleus formation
In the language
is
of perturbation theory, the quantity
irregular variations, instead of being constant.
probability would behave
like
ER)
1/(E
KlHio
2
Z ,
WFO
\HFO
2
will
\
have some
Near a resonance the transition
since
27T
PF
E = Ec
levels the probability
smaller.
EC
h
Pr
.
(XIV. 6.1)
W
FO very large. At the resonance energy, is infinite. The infinity results from disregarding the short lifetime of the compound state, for formula (XIV..6.1) does not take this into account. The correct equations describing the decay of a compound state can be obtained
Near
(resonance),
is
from the integral equation for the unitary operator U(t)
There are three possible groups of
(a)
The
particle (b)
+
The
:
states.
initial states \Eo) with energy
EO
refer to the
unperturbed states of the
nucleus. final states \Ep) with energy
Ep
and
the
be discrete states) with energy
EC
refer to the residual nucleus
emitted particle. (c)
The compound
refer to the
compound
states \Ec)
(assumed
to
nucleus.
The above groups of states arise, for example, in a radiative capture of neutrons. The energy levels of the initial and final states form a continuous spectrum. From a group of one.
We
shall
initial states (0)
assume that no
states (F) take place.
the experiment will pick out a particular
direct transitions
Thus the
representatives
from
Hoo
initial states (0) to final
f ,
HFF',
HOF
of the inter-
Resonance Transitions and the Compound Nucleus
[XIV.6.]
H
407
r
shall be neglected. If we are interested in one level we can action energy assume that there is only one intermediate state in the resonance transition.
Under the above assumptions the transition probability amplitudes from one another, from an initial state to a final state, or from an initial state to a compound state, are respectively given by initial state to
=
(Eo\Eo>}
UFO =
(EF\Eo>)
Uoo>
>
Uco>
=
-
H'o
I
H'pc
-
(EC \E
e W<*'-***Uco> dt',
J*
H'co
>)
~
I
ri
Z F
B'cr
,
['
fn Jo
(XIV. 6. g)
e<*<*-**'WUoo>
eM<*<-*>WUr ,o>
dt'
dt'.
last equation for the transition amplitude we took into considthe fact that transition from an initial state to a compound state eration
In writing the
cannot be independent of a certain set of initial and final states, since a compound state amplitude Uco can disintegrate either to the initial state (0) or to the final state (F). At time t = the states
=
and O are the same and there are no compound f
UFO
=
=
0. From the three-coupled set of 6. we see that we can substitute for Uoo and UFO* (XIV. 2) equations integral in the third equation, from the first and second, and obtain a single integral equation for the compound state probability amplitude Uco The kernel of
states, so J7oo'(0)
1
and
>
Uco'
f
f -
the resulting equation involves both first-order and second-order terms in the The boundary conditions above imply that the set of interaction energy
H
equations
(XIV. 6. 2} Uoo>
=
f
.
can be replaced by 8 o>
UFO = ft
co "
v h F
f*
eMW'-wUco'W df,
eWW'-wUcoW
H'FC
'
f
-1&OC
dt',
(XIV. 6.3)
e ->
yo
,
Substituting for TJo"o and Uyo' in the third equation, from the first and second equations, we obtain an integral equation for Uco> in the form f
a(0 =
-r jOf n>
(0
^ ^ l^o /i
Fewwu**-*<w,
JO
(XIV. 6 4)
Approximation Methods
408
[Chap. XIV]
where
= UCO '(fl,
ac (t)
T = -
=
T
T
+
(XIV.6.5)
IV,
iH'col'-eWt'-'"^f "E Q
dt',
(XIV.6.6)
dt'.
(XIV.6.7)
hJo
IV
=
I''
fl
JO
Z) !#^|VWt'-n<Ec-M p
For large i' that is, when EO (XIV. 6. 6) and (XI 7.0.7) by
To
Ec
near
is
we can
replace the definitions
= ij^
\H'co\*8+ (E c
- E )e-WV<'-
(XIV.6.8)
Z
\H'cp\**+(Ec
- Ep)e-WV-*'\
(XIV.8. 9}
o
and IV
=
*
F
B+(x) is defined by (III. 8. 19). It is clear from the definitions (XIV.6.8) and (XIV. 6.9) that T has real and imaginary parts. The real part of T is %h times the total transition probability per second from state (C) to all other states. The imaginary part of r is related to the "self -energy" of the state (C) it will not be discussed in this book.* The real part of T is independent of ".
where
;
The
integral equation /,\
ac (t)
-- -_ _
(XIV. 6.4)
=
is
solved
^ *77^[>-(r/ft)* 1Ilco\. e
p (i/fi}(Ec-Eo)t~\ 6 J.
JT^ ^(&c
We introduce a dispersal time
ITT
N
&o)
r for the
Tr
by
/"F7T7/? m\ (A* V'0.10)
l ~r r ,
compound
state (C)
= P,
by (XIV. 6. 11
where, in accordance with the uncertainty principle and the experimental data on the energy widths, T will be very small; therefore the term exp [ (F/A)i] = exp ( t/2r) can be neglected. In this case the formation of the
compound
state is proportional to
which has the form
of
a resonance or intensity distribution with an energy
width given by the real part
r
=
of F,
\H'coWc -Eo)
TT
O
+
^
2
\HcF\ d(E c
- Er
),
(XIV. 6. 13)
F
where the imaginary part of T is neglected. Formula (XIV. 6. IS) was first derived by Breit and Wigner. f The
*
t
maximum
See W. Heitler, Quantum Theory of Radiation, Oxford Univ. Press, Oxford, 1954, p. 168. G. Breit and E. P. Wigner, Phys. Rev., 49 (1936), 519.
Resonance Transitions and the Compound Nucleus
[XJV.6.] of
X;
2
occurs at the incident energy
E = Ec
,
and
it is
409
given by
'
Hence the energy width at
half
maximum
is
AS = 2F = T = is
7*,
y R the quantity T
so in terms of energy width
where (l/K)y R
given by
defined
is
by (XIV. 6.16)
IT*,
the disintegration probability per unit time.
XIV.6.A. Transition Rates and Resonance Phenomena In accordance with formula (XIV. 4-^ and expression (XIV. 4.27) for the of final states, the number NF of transitions per unit volume per unit
number time
is
given by
Np =
\H'FO -\i TT/i
The number NF can be used
Z \
& (2SF + up
1)
(2IF
+
1).
(XIV. 6.16)
to introduce a quantity VOF, called the "reaction
cross-section per nucleus" for processes involving scattering of particles from (A general discussion of scattering theory is given in Chapter XVII.)
nuclei.
Thus
VNF =
n
v
(XIV. 6. 17)
where
= V=
noVo
With
V=
the
number of incident
particles per unit area per second,
volume of the space.
and an assumed density n Q
1
TTfl
=
1
per unit volume,
OF
4
1).
we
obtain
(XIV. 6. 18)
The quantity
H
(C) can either return to the initial state or
make
a transition to a final state.
the ..decay to the final state results in the emission of a photon, then the probability of transition to the final state per unit volume per unit time, If,
neglecting spin,
is
^ where p y
=
=
^ IW ? = ^ IW $
(XIV 6 -
-
19}
In the same way the probability of transition state per unit volume per unit time is
hu/c and
to the initial
=
vy
c.
Approximation Methods
410 1TO
where po
= mvo and
=
\HOF\
Z
=
Using (XIV. 6.11} we define
Ty Hence the
=
total energy width
1
=
^=
2
I#OF -|IT/I*
-f-! ITU*
Vo
[Chap. XIV]
(XIV. 6.^0)
\H'OF \*vo,
2
l-HVo|
.
partial widths
by
\^ A Ty
=
Yo
3
Jj
(XIV .6. 21}
TO
is
=
2r
2
or
-
=
T
+
(XIV. 6 M)
TO
Ty
we
neglect the growth of the compound state, then the corresponding probability amplitude for the state (C) is
If
n t^C
where TO and rY are the
~~~
life
/> *^
n/ft
"~~"
(YTV ^^XJl.
z>~(f/2)(i/rY +i/Tc,) " j
r
of the corresponding states against the
^\j
fi .
decay
of
the compound state. However, the formula that takes into consideration the occupation of the compound state arising from the transition of the initial state
is correctly given by (XIV. 6. 10}. In this case the number of nuclear reactions* per second per unit volume
O-OFVO
=
a
2 fi
|
^
(XIV. 6. 24}
TF
>where the reaction takes place in the order initial state compound For the it state. reaction where follows the order initial final
compound volume is
state
*-
initial state, the
number
0-00^0= ac
of reactions per
state
>
state
>-
second per unit
2 |
TO
Hence the corresponding
is
(XIV. 6. 95)
cross-sections are
If the incident particle
happens to have a velocity such that we can write
it
hits the
resonance exactly, then from (XIV. 6.20)
*
E. Fermi, Nuclear Physics, Univ. of Chicago Press, Chicago, 1950, p. 156. Notes comby J. Orear, A. H. Resenfeld, and R. A. Schluter.
piled
Resonance Transitions and the Compound Nucleus
[XIV.6.]
411
the resonance velocity of the incident particle and TOR is the value of the partial width at the resonance velocity V R Combining (XIV.6.27) and
where V R
is
.
(XIV. 6.28) we obtain the cross-section a
o in the
form
(XIV. 6.29)
=
where X# a OF
h/mvn. Similarly, the Breit-Wigner formula for the cross-section
is
=
* or
*****
(EO-EJ' +
= h/mvoIn an atomic collision inelastic processes are
(xiv.6.so)
T*'
where X
rare, since the interaction be-
tween the incident particle and the individual electrons of the Therefore,
an
incident particle in
most cases
atom
is
small.
through the atom with-
will pass
out losing any energy, its deflection being produced, essentially, by the average field of the atom (elastic scattering). The cross-sections for emission of radiation (inelastic processes) are small for
atomic
collisions.
However,
for
nuclear scattering phenomena the interaction between the incident particle and the nucleus is strong, and therefore the former cannot go through the
nucleus without transferring some of its energy to nuclear particles. In a short time the incident energy will be shared by the nuclear particles plus the incident particle. After a comparatively long time there will be a probability that one of the nuclear constituents will absorb more energy than the rest of
escape from the nucleus in a state differing from residual nucleus is left in an excited state. If the escaping
the particles, leading to its initial state.
its
The same kind
particle is of the
as the incident one
and
if
the internal state of the
a nuclear elastic process. In contrast in atomic nuclear to collisions, scattering the probability of inelastic processes is much larger than the probability of elastic processes to occur.
nucleus
is
not changed, then the result
is
In general, every nuclear process must be treated as a many-body phenomenon. In particular, a nuclear collision cannot be described as a stationary state where the energy of the system is sharply defined. The many-body aspect of nuclear collision phenomena leads to a "quasi-stationary state" of a compound nucleus where the energy is not sharply defined. Such a state arises
from the
finite lifetime of
the
compound
state,
which
is
very large compared
to the time required for the incident particle to pass through the nucleus. The stationary states of a compound nucleus, besides being the cause for elastic, inelastic,
and other
The knowledge
collisions,
are responsible for resonance phenom-
can be utilized to obtain the spacing between neighboring levels of the compound nucleus. The spacing between levels is a function of the mass number A and the excitation energy of the ena.
of resonances
Approximation Methods nucleus.
Thus the knowledge
of spacing
is
[Chap. XIV]
important to the study of nuclear
structure.
XIV.6.B. Problems Assume that the only compound nucleus
1.
= 8A
to Sc
spin states that resonate refer
By counting the initial parallel and antiparallel spin states, that the probability that incident particle will form a parallel spin state
show
=b |.
with the nucleus is (SA + l)/(2&i + 1) and that the incident particle forms a state with antiparallel spins is SA/&SA +1). 2.
Using the results of problem
prove that the probability that the beam will be found to
1,
incident particle and target nucleus in an unpolarized have the spin Sc of the compound nucleus is
+ Hence show that the
1)
and (XIV.6.30) must be weight g(Sc) of the compound nucleus spin state. 3. A charged particle moving in a plane normal to the lines of force of a spatially uniform but temporally varying magnetic field B obeys the equation multiplied
by the
cross-sections (XIV.6.8&)
statistical
d zu
+2
.
,
t
^+ du
.
.da l
-u =
t
0,
where
u co
dt*
co
The
=
x+ exp 2
#+
=
-
(
0,
J
i
I
co
#+
d),
=
Xi
+i
Larmor frequency.
probability amplitude a c for the formation of a
obeys the equation
where
^
*"
h
2
>
and the amplitude ac can be written as ac
with /72/t.
=
12*
=
=
V
compound nucleus
[XTV.6.]
Resonance Transitions and the Compound Nucleus
Comment on 4.
413
the oscillators with real and complex frequencies.
The amplitude
of
a
harmonic
classical linear
oscillator
with a reaction
force
ws = satisfies
^
3c 3
2
the equation d*x
.
dx
.
2
^+
o*
+ T^
where
Show
that the amplitude of the oscillator
x
where
b satisfies the
=
fce-C
1 /2
is
given by
^,
equation
s "- * with
2 *>
=
Thus 1/7
7
COQ
is
2 .
The energy
of the oscillator averaged over one period is
the lifetime of the oscillator. Therefore the electric vector of the
emitted radiation can be taken to be
E= It represents
Etf
a certain intensity distribution
where
E= / J(oj)
5.
=
dw
=
Jo
7
=
width at half maximum.
total intensity,
For a periodic perturbing energy
H' show that the
Then
Ac-^
+ B^*,
transition probability amplitude to the first order
is
calculate the corresponding transition probability per unit time. Discuss
the cases where
Ep = EQ
+ ^co and Ep =
Eo
ftco.
CHAPTER XV
INTERACTION WITH
RADIATION XV.
1.
The
Einstein Coefficients
transition of
an atom from one
state of energy to a state of lower
energy leads to emission of electromagnetic radiation. The reverse process of absorption of radiation is the result of an upward transition caused by the action of a radiation field on the atom. For historical reasons further illustration of Einstein's unfailing
we
shall briefly discuss his
judgment
and
also as a
in all paths of physics,
theory of spontaneous emission of radiation
of atoms. This constitutes
one of the
by a
important examples in physics where indeterministic reasoning plays a fundamental role. Consider two stationary states of an atom, a low state B and an excited system
state A. Einstein
a
assumed that
if
the atom
probability of transition to state
VAB==
B EA
is
first
found in the state
A
}
then
it
has
by emitting a photon of frequency
EB
~h~-
In a large assembly of such atoms the number N(A) of excited atoms that B per unit time is proportional to their number N(B) in the
return to state initial state.
The
reverse process VAB>
The
B
resulting radiation will produce a certain probability for the >-
A which represents absorption of a photon of frequency
latter probability is proportional to the radiation density of the
corresponding frequency. The emitted radiation will influence not only the absorption process, but also the emission process itself that is, the transition A *- B. This is called "induced emission/' and it is proportional to the radiation density for the frequency VAB* These assumptions, together with the use
Maxwell-Boltzmann distribution, gave Planck's formula. For the calculation of a relation between emission and absorption rates
of the
we
can proceed in reverse order, by making certain assumptions for a state in thermal equilibrium. 414
Einstein Coefficients
[XV.l.j
*kte>
TI e numbers of (a L ,l the Maxwell-Boltzmann law,
atoms belonging
N(A) =
g (A)e-
to different levels are
E*'< T
given by
(XV. 1.1)
,
where g(A) is the statistical weight of the level A. (b) The radiation density is given by Planck's formula:
vw _ To
these two assumptions
we
shall further
-f
add the "principle
(XV.1.2) of detailed
balancing." (c)
In
statistical
equilibrium the rates of each elementary process and
its
inverse are equal. (See Section
XVIIL4.) The "induced" and "spontaneous" emissions together are regarded as one elementary process and the one kind
Now, let a (A, J5) be the probability per unit time a making spontaneous transition with emission of radiation to each level of lower energy. The number N(A) of atoms in the excited level will
of absorption as its inverse. of
B
decrease
by radiation according to
= -
a A 5 ) N(A) J (
[X)
dt
>
(XV. 1.8)
where the summation
is taken over all states of lower energy. If only spontaneous emission occurs, then the system will decay according to
where TA
is
the
mean
lifetime of the
atom
in the state A. In general, rA
is
given by 1
the radiation
_ X
assumed to be isotropic and unpolarized and have spectral energy density p(v/c] dv/c in the frequency range dv at v, then the calculation of absorption and induced emission rates can be done in a rather If
field is
C denote an energy level higher than A; then transition via absorption takes place at a rate
simple way. Let
from
A
to
C
N(A)(*(A,
where p(A, C)
is
OP
Q>
(XV. 1.6)
the probability per unit time of making transition from A to emission resulting from transition from C to
C by absorption of radiation. The
A
as induced
by
radiation
itself
takes place at a rate
N(C)fi(C, A) P \c/
The
probabilities
a and
ft
(XV.1.7)
are closely related to the interaction of the
atom
416
Interaction with. Radiation
[Chap. XV]
with radiation field and they are independent of a particular equilibrium
From
the principle of detailed balancing
N(C)
[a(C,
A)
=
+ fl(C, A) P
state.
follows that
it
N(A)fi(A,
(j)]
OP
Q-
(XV. 1.8)
Thus, from (XV. 1.1), (XV. 1.2), and (XV. 1.7), we obtain
g(A)0(A, C)
=
g(C)P(C, A),
A)
=
8r
,
Hence the
7
total rate of the emission process Pemission
From (XY.i.)
it
=
(
C
(XV. 1.9)
(Z7J JO)
^)-
is
+
N(C)Q,(C, A)
(Z7J .11)
l
follows that c 2p
is
the average
number
of photons in the energy range hv.
We may,
therefore,
write the total emission rate as Pemission
=
N(C)Q,(Q, A)
\ft(v)
+
1].
(XV.1.1&)
This result will be compared with quantum mechanical calculation.
XV.2. General Formulation of the Radiation Problem The
Hamiltonian of a system of charged particles interacting with the radiation field and between themselves is given by total
H
= Ho
+ H',
(XV. 8.1)
where n
1=1
and
a.i
and
The
jS t-,
for i
=
1, 2,
,
n, are
Dirac matrices corresponding to parti-
sums The Coulomb energy of the system is represented by the second sum in (XV. 2. 2), where the first sum represents the total relativistic cles.
last
in (XV.%.%) represent the energy of the radiation field
[see (XI. 2.20)].
kinetic energy of the particles.
We
are assuming that the vector potential is an operator satisfying the equation
A(ri) at the position r t of the ith particle -
V-A =
0.
In this case the scalar potential
A
4
of the radiation field
can be set
General Formulation of the Radiation Problem
[X.V.2.]
417
equal to zero.* In this book it will not be necessary to explain the full construction of the radiation theory; we only need to know some of its direct results.
Therefore the discussion based on quantum electrodynamics will be
omitted.
The
general state of the system will be described by a state vector \&}
satisfying Schrodinger's equation:
ih
1*>
=
(fTo
+ #')!*>.
(XVJt.fi
the particle field is not quantized, then j^} depends on the coordinates of the particles and the variables used to describe the radiation field. For a If
system of identical particles the state
jtP)
must be antisymmetric
in all
particles.
Hence, neglecting spin, the interaction
is
given by
For the vector potential A(r) we can use an expansion similar to (XI.2.12),
W, where us (r) are plane waves
of the
form
us (r) = V(47ry The matrix elements
of q\ 8 (n
,
interaction energy
\
For a transition zation (X
given
=
1
es .
by
-^1
-^+D1 -
for
one electron can be written as
s
X
r^
=
(XV&.S)
H/
(/c *' r)
as follows from (FT 1. 1.57), are given
(gx.)n+l,n
The
(XV. 2.7)
2
(nXs
I"
/TT7^ (XV. 2.8)
= ~
where only one photon with a definite polarif emitted or absorbed, the matrix elements of are
of the electron,
or
1) is
H
by
(a, n.\H'\b,
na
+
1}
(XV.8.10)
*
W.
Heitler,
Quantum Theory of Radiation, Oxford Univ.
Press, Oxford, 1964, p. 125.
,
418
Interaction with Radiation
where the initial
and
|^}
and
final states,
nonrelativistic limit (a, n.\H'\b,
n8
wave
are the electron
E78
[Chap. XV]
Coulomb potential in the photon energy in mode s. In the
functions in a
Jiu8 is
we have
+ 1} =
In both cases we have ks e (a)
0, so
momentum
the
operator
commutes the wave
with the plane wave. We further note that in formula functions are normalized in a volume V, so they have the dimensions
(XV. 2. 11}
of (length)- 3 / 2 .
XV.2.A. Dipole Radiation The a
interaction energy between a radiation field
transition of the unperturbed
This can occur in the form of
and an atom can cause
system from one state of energy to another. emission or absorption of radiation from the
atom. The atomic energy levels are low compared to the rest mass of the electrons. Therefore, for most purposes a nonrelativistic treatment of the f = problem is justified. In this case the interaction energy is of the form
H
(e/mc}p-Aj where the term proportional to A* can be neglected, giving rise to transitions in
Let
which two quanta are involved.
Eo and EF be any
initial
and
final state energies of
we
case of emission of radiation of frequency
the system. In the
have, from the conservation
of energy, the relation
EQ = EF The
~h /wo ==
EF
Hr Ey.
transition probability per unit time is
WFO - y where HFO
is
given
by (XV. 2. 11} e HiFO = ~~m
in the
(XV, 2.12}
pr,
form
/r2irft*(n,
VI
2
HFO
Eya
+ in RFO
>
J
where
RFO
The number given
=
I
i#[e
w 'pe ik
of final states consists of the
*' r
]\f/o
d*x.
number
(XV.2.14} of radiation oscillators,
by
We have
assumed that
all
the light quanta have the same frequency (within
General Formulation of the Radiation Problem
[XV.2.]
419
dEy ),
the same direction of propagation (within the solid angle d!2), and the same polarization. Hence the transition probability per unit time for the emission of a photon of energy h& in a direction within dO is given by
WFO where we replace ns
_
&"
=
in (XV.2. 13)
Ci)
CL\L
i
^-.
I
,.
-
,
by the average number n
of light
quanta
and propagation vector k. This is per radiation oscillator of frequency in to order make sure that the emitted radiation is independent of a necessary Result (XV.2.16)
particular oscillator.
is
of the
same form as
Einstein's
formula (XV. 1.1$).
An
elementary application of formula (XV. 8 .16) comes from the following simplifying assumptions.
The dimension
atom is negligible compared to the the is, perturbing energy is constant in the direction of the emission, so it is a constant and changes significantly only over a distance of the order of a wavelength of the light. (a)
of the emitting
wavelength of the radiation.
(b) If
we
That
E is the energy of the atom and X the wavelength of the light,
then
assume that X is of the order of hc/E. The dimension of the atom, estimated from the potential energy E = e 2 /a, gives the result a/X = e z /hc, shall
the fine structure constant.
With these assumptions the factor exp (ik-r) is effectively unity in the region where $F and \f/o are different from zero. Putting p = mv and v-e = v
cos
0,
we
obtain from (XV. 2.16) the result
WFO dO =
cos 2 6
dtt\v FO
2 \
(n
+
1),
j
(XV.iB.17)
where IT^ol
The components
2
=
VIFO
+
VIFO
+
of the oscillator velocity
so the transition probability
WFO
dtt
=
7!*,,
Vpo
satisfy
becomes
~
cos 2 B dti\XFO 2 c(n \
+
1).
(XV. 2.18)
now be compared with Einstein's formula (XV. 1.12). The term in (XV. 8. 18), with n == 0, gives rise to spontaneous emission and is, course, independent of the intensity of radiation. The second term is
This result can first
of
proportional to the intensity of the radiation
n
of frequency w,
which was
420
Interaction with Radiation
[Chap. XV]
there prior to emission process. This term corresponds to Einstein's induced emission of radiation.
The
total intensity of radiation per unit time
can be obtained by multi-
plying (XV. 2.18) by integrating over the angles. Thus if a is the angle between the vector (the position of the electron with respect to nucleus) and the direction of propagation fe, then for the spontaneous radiation we fu>
and
X
obtain
I do =
|5t |XFO
2
sin 2
a
|
(XV. 2.19)
dfl,
where
=
dQ,
by
sin
a da dp
replaced by sin a. The total intensity follows from (XV. 2 .19) to 2?r), as to v and j8 from integrating over all angles (a from
and
2
cos 9
2
is
I
=
|X|X|
in ergs per second per emitting atom.
going from
to
F is
tained
(XV. 2.20)
total transition probability for
obtained by dividing (XV. 2. 20) by
AFO =
The
The
2
|X,
~J
all final
(XV.2.21)
|,
vacated by emission can be obstates with energy less than that of initial
total probability that the state
by summing over
ha>:
is
state,
Bo =
^
APQ.
(XV.2.22)
This result can be used to define the reciprocal of the state
mean
life
of the initial
by TO
which
is
=
(XV. 2.23)
^p
of the order of 10~9 sec.
XV.2.B. Absorption
of Radiation
Light can be absorbed from any of the radiation oscillators. The intensity beam for an initial state with average number of n per oscillator is
of a light
/o(u) dco
=
nhucpp
= n (e>
^ -j da do;.
(XV. 2.2$
In the discussion of absorption transition we multiply the transition matrix element by the number po of initial states instead of pp, the number of final states used in calculating transition probability for emission process.
General Formulation of the Radiation Problem
[XV.2.]
The
transition probability per unit time,
Hence the
ratio of the emission
and absorption
Demission *
from n to n
_
71
1, is
probabilities
421 given by
is
T" 1 W-
absorption
repeating the previous calculations per unit time the result
By
we
obtain for the absorbed energy
(XV A
XV.2.C. Problems the direction of polarization is resolved into two components, with one ei, perpendicular to F o, then the other one e2 will lie in the plane determined by the propagation vector fc and F o at an angle (ir/2) a. Show that If
1.
X
X
light of polarization ci is
not emitted, and light of polarization e2
is
emitted
with an intensity given by (XV. 2. 19}. 2. Result (XV.2.20) is almost identical with the formula obtained for an oscillator in the classical theory. classical
in deriving
Consider a Hamiltonian for a Z-electron system of the form
3.
and
Compare the assumptions made
and quantum mechanical radiation formula.
let
n
nr
P =
r-X><, tl
I>. 1=1
(XV. 2.27)
taking the representatives of
By
ih
show that
PFO
=
where UFO 4.
By
defined
-
= EQ
using the results of the problem
3,
.
show that the
.
oscillator strength,
by >o|,
(XV. 2.291
422
Interaction with Radiation
[Chap. XV]
can be expressed as ,
IPO
2*'
=
T"
Then prove that
where
a;
and p x are given by
(.XT. #.#7), so
z
Thus
= 2
(X7.0.30)
-
5. Prove that if emitted light were polarized would not appear in the radiation formula.
#2 or
x$ directions
then XIFO
Derive the transition probability for two-photon emission and show that
6. it is
in
very small compared to one-photon emission.
XY.3. Applications of Radiation Theory XV.3.A. Angular
Momentum
Selection Rules
X
and Dipole radiation depends on the representatives of the operator rules be can conveniently formulated therefore angular momentum selection in terms of the representatives of #3
and x+
=
Xi
+ ix%,
X-
=
x\
ix^.
Thus
(E,\x*\Eo>
=
= It vanishes
f
(EF \r)(r\E
m
d*x
r 2 dr Rn>i>(r)Rni(r)
Jo if
)x,
f
-^
m
}
fj
dd
j^
d
so the selection rule for
Am = m - m =
Yi> m >(0, 0)
Yim (d,
cos 6 sin
magnetic quantum number
6.
is
f
If this selection rule is fulfilled,
provided the angular
OL.
we
then the integration over obeys the selection rule
=
does not vanish
v
- = I
1.
(JJT.3.*)
obtain the selection rule
Aw = Thus
in this case the radiation
From
the
6
(XV. 3.1)
momentum AZ
For x+ and
0.
integration we
m'
-m=
dbl.
(XV. 8. 8)
polarized parallel to the XY- and xa-axis. again obtain the selection rule (XV. 8. 2). All these is
selection rules are in accord with the general discussion of Section XIII. 3. B.
Applications of Radiation Theory
[XV.3.]
423
XV.3.B. Parity Selection Rules From anticommutation anticommutes with the
of the parity operator with
electric dipole operator
follows that
r, It
and that the
electric
(P
moment
can have nonvanishing matrix elements only between states of opposite parity.
XV.3.C. Magnetic Dipole and Quadrupole Radiation If
we
in the
define the current vector f
=
(e/m)p,
'
we assume
^
= ~
HF If
by f
we may
that X
=
l/|fc
is
eik-r
As shown
much
very
expand the "retardation factor" e
2 greater than e /E,
we may
=
term
(fc
X
we may
then
'
:
+
2
i(fc-r)
-
*
.
of this expansion gives rise to the electric
tribution arising from the term ik-r. )
(XV.3.4)
ik r
= l+ik-r -
before, the first
'^"Wo d*x.
c(S)
dipole radiation. If dipole transitions do not exist, then
HFFO
write (XV. 2.18)
form
By
we
calculate the con-
using the vector identity
e)-(r
X
+
f)
(fc-f)(
w -r),
/f
write (XV. 8. 4) in the form I[~2ir?i?(n 3
~VL
K
+1)1
$F(T
where the
first
J
X
f)^o d
term on the right
is
x
~f~
cki^j
I
J
the magnetic dipole term and the second
refers to the electric quadrupole term.
The
from 8) and
intensity of radiation arising
magnetic dipole and electric quadrupole interaction, using (XV. 2. (XV. 2.12} can be written as ,
2 2 |[X(fc.X);UI sin a dQ.
(XV.8.6)
This radiation exists only between states of the same parity.
XV.3.D. Nuclear 7 Ray Emission It follows
from the above
relations that the ratio of dipole
radiations are of the order (a/X)
2 .
In nuclear radiations
it
and quadrupole
has been observed
424
Interaction with Radiation
[Chap. XV]
that the intensity of dipole and quadrupole radiation are of the same order. From (R/X) 2 for 1 Mev y rays and nuclear radius
R= one obtains the
1.33A 1
/3
X
10~ 13
,
ratio
/72V
=
\X /
ri.33A^
X
2
L
3
X
1Q" 13 1 2
10-"
J
^
~~ J. 62 625
where we take A = 238. Thus the quadrupole radiation should be about 625 times weaker than the dipole radiation; thus there is a sharp disagreement with experiment. In general, nuclear dipole radiation is very weak. This can be explained by noting that a system of particles having the same "specific charge" (= charge per unit mass) will not emit any dipole radiation but will emit quadrupole and higher multipole radiation.* From the above formulation, the dipole and quadrupole transitions for several charges are respectively proportional to
and
functions fa and ^o will depend on the relative coordinates of the with respect to the center of gravity. The coordinate of the center
The wave particles
of gravity
and the
relative coordinates are defined
by (XV. 8.7)
Let tively.
X{,
Xij
and
m be the
x components of the vectors riy R, and s t respec-,
Then
Wo dXi
^ Wo ^
2^
QULL
dx
_ - Wo - mi^Wo *
.
Qn
M
^.
,
and i
where
Se t and
=
Mj
d^ji
the apparent charge of the particle different from e^ Neglecting the proton-neutron mass difference, then for a and nucleus with mass number A we have e
-
e\
ei
(em;/M)
is
M
k
H. A. Bethe, Rev. Mod. Phys., 9
(1937), 221.
mA
Nonrelativistic Calculation of the
[XV.4.]
where
rg is
Lamb
'
Shift
the isotopic spin component of the nucleus which
is
425
+1
for protons
and we will consider neutrons to have a negative effective charge equal to about half an elementary charge and protons to have a positive effective charge equal to half of their true charge. For a system whose particles have all the same specific charge we have and therefore for such a system dipole moment would vanish identically. e = However, quadrupole and magnetic dipole radiations need not be small. 1
for neutrons. Thus, for dipole radiation
XV.4. Nonrelativistic Calculation of the Lamb Shift
A
Lamb shift was given in Section XIII.3. and somewhat phenomenological derivation of
qualitative discussion of the
Here we give a
nonrelativistic
shift, first obtained by Bethe.* According to Dirac's theory the states 2 2$i/2 and 2 2Pi/2 exactly coincide in energy, the latter being the lower of the two P states. In the absence of external electric fields the S state is meta-
the
stable,
and radiative
selection rule AZ
=
transition to the
ground state
! 2$i/ 2 is
forbidden
by the
shows 2 2 state Me. than the about Kemble 1000 Pi/ 2 higher by and Presentf and PasternackJ have shown that the shift of the 2S level cannot be explained by a nuclear interaction. d=l. Actually, contrary to the theory, experiment
that the 2 2 $i/ 2 state
is
was suggested by Schwinger, Oppenheimer, Weisskopf that the shift might arise from the interaction of the electron with radiation field. The field It
have always led to an infinite value for the shift. Bethe has shown that the infinity arising in the level shift calculation can be associated with an "electromagnetic mass" of a bound as well as of a free
theoretical calculations
electron.
He assumed
this effect; that
mass
is,
that the observed mass of the electron
must include
the infinite mass here must be regarded as the observed and must actually be set equal to it (mass renormaliza-
of the electron
tion).
From
the remark following formula (XIV. 6.9), the self-energy of an electron in quantum state n, arising from the emission of virtual photons in intermediate states, can be written as
where
En = E n '
tion element |Hi'| *
2
E7
and
is
given
Ey =
"the virtual photon energy." The transi-
by
H. A. Bethe, Phya. Rev., 72 (1947), 339. E. C. Kemble and R. D. Present, Phya. Rev., 44 (1932), 1031. t S. Pasternack, Phya. Rev., 54 (1938), 1113. t
426
Interaction with. Radiation
[Chap. XV]
where Integrating over
photon emission we write for the
all tlie possible virtual
self-energy
where
X is a certain upper limit for the virtual photon energy.
For a
free electron
This represents the change in kinetic energy of the electron for fixed momentum and is due to the addition of the electromagnetic mass of the electron to
its
pee
This electromagnetic mass, being contained must be subtracted from
(or bare) mass.
in the experimentally observed electron mass,
(XV 4.2}
to yield the relevant part of the self-energy.
For the bound
from (XV. 2.29} we
electron,
V /
= (v^ \
\vnn'\>i 2
'
'
j
I
may
write
*
jwt\>'*
nr
Hence
This
is
to
be regarded as the actual
with the radiation Integrating over
shift
due to the interaction
of the electron
field.
Ey
s? one assumes that
(XV. 44} and assuming that
in
to all energy differences
K
En
*
we
En,
i^'i
= me
2
K
is
large
compared
obtain
^' -
2
then the logarithm in (XV 4-5} is very large; can therefore be regarded as independent of n'. In this case we have to evaluate a sum of the form If
,
it
A = X) n
lp'|*(JSn'
- En
}.
f
But from
pH - Hp =
[p,
HI =
-&
we have (En ,\(pH
-
Hp)\En) = -iUnn'
(XV 4-6}
[XV.4.]
Lamb
Nonrelativistic Calculation of the
Shift
427
or Pn'n(En'
~ En =
-tJiS
)
m ,W.
(XV 4-7)
Hence
~ E n yPn n = -Mp M >6*"VV, - En = \Pn'n\*(E n -ihp M -VV. '
(E n
>
,
nr
\*
)
Using (V.I. 25) we obtain
X)
2
ipn'n!
(#,'
~ En = " )
n'
We use the
equation
V*V = ^Ze^(r) and obtain
4-9) in the
form
where we use the relation
=
(XV
(XV 4.1
(XV4.H)
2)^(0)1^ \//
n
(En \r) and carry out a
canceling the surface integral term. The = we have with I 0, vanishes. For Z
partial integration
sum (XV4.11),
for
any
electron
^
where
AT
is
the principal
(XV 4.11) and (XV. 4.! 2)
quantum number and a
we
is
the Bohr radius. Using
(XV 4.6) Z 4^, = s; -F- log in
obtain
8 /e 2 \ 3
U)
where
the ionization energy of the ground state of hydrogen. The average excitan )} for the 28 state of hydrogen was calculated by Bethe as 17.8 R y leading to 7.63 for the logarithm in (XV 4.1 3). The shift is
tion energy ((E n
*
E
,
comes out as
AF =
" 136 log
=
1040 me,
(XV 4-15)
is in good agreement with the experimental results. Another interesting explanation of the Lamb shift was given by Welton.*
which
He
pointed out that in a quantized electromagnetic field the lowest energy state does not correspond to a zero field but that there exist zero point oscil*
T. A. Welton, Phys. Rev., 74 (1948), 1157.
428
Interaction with Radiation
The
lations.
zero point energy associated with each oscillation
so the energy density of the field
where dN(k) field
[Chap. XV]
=
(1/47T
3
)
d*k.
energy corresponding to
is
is
just \h 9
determined by
Thus the Fourier component wave number k is
of the electric
The existence of these fluctuating electric fields will induce some rapid variation of position. This leads to an extended picture for the point electron of conventional theory. The attraction at short distances of an extended electron
by the nucleus can be expected
to be
means that the
tion of a point electron. This
somewhat
less
than the attrac-
state of zero angular
momentum
than other angular momentum states having a finite probability of pushing the electron near the nucleus. Now let dr be the change in the electron's position induced by the fluctuat-
will lie higher in energy
ing
field.
Then
the
new
of dr is of course zero, for a
bound V(r
+
position of the electron
electron the potential energy Br)
=
7(r)
+
(dr-V)V(r)
On taking the time average of out to zero and in the term
is r
>
The average value need not be zero. Thus
is
+ 4(3r- V)
(XV 4*16)
+ 8r.
of (dr} 2
but the average value
2
---7(r) H
.
the terms linear in dr will average
only the diagonal terms,
will contribute.
Thus from {(&*)*>
we
=
2
<(5z2 ) )
=
2
((5z 3 ) >
=
obtain
(57)
The spread
Hence
dr can
for the feth
=
(V(r
+ dr) -
7(r)>
=
2
i<($r)
2 >V 7.
be assumed to obey the equations of motion:
component
(XV 4.16)
of the forced oscillations
.*&,
we
find
Nonrelativistic Calculation of the
[XV.4.]
Lamb
Shift
429
so 2
<(Sr) >
=
Jf
IT
and
ki
k% assure the
potential energy
is
The average value
This statement
is
he
/ftV
dV
^
f J
^
COj
rfe dfc^'
\mc)
k
Jki
convergence of the integral.
therefore given
of
=
dfc
771-
2 e^
= where
<(8r t )*>
The change
of the
by
for the state
^
leads to a shift in energy:
same form as that obtained by Bethe.
of the
XV.4.A. Problems 1.
The
retarded potentials of classical electrodynamics are given
At=f&^-dsX A= where
JR
density
=
|r
/ are
r'[
and
r
t
=
f R
t
c
stationary; that
^R^
by
(XV4*19)
',
d3 *''
(XV4*0)
Assume that charge
density p and current
is,
and /(*o
Then show that the
f [(l
-
j
scalar potential can
pP
-
=
3 rf
*'
+ ik ( l ~
be expanded according to
%;)
p f ^'
i
^ 3*'
|
where 7 fc
=
-> c
a).
(XV 4.*
430
The
Interaction with Radiation spherical Bessel functions ji
[Chap. XV]
and n are defined by z
= -J 2
=
hi(kr)
and
a.
refer
+ ini,
ji f
For a distribution where the total charge term in (XV. 4.8!) vanishes. The second and third terms to the electric dipole and electric quadrupole radiations, respectively. is
is zero,
The
J
the angle between r and r
the
.
first
electric dipole
and quadrupole moments are defined by
D=
Q=
pr d*x,
(XV 4.23)
prr d*x.
Compare expansions (XV. 4.21} and (IX. 1.18). 2.
Using the definitions of the Legendre polynomials, show that
where
Q
s
is
the invariant
sum
of diagonal terms in
Q
and
r
is
a unit vector
in the direction of r. 3.
For a stationary charge distribution the equation for charge conservaform
tion takes the
V /=
ikp,
where p and J are assumed to vanish outside a
finite sphere.
For an arbitrary
function F, prove that ik
Then show
that f or
F=
r
I
P F d*x
and
F=
= rr
j J-VF we
d*x.
obtain
(XV 4
*x,
Qa =
^/
(xJs
and the vector potential can be expressed
(XV. 4.84)
+ xJi}
d*x,
(XV. 4
as
xm-|*(l-i)r.
Nonrelativistic Calculation of the
[XV.4.]
where
Shift
431
m is the magnetic moment of the electric current distribution: m=
i
(r
the expressions for 4 4 and the field can he written as fields of 4.
Lami
From
_
nr>
oV
Aw
1
\
A
J)
rf
3
*.
show that the
/>.
P=
-Jt-
-7T
(g
c
X
3C*
and magnetic
I .
Hence show that the time-averaged radiation over 1 c -
electric
VU Tl\
- H H* r/
&r) / Z~2
,j{(a>: O
r
X
large distances
is
given
by
4
+ 8* X 3C) = O7T ^--5 IT) fr
2
(r*D)
2
(ZF.A^)
]r.
7""
Show from the problem 4 that for transitions in which Am = the dipole moment is of the form Dk, directed along the z-axis, and the radiation field 5.
is the same as a simple linear oscillator. By comparing with (IX. 1 .18), find the expressions for the corresponding electric and magnetic field vectors. In this case the radiation is linearly polarized with the electric vector in a plane
determined by r and angle 6 to the 2-axis is
/).
The corresponding
intensity in a direction at
an
P-j>'ain/. Hence the
total rate of radiation through a large sphere is given
J
p.dS =
= J
6.
Show
'wtede =
D* JO
T:
that for an atom with
Z
^
~
form (l/v 2)J>(e 1 tions.
In this case
d= ^62),
D
where
Am =
are
Q = -e ]T) r
tf ,
ei
and e2 are unit vectors
in Xi
D and
is
of the
x% direc-
can be written as
D = ^ \D\(ei cos Thus
moments
z r
where the S- are the spins of the electrons. 7. Prove that for transitions Am = 1 the dipole moment /
(XV4.88)
C
electrons the corresponding
z
D = -e
O
by
coif
Te
2
sin
coZ).
plane (as viewed from the positive 1. #3 direction) in the clockwise direction and in the opposite sense for Am = for
1 it
rotates in the
(xi, x%)
432
Interaction with Radiation
Show
[Chap. XV]
that in both cases the intensity in a direction making an angle 6 with
the #3 direction
and that the
is
given
by
total intensity over all directions
is
the same as in
(XV. 4.38).
correspondence principle, show that the quantum mechanical forms of the above transitions are as given in the text. Thus the transi8.
By using the
tion probability from a higher state
&(A, B)
Hence show that the dipole radiation
A
= |
to a lower state
^
\(A\D\B)\*.
B
is
(XV 4.35)
transition probability per unit time for the magnetic
is
Om (A,
B)
=
\(A\m\B>\*.
(XV.4.36)
CHAPTER XVI
MANY-PARTICLE SYSTEMS XVI.l.
The Hartree-Foek Method A
All systems of physical interest are, actually, many-particle systems. single isolated particle does not correspond to a quantum reality. By a oneparticle picture
we
assumed to be at
usually
rest.
mean a
An example
tion of the proton being at rest
is
particle interacting with another particle
the hydrogen atom, where the assumpa good approximation. In reality there exist is
mass corrections to the energy level structure from the motion of the proton. In a many-electron atom, electrons are
Coulomb
of the
hydrogen atom, arising
we assume that the nucleus is fixed and The electrons are attracted to the nucleus by
also,
moving around and electrons themselves it.
forces,
will repel
one another with similar
forces.
In practice, for
many
electron atoms, one aims at a reduction to a system
of noninteracting particles. This
is
zero-order approximation. For weakly is to neglect all three-
interacting particles the next order of approximation
body and higher-order particle processes and assume instead that only twobody encounters or two-body interactions need be considered. A further approximation is to assume that the various electrons of an atom move approximately as if each were acted on by a central field produced by the average motion of
One
the other electrons in the atom.
quantum theory or wave mechanics lies in the not closed. The most important mathematical entity of the the Hamiltonian operator. The knowledge of the Hamiltonian via
of the basic features of
fact that
theory
all
is
it is
Schrodinger's equation, together with the boundary conditions, determines the solution of a physical problem. In general there exists no special formalism
For any given physical problem we are guided by the experiment, the symmetries, and above all by intuition, to guess at a to find the Hamiltonian.
433
434
Many-Particle Systems
[Chap. XVI]
reasonable Hamiltonian for which the corresponding Schrodinger's equation possesses solutions. In treating a many-body problem we shall, as we did in
the one-particle field, postulate the validity of Schrodinger's equation. The wave function will be assumed to be a function of the coordinates of N particles and also of time t (In relativistic theory, however, we have to use different
time coordinates for different
particles.)
It is natural to expect that the
]V-particle Hamiltonian will be a function of
of
N particles.
An
3N coordinates and 3N momenta
N-particle wave function ^ can be used
to set
up a prob-
ability interpretation, thus
*K1, 2, is
,
the probability of finding at time
N, t
t)
d^xl
d*xN
-
the particle 1 in the volume d*xi, parti-
and particle N in d*XN. For noninteracting particles the wave function is just the product of the wave functions of the individual particles. For example, if we neglect the mutual repulsion between the two electrons 3
cle
2
of
an atom, the resulting motion can be described by a wave function which
in
<2
#2,
,
a product of the individual wave functions of the electrons. ing Hamiltonian can be written as is
The correspond-
72 (r2 ) refer to potential energies of the electrons with the nucleus. However, if interaction between the electrons is not respect to neglected, then the probability of finding one of the electrons at a given where 7i(rO and
region will not be independent of the probability that the other electron at the
same region or
in a nearby region. Actually the probability will
is
be
the other electron happens to be near the region in question. In this case there is a correlation between the motions of the electrons and smaller
if
therefore the corresponding Hamiltonian will
as follows particles.
from the
first
have to be modified to take
The
required extra term in the Hamiltonian, principles, is a function of the coordinates of the
this correlation into account.
To find a consistent approach to the we shall reconsider the one-particle
action term
calculation of such
an
inter-
The average motion can be described by the
picture.
of one particle in the presence of other particles
coupled set of equations
t = Hf,
(XVI. LI)
QA = -/a,
(XV1. 1.2)
ihj-t
where
The Hartree-Fock Method
[XVI.1.]
For
static fields
we
set .4
=
435
and replace (XV 1, 1.2) by
V
2 .4. 4
=
4wp, so
the Hamiltonian can be written as
H =
+ ^4,
(XVI. 1.3)
where
^=
e
Hence
and with ^
=
4>(r)e~
W> Ei we
f -^^^W.
(XVI.1 4)
have
Thus the average motion o f one charge in the presence of linear problem. In this case the function ^ must be regarded "reduced probability amplitude," The is
relativistic
others
is
a non-
as a one-particle
form of equation (XVI. 1.6)
given by
Df (x ~ where J^(x)
=
f
x )J&')
D F (x
aX^T/*!^) an d the
HDP (x
-
x
iv + mc|^> -
r
)
=
^
;
)
0,
function satisfies the equation
-4:ir8(x
-
(XV1. 1.8}
re').
For an A^-particle problem we can approximately volume density of electric charge resulting from the
set
up a continuous
particles.
Using
this
charge density we can calculate the corresponding electrostatic potential as a function of position. This is the average field where, in a first approxima-
each particle moves independently. Now let us suppose that we do the motions of the electrons in the average field. In this case we can calculate the corresponding charge density. To keep our assumptions con-
tion,
know
must be the same as the charge density Such a requirement the equality of charge dencharacterizes the "self-consistent field method." sities In practice one makes an initial guess as to the charge density of a manysistent the calculated charge density
assumed
at the beginning.
and solves SchroOne then repeats the same
particle system, calculates the corresponding potential,
dinger's equation to find the final charge density.
and so on. If these successive apassumed charge density (or to any one
calculation with the final charge density,
proximations approach the
initially
then the correct solution of the problem is found. applied by Hartree and was later generalized by Fock.*
of the calculated densities)
This method was
first
*
D. R. Hartree, Proc. Cambridge Philos. Soc., 24 (1948), 89. V, Fock, Z. Phys., 61 (1930), 126.' ,,
436
Many-Particle Systems
[Chap. XVI]
In the Hartree-Fock approximation one assumes a many-particle Hamiltonian of the form
EH
B=
*
+
i
where the r
t-,
for i
=
W,
(J07.0)
are position vectors of the particles in the
-
1, 2, 3,
Z7
tvy
t
,
system, and
the pi being the momenta. Each particle is moving in a field resulting from the sum of the external field F(r,) plus the field produced by the sum of pairwise interactions of the particles, the second sum in (XV 1. 1.9). The twobody interaction energy V i3 is a function of the position vectors r- and r, of the ith and jth particles. Hartree's idea was to obtain a reduced interaction in such a
the ith particle would
on
rt
its
way that own position
.
Therefore the general behavior of potential, so the resulting field
problem
can, in principle,
N
yV^. V
i3
is
in a potential
depending only an "effective potential/' must lead to this effective
called
all particles
In this
self-consistent.
is
way a
many-particle
be reduced to a one-particle problem. particles having a total Hamiltonian
Consider a system of by (XVIJ.9). In the self-consistent
energy \
move
Such a potential may be
field
constructed as a
H as
defined
approximation, the pair interaction
sum
of average pair interactions of the
form
H( =
(XVI.1.10)
<*/|ff ,
j^i
where H(^ for an atomic system,
=
TT/
Htj
^**
l^j) is
or, in
r#
> '
and
refers to pair interaction
=
1 -
|r t
I
ry|,
ij
a single-particle eigenstate. Thus for
terms of wave functions,
-
energy of the form
we
H( we can
write
get
H i
It satisfies the differential equation
The
state vector
|>fy)
satisfies
a Schrodinger's equation of the form ),
(XVI.L1S)
[XVI.2.]
Statistical Description of Many-Particle
where the Hamiltonian Hj
The
state vector |,}
is,
is
now
437
Systems
defined as
of course,
assumed to be
of unit length:
Fock and Slater* have derived the wave equation (XV7. 1.13) from a tional principle.
One
varia-
postulates *Xi
d 3 *v
=
0,
N. The subject to the subsidiary conditions (,-1^) = 1, for i = 1, 2, state vector must be defined |^} JV-particle according to the symmetry principles of identical particles. Thus, for an ^-electron system, for example, we -
-
,
have
where
= +1
an even and
1 for an odd permutation. The antihere can be to include spin in the case of extended symmetrization implied electrons and isotopic spin and spin in the case of nucleons. The application
CP
of Hartree's
for
method (with a number
of modifications) to nuclear force prob-
lems has recently been proposed by Brueckner. f
XVI.2.
The
Statistical Description of
Many-Particle Systems For a many-particle system of high density and low temperature a quantum statistical description based on a distribution function, where symmetry or antisymmetry of the wave function plays a basic role, can be used to study many-body phenomena. For example, the electrons in an atom can be regarded, together with the nucleus, as a "bound plasma," in comparison with a free plasma, where the atom is completely stripped of its electrons. By
N
particles, we analogy with Liouville's classical distribution function /# for can define a quantum distribution function FN to describe an electron plasma.
Such a distribution function can also be used to describe "ionization and recombination phenomena" in an atomic gas as a process of charge creation and annihilation. The same function can be applied, with appropriate sym*
C. Slater, Phys. Rev., 34 (1929), 1293. Phys. Rev., 96 (1964;, 508; 97 (1955), 1353; 100 (1955), 36. K. A. Brueckner, R. J. Eden, and N. C. Francis, Phys. Rev., 99 (1955), 76. J.
fK A. Brueckner,
438
Many-Particle Systems
[Chap. XVI]
These two systems are typical quantum liquids that could not possibly be described classically. The formalism to be developed in the following can also be applied to various
Hi and
metries, to the description of liquid
liquid Hi.
solid-state problems.
We begin by by
assuming that the state
a particle system can be identified
of
the density operator p as defined in Section VI.2.A.
A
quantum
distribu-
tion function FN, closely related to Liouville's distribution function fN
,
can
be defined as a Fourier transform of the function
t\
P \r",
(XVL2.1)
t)
with respect to the relative coordinates, where the state vector fined
|r', t) is
de-
by k',
=
*>
-
-
|i,
|r{, *>|ri, t)
t}.
It satisfies Schrodinger's equation
^|
=
r', t)
Hok',*>
(XVIJB.8)
f
Cut
where Ho
The
is
a free ^-particle Hamiltonian,
state vector
|r', t) is
normalized according to (r', *|r", t)
=
f
5(r
-
(XVIJt4)
r"),
where 5(r'
and
is
an eigenstate
-
r")
= j[ tl
-
=
r' r', t}.
distribution function for -|3AT
[1 2^J
N particles can now be defined by
r
/
r and p represent the 3N coordinates and and the vectors X and r are defined as
where
X = The exp
integration [
(i/K)p-\]
Putting X
==
hrj,
is is
r")
of the position operator q, q\r', t)
The quantum
*(ri
r'
-
r",
r
=
(r'
/>** dA,
3N momenta
(XVI.2.5) of
N
particles
+ r").
taken over the TV-particle space, and the phase factor to be understood as
the distribution function can finally be defiiied as
Statistical Description of Many-Particle
[XVL2.]
\ 3Ar r
2^) (1
The complex
conjugate of (r,
p,
f)
+
(r
j
F&
is
lto|, *!p|r
given
m
=
(r
-
*>-***
|ftif,
43Q
Systems
drr
by
-
>which with the transformation tj 17 under the integral sign leads to = (XVI .6). Hence F*N FJV and therefore the quantum distribution function is real. However, because h has a finite value, FN is not necessarily positive.
The classical from
(XV
FN was
distribution function /y (Liouville)
1. 2. 6}.
first
Definition
given
(XV
by Wigner.*
1. 2. 6) of
It
is,
the
is
obtained as/v
quantum
= lim^o FN,
distribution function
a probability function and
essentially,
can be used to calculate average values pertaining to the quantum properties an JV-particle system. Its main difference from the classical }N is that it
of
describes
an
JV-particle
and
also
system whose members possess wave and
particle
(Fermi-Dirac or Bose-Einstein). For a system of particles obeying Bose-Einstein statistics the state vector free particles is expressible as a symmetrical combination of |r', t} of the
properties
obey quantum
statistics
N
the single particle state vectors has the form ir' f *>.
= y:
where ]T) means the sum over p
rj, t}.
y \r r
all
I
'a?
Thus, a general symmetrical state
Alrv vr&j
A
r
\r
-
(YVT 9 7} ^!L K JL.&*1 )
A
1*5, 6/j
l>/
permutations of
P
[see (VI.2.5}}.
For a
given total energy E there is only one symmetrical state. Thus for a Boserr f Einstein system the representative (r t\p\r t) does not change its sign under ,
the operation of interchange of particles. For a Fermi-Dirac system of particles
,
we must
use an antisymmetrized
combination of individual particle state vectors as a,t)
=
det
In this case the antisymmetrical state |r', resents a state where the particle states a, *
(XVI.8.8)
)A of the ]V"-particle 6,
,
assembly repand where
q are occupied
E. Wigner, Phys. Rev., 40 (1932), 749. Mayal, Proc. Cambridge Philos. Soc., 45 (1949), 95. J. Irving and R. Zwanzig, J. Chem. Phys., 19 (1951), 1173. Y. L. Klimentovich and V. P. Silin, DokL, Akad. Nauk. SSSR., 82 (1952), 361. J. Bass, Sci. Rev., 3299 (1949), 643, J.
440
Many-Particle Systems
[Chap. XVI]
occupation of any state by each particle is equally likely. The occupation of any of the two states [r, t}, |r, t} by any two particles, where \r'a t) and |r, t) ,
are the same, does not correspond to a state for the assembly since in this case |r', t) A = 0. already know that none of these symmetry properties
We
of
FN
are possessed
by
the classical distribution function /#.
XVI.2.A. The Time Development of the Quantum Distribution Function In any practical use of FN we need some method for
its
actual calculation
for a given system. This can be found
(XV'I.,2..2) and
by using Schrodinger's equation the equation [see (VI. 2. 17)] for the density matrix,
=-D>,flX
iftf
where we assume that the Hamiltonian
(XVI. 2.9)
H for an ]V-particle system
is
of the
form
H= =
for i
H,
H -
+ H',
Q
H,
+H +
-
-
z
+ Hy,
H,=
^
,N, and
-
1, 2,
ff'
= ]T
V(\x i8
-
(XVL2.10)
3*1).
i^l If
we
wish,
we can
also include in
H
Q
the action of an external field and write
The term
^ t>|) in T' refers to a centrally symmetric potential and V(\Xi 8 the interaction of one particle with each of the remaining ones. represents the distribution function FN as defined by (XVI. .6) Now, differentiating
and using equations (XV7. 2..) and (XV1. 2. 9), we obtain
f
+ in, *|[ff, p]|r -
Hence
? y^jy
**
If
=
W /
I
\
32V
we assume
/
/
(r
+ ***'
' 1Cff ''
p]|r
~
**"' *)e
that the interaction energy
H'
is
~ i"' P a function of the position
Statistical Description of Many-Particle
[XVL2.]
operator g, then (XVI the unit operator,
.18)
441
Systems
can be simplified by inserting in the integration
and we obtain dF*r ih
/
=
~df \ 3JV
(1 2^J \
m
+ Itoi,
r
I H'(r
(1 ^J \
r
/
w
f
y H'(r
(1 ^J
<|p|r', <>
+ ihn -
drr
-
t)(r', t\p\r
,
-
+ fa,
yun, t)(r
-
t\H'\r
(r',
fa, t)e~^
Jft,, t)3(r
t\p\r', t)d(r'
Hence ih
1?
=
(^)^ /
[H
+ *^ ~ F (r ~ '
'
(r
* aij)]
}
X
|r
-
PTJ,
-'"^P,.
(XVIJUS)
f)*'"' dTy.
(XVIJMfi
i}e
Using the definition of ^V by (XVI. 2.6) we get
+ PJ,,
i|p|r
-
}fef,
Thus the time development
of
>
FN
X
=
y Far(r, p',
obeys the integrodifferential equation
F^(r, p', )e*VCp'-p) dp, dry.
XVI.2.B. The Classical Limit
Quantum
We
first
of the Distribution Function
divide both sides of
(XV 1. 2. 15) by
h and set h
that
= 71
we
=
obtain ajN dt
where
1=1
The equation for/# can
further be written as
=
0.
Thus, noting
442
Many-Particle Systems
=
[Chap. XVI]
~
Hence, using
we
obtain
which,
is
the Liouville theorem of classical statistical mechanics.
XVI.2.C. The One-Particle Distribution Function symmetrical with respect to the interchange of rN of the the coordinates n, ra particles. In complete analogy to the classical case, we can define the reduced distribution function for k particles,
Let us assume that FN ,
where k
<
2V,
is
N
,
by
xfa, r2
-
-
-
,
,
rN) px,
p
-
-
2j
,
dl*
pNj
,
(XVI.8.1T)
where
integrating (XV 1. 8. IS) over phase space of the (N obtain for the one-particle distribution function the result
By
X Fu (n, where Fi2 limit h =
where of
In.
is
,
,
0,
(XV1. 2. 18)
is
we
<)^ &*-" d
the two-particle distribution function and
Vi$
-
r2 p{, P2
l)th particle,
$ =
(AT
1)
V
.
In the
yields the classical equation
the average force acting on the particle
1,
and $
is
a function
r2 (.
In classical transport theory, equation (XVI&.1&) can be approximated,
under certain assumptions, by the Boltzmann equation:
f=
f
dO |p
- p^(|p -
Pl
,
ff)(f'fl
- JjfO,
(XVI JiM)
Statistical Description of Many-Particle
[XVI.2.]
Systems
the relative momentum and the prime and the subscript to the / refer the momentum variables alone that is, /i = /(r, pi, f) = /' f( r P', ) The four momentum variables p, 1; p', p{ refer to the
p
where
is
p
,
menta
443 1
of
and mo-
of the binary collision,* (P? PI) ~*~*~ (P'J PI)
P
?
PI]
=
lp'
pil,
dl
and v(\p
the differential cross-section for a collision in the pi |, &) = 8 c?0 sin d6 solid angle ( d
from (XVI. 8. 18), by truncating the two-particle distribution function F12 as a product of the single-particle distribution functions F\ and F<> and a pair correlation function g^. The execution of such a laborious task is outside the scope of this book. However, some comments on the use of such a quantum transport equation
in order.
is
In a many-body system (for example electrons in solids) a perturbing interaction does not lead to a set of stationary states whose energies are sharply defined; instead, it produces a relaxation process. This means that the particles will suffer collisions with one another during a mean free time T. Classically (/
this
/O)/T,
leads to replacing the
where
collision
//T describes the removal of
term in (XVI. 8.20) by particles from some parts
collisions, and /o/r describes the reappearance of the some other parts of phase space. The occurrence of the two
phase space due to
of
same
particles in
processes together insures that the initial distribution will tend to equilibrium during a time of the order of r. The quantum analogue of this procedure was
proposed by Karplus and Schwingerf in terms
of
a density matrix p where the
interaction Hamiltonian contains a part causing the interparticle collisions
and hence making the distribution tend to a thermal equilibrium. In their formalism it is assumed that the change in p, as caused by collisions, is of the form ihl
_-
_
AP~
Po
r
is the part of the interaction causing the collisions and p is the equilibrium value of p. A useful application of this formalism can be found in systems where the influence of relaxation processes upon transitions between
where Hi
various energy states
is
a dominant and nonclassical feature of the many-
particle systems. *
S. Chapman and T. G. Cowling, The Mathematical Theory of Non-Uniform Gases, Cambridge Univ. Press, Cambridge, 1952. J. G. Kirkwood and J. Ross, in International Symposium on Statistical Mechanics, Edited
by
I.
t
New York, 1957. Schwinger, Phys. Rev., 73 (1948), 1020.
Prigogine, Interscience,
R. Karplus and
J.
444
Many-Particle Systems
[Chap. XVI]
In an equivalent but more general approach for the study of relaxation phenomena, the use of the quantum distribution function may be found more realistic, since there
random
can be no ambiguity with regard to a representation of This procedure has a direct
collisions causing the relaxation process.
classical analogue; that
h
is, f or
=
we can
get a
Boltzmann equation.
XVI.2.D. The Quantum Distribution Function of a Harmonic Oscillator To illustrate the basic difference between classical and quantum distribution functions, we shall solve a related problem. Let us first compare three diffusion type equations:
= Dv
2
ot
(XVI.t.81)
P,
(XVI. 2.23} These equations are solved by /
P=
(
1
\ 3/ 2 e -(r-ro)V4Dt
~^\
_2M\
3/ 2
e
(<m
- mv */2K T
(XVI.S.84)
(XVI.Jt.t6)
3/2
(XVI.Jt.S6)
where P(r,
and
r
+ dr
(y.5.Jf)j
r at
dzr
the probability that the particle will find itself between r / is the Maxwellian distribution and \f/ represents the probability amplitude of the position operator q having the value f)
time
is
after time t;
t if it
had the value
r at
time t
=
(where q is diagonal). a transition from a single-particle quantum path description to a quantum ensemble description by the substitution [comparing
We
can
now make
(XVI.1t.96), (XVI.8.84),
and (XVI AM)] t
=
ih$,
ft
=
^
Boltzmann's constant and T is the temperature of an equilibrium can apply this transformation to the transformation function of a harmonic oscillator [see (V.5.59) and (V.5.60)] and obtain the density func-
where
K is
state.
We
tion for a harmonic oscillator in coordinate representation:
Statistical Description of Many-Particle
[XVL2.]
exp
=
p
nrW(z,
exp
=A
ar
exp
ify
,
-iK) dr
W(ar, TO,
-
7
|- |jr
[(x
445
Systems
J
x
2
coth
)
+
(f^)
(x
+z
2 )
tanh
where
From (XVI. 8.6) and we obtain F(X,
the substitution x
p,t)~A exp -
XQ
=
^
and |(x
+X = Q)
Z,
tanh
X
~
jj^ exp
-
2 1?
coth
(JjSftw)
exp
(
irjp) dq.
Hence
F= ~
tanh
(IjSftw)
exp
-
tanh
where
is
the energy of the oscillator. The classical distribution function of a free harmonic oscillator follows by
letting
h
*
in
-E/K T
(XVI. 8.88):
=
{
-1
/
A
p-p
\2TrmKTj
where /
is
As an
normalized by
of
application
(XVI. 8.88) we oscillator. Thus 9
I
f dp dq the
=
oscillator
shall derive Planck's
(E)
=
r
y_oo
dx
r
J
1.
oo
quantum
distribution
function
formula for the average energy of an
EFdp~
Jftwcoth(ijltf).
(XVI 30)
This result can be rewritten as
where the
first
term
i'
(XVI.2.S1)
refers to zero point oscillation energy.
The form (XVI. 2.80)
of
the average energy of a free harmonic oscillator
is
446 of
Many-Particle Systems
some physical as well as
historical significance. It
for the specific heat of solids.
same molar terms of a
specific
heat for
The
all
classical
monatomic
law
[Chap. XVI]
was derived by Einstein*
of Dulong-Petit leads to the
solids.
This law
is
formulated in
theory of lattice vibrations. From the equipartition of found that there are 3 AT" (N = Avagadro's number) normal modes classical
energy it is in one mole, each with energy K.T. Later experiments have demonstrated that the Dulong-Petit law does not hold at low temperatures; K.T then becomes less
than the spacing of the quantum energy depart from the equipartition of energy.
levels
and the
specific
heat begins to
The lattice vibrations of a solid are to be represented by uncoupled quantum harrnonic oscillators with energy levels n = [n |]^co. In this case the each mode is of given by (XVI. 2.30) instead of corresponding average energy K.T (corresponding to h = 0). However, later studies have shown that the
E
+
agreement between theory and experiment was more of a qualitative nature,
(XVI. 2*30) drops off rather steeply and causes a quantitafrom experiment. It was found that for solids the uncoupled quantum harmonic oscillators, representing lattice vibrations, ought to have a certain frequency distribution. The correct result was obtained as the aversince the expression tive deviation
age of (XVI. 2.30) over a given frequency distribution,
E= where
is
\Ji
[
co(?(co)
coth
the cutoff frequency and (?()
is
a certain frequency distribution
function. * A. Einstein,
Ann. Phys.
Leipzig, 22 (1906), 180, 800.
CHAPTER XVII
THE ELEMENTARY
THEORY OF SCATTERING XVII. 1. The Cross-Section for Scattering Scattering of waves or particles from a force field can be used to obtain
information about the nature of the force interaction of the
two systems.
By
field,
the scattered particle, and the
scattering charged particles from atoms,
we can
explore the properties of the atoms as highly localized charged nuclei surrounded by planetary electrons. Scattering of electromagnetic waves from charged particles gives information about the interaction of matter and
The scattering of mesons from nucleons is an effective way to probe the interactions of mesons and nucleons and hence to get some information about the nature of nuclear forces, and so on. radiation.
Classically a cross-section is defined as the ratio of the amount of energy emitted by the scatterer in a given direction per unit time to the energy ux density of the incident radiation. For example, for the scattering of electro-
magnetic waves by charged particles the effective defined
differential cross-section is
by da-
=
^-^ av [S]
(XVII. U)' v
the energy radiated by the system of charges into the solid, angle
where dl
is
We
have a dynamical system consisting of a scattering center and an incident wave which may consist of a beam of localized particles in. a pure state; each particle is sufficiently separated from the others that we may neg-
itself.
447
The Elementary Theory of Scattering
448
lect interparticle interaction in the
a number
and
of stationary states,
[Chap. XVII]
beam. The scatterer by itself may have initially it is in one of these states. After
the incident particle is scattered it can be left in a different stationary state. This means that the incident wave may induce a transition in the scatterer.
We
first
and show that the
calculate this transition probability amplitude
cross-section for scattering
proportional to the transition probability.
is
In a scattering process for any state of motion of the system the particle spend most of its time at infinity. The probability for such a state of
will
motion and
its relation
wave function infinity,
to the scattering cross-section are contained in the
most
of the system. Since the particle spends
of its
time at
of the wave function Under these circumstances, the plane plus an outgoing spherical
we shall be interested in the asymptotic behavior
at large distances from the scattering center.
wave function
will consist of
an incident
wave:
^
s
e
+ I e *rf(fi
a
(XVII.L2)
9
where e^z represents a stream of electrons, say, and k large distances from the scattering center the Hamiltonian to a free Hamiltonian, which permits plane
wave
=
p/h
of the
=
mv/h. At system tends
solutions together with
outgoing spherical waves. (For ingoing waves see Section XVIII.3.) ikz e represents a density of electrons of one per unit volume, or
=
ik
\e
and hence a flow
*\*
of v electrons across
The wave
1,
a unit area per unit time. The function
/(0) has the dimensions of a length. It can be related to the scattering crosssection as follows: the number of electrons in the scattered wave crossing an
element of area dS at the point
(r, 0,
<)
is
'f I/WI' per unit time. If the incident beam is such that one electron falls on a unit area per unit time, the number tr(0) d2 scattered into a solid angle dQ per unit
time
is
equal to
and so
The numbers
of electrons scattered
=
(XVI1. 1.3)
|/(fl)|.
between the angles
sin 6 dB
=
2
2*-|/(0)|
9
sin 9 dd.
and dd
is
given as
[XVII.l.j
The Cross-Section
XVII. 1. A. Calculation
for Scattering
449
of the Scattering
Cross-Section
We
begin by defining a stationary-state Green's operator
- HQ)G =
(E
(?
by
1.
Q
(XVII. 14)
In order to impose an outgoing wave boundary condition it is sufficient to assume that E has a small positive imaginary part, ie. The Schrodinger equation
H\*) with
H=
HQ
+ H',
E\*}
(XVII.1.S)
}
can be written as
-
(E Hence, using (XVII.1.4),
where the state vector |^
=
(XVII. 1.6)
#'!*>.
derive the integral equation l*o>
+ QJ3'\V),
(XVII.1.7)
represents the initial state satisfying the equation
}
(E
E
F )|*> =
we can I*}
where the energy
=
-
#0)1*0)
=
(XVII.L8)
0,
can form a continuous spectrum. The
size of e is closely
related to the spread of the energy of the system; the smaller e is, the sharper the energy. Furthermore, regarding the incident particle as a wave packet approaching the force field or the scatterer, it is important that the size of the
not smaller than the range of the force field. The latter also is closely e. With these remarks the integral equation (XVII.1.7) well defined and can be used for the discussion of scattering problems.
packet
is
related to the size of is
Let us this
first
consider a conventional discussion of equation (XVII.1.7). For
we use Schrodinger
(XVII. 14) to write
it
J
s
representation and take the representatives of
as
or
where* Jfcg
To *
=
|rE,
integrate the equation
In view
we
0o(r,r')
r'.
|r'>.
use the Fourier representations
of the translation invariance of
of the relative coordinates r
=
(XVI 1. 1.9),
Green's function
is
a function
The Elementary Theory
450
(7o
of Scattering
[Chap. XVII]
=
and obtain
where
R=\rCarrying out the angle integrations
r Go
_ ~
2m
1
ft*
4x*
first,
1
d
r'\.
we
obtain (XVII. 1.10) in the form
r-
e**dk
Because of the small positive imaginary part
U
(XVII V1L1111}> (X
RdRJ%- k>
-
of
fco
we can
write
(XVII.1.11) as
G =
f
m,_
/
where in the second
line
e
atie
*
we have
'
to integrate over
a
are nonnegative parameters the integration over
first; /3
since
first
both a and
/3
would yield the
substitution
p= a- R, easy to see that the method we are employequivalent to the usual complex integration. Noting that
which could become negative. It ing
we
is
is
finally obtain Green's function for the free particle:
On
taking
the
representatives
of
(XVII. 1.1$), we get an integral equation
equation (XVII. 1.7) and using for the wave function describing a
scattering process: '( r
The second term on
W)
rf3 *'-
the right represents an outgoing wave. For
(XVII. 1.13}
^
to have the
The
[XVII.I.]
Cross-Section for Scattering
asymptotic form
(XVII. 1.2} we must
451
choose for the incident wave fo(r) the
form *o(r)
=
e***'
r
(XVII. 1.14)
.
At this point it must be remembered that for Coulomb interaction the incident wave is distorted by the nucleus even at infinity; therefore form (XVILL14) is not suitable as an incident wave. If
H (r f
f
decreases faster than 1/r', then the contribution to the integral in
)
r
(XVII.I. IS) from large values large
In this case we can write, for
r,
R = where r
is,
[r
2
_
/2 4. r
2rr' cos 0] 1 / 2
^
a unit vector in the direction
is
*
*(r)
This
of r is negligible.
* -
9<m pilw
1
TT
j- H" 47T
of course, exact.
r /
r
J
- r V,
r
of
r.
4 K
Putting fkQ
e-*k'-''H'(r')t(r')
In principle the solutions
fflxf.
=r
~'
=
feo
we
obtain
(XVII.1.1S)
of the integral equation
(XVII. 1.15) correspond to scattering states of the incident particle from the force field
H
r
The
scattering amplitude can be expressed,
by iteration of Each term represents a particular physical process in decreasing order of strength. If we neglect all the terms and retain only the first term of the series, then the wave function ^, as follows from .
(XVII. 1.15), as an
(XVII. 1.15),
is
infinite series.
of the
form
*(r)=*" +
piker
where f(g)
j-^ TcTT
Tl
the scattered amplitude. The above treatment of a scattering process is called the "first Born approximation" and its validity depends on the fulfilment of certain conditions. For is
a centrally symmetric
field [H'(r')
=
= _^3 nr JO /
where
K=
fcol;
|Jc
=
= h/mv and
2fc
sin - ==
(|fco|
sin
K
=
(XVII.1.1T)
fco|)
it
can be written as
A
6,
the angle between the initial and final intensity scattered into a solid angle element dQ, around kg is
27r//c
momenta. The
H^T Kr F(r)r2 dr>
for elastic scattering
K= where X
V(r')] f(8) can be written as
is
sn
2 7
dr
dQ.
(XVII. 1.18}
The Elementary Theory
452
of Scattering
[Chap. XVII]
For the Coulomb potential energy, or
7=
^, r
between
particles
integrated.
with charges Ze and
Ze r
}
the scattered amplitude can be
Thus sin.
where
"
e"^
7
with
r;
^
was introduced as a convergence
factor.
Hence
differ-
ential scattering cross-section is
This
is
The
the well-known Ratherford collision cross-section. validity of the
Born approximation
requires
2m A2
4?r
or
ZZ'e* hv
(XVII. 1
1,
Born approximation can be expected to give For Coulomb scattering of charged particles we have the fol-
so for high-energy scattering the
good
results.
lowing special features.
The cross-section is independent of h. (b) The exact quantum mechanical calculation leads to the same result the one obtained by the use of the first Born approximation. (c) The classical form of the differential cross-section is the same (a)
(XVII. 1.19). This might be expected from
the absence of h in
as
as
(XVII. LI 9).
Because of the infinite range of a Coulomb force, all angular momenta are needed to represent scattering. In this sense the scattering of charged particles from charged scatterers is more like the scattering of wave packets or to a situation where the particlelike aspects tend to be emphasized more, as compared to the wavelike aspects. However, if only a small number of phases or angular momenta are involved, then neither the classical nor the Born scatter-
ing can provide a correct description of the scattering process.
The
[XVII.l.]
Cross-Section for Scattering
453
XVII. l.B. The Momentum-Space Representation of Scattering
We shall use the notation Schrodinger's equation
wave boundary
condition.
is
now
E and write
as an integral equation with an outgoing have, from (XV 1 1. 1.5),
We
=
\E)
It
\E) for a stationary state with energy
(XVILI. 5)
+ E + ^_ HQ H'\E).
\EI)
(XVILUtf)
easy to obtain momentum-space representation. Thus, taking
representatives in
--
momentum-space we get *+ (p)
=
*f (p)
=
(p\H'\E)
=
+
*o(p)
~
2
(P\H'\E),
where
Hence *+(p)
=
+
*o(p)
UP) =
--
f
(p\H'\p }t+ (p'} d*p'
Jf
,
E + ie-^ 2m is
(XVILLtS)
the required integral equation for a scattering process.
The
factor [E
+ ie -
1 (p*/2m)]- in (XVII.1.2) can be written as
2m A2
where
= 2mE/h
fcg
2
and
fe
2
=p
1
-
fcg
2
A
fc
2
+ ie
The
2 -
function (l/27T 2 )[l/(fc
-
2
fe )]
is
the Fourier transform of the Green's function (I/R)eikoR so the scattered 2 2 fc )] in the asymptotic limit. amplitude is the coefficient of (l/27r )[l/(/co ,
In terms of the wave vector (p\H'\p'}
where
H (k
we have
= f(p\r)d*
r
fc')
fe,
is
X (r\H'\p')
the Fourier transform of H'(r): Hence the integral
equation (XVII.1.22} becomes
H'(k If in
the asymptotic limit of
fc
-
0,
-
H'(k
fcOiMfcO fc')
dW.
increases
(XVILUKS)
more slowly than
The Elementary Theory
454 2
1/fc
over
then
,
fc',
it is
of Scattering
f
k and can be taken
nearly independent of
[Chap. XVII]
outside the integral
so
^ h + ~~r H'^~^
t+
Hence the
scattered amplitude in the
I
iM
fe/ )
f
f
2
where the incident plane wave
^o(fe) is to
(XVILI .24)
Born approximation
first
= -2**~H'(k - k ) n
J(e)
d * k '-
ifo(fc')
d*k
is
r
(XVII. 1.25)
,
J
be represented by a delta function,
Hence =
_ 2T
^
2
#>( fe
_
(XVILLSe)
fe Q.
For Coulomb scattering we have
H'(k
-
fc')
=
lim
_ 2r 2
-
\k
7 !
where, from the conservation of energy, \k
and
-
2
fe
=
fe'|
we have 2fcsini0,
thus, as before,
ZZ'e* 4:E sin 2 1^
XVII.2. Partial
Waves and Phase Shift
Consider the partial wave expansion (IV. 3. 22) of a plane wave: oo
1
The
case
I
=
1
=
corresponds to
I
m=~l
P
waves, >s
which
for small r
behaves
mum approximately at the particle will get
fcr
much
like ^i
=
1,
~
after
kr
2
sin fc^
cos
^.
which
it
fc
r
closer to the origin
The quantity decreases, so
2
|^i|
it is
has a maxi-
not likely that
than the distance
7c
P
particles of unit angular momentum are not likely to be nearer to the origin than the distance r n where their classical angular momen-
Thus
waves or
,
[XVII.2.]
Partial
= MTO would be of the order of
turn pr
one can show large
Waves and Phase Shift
is
given
455
For the present case of free waves, way that the minimum distance at which |^j 2 is
in a similar
fi.
by pro
S hi
(XVII. 2.2)
In the presence of a strong attractive force,
let
us write
and
= V2m(E -
p
Hence the
criterion for the
V).
minimum probable
radius
is
W V
is large and negative, the wave may be pulled fairly close to the origin 2 despite the repulsive effects of the centrifugal potential 1(1 l)/r arising
If
+
from the presence
The
P wave
is
of
an angular momentum.
and asymptotically waves. Near the origin
proportional to cos
it is
just the
sum
and outgoing spherical it has a complex behavior and does not hit the origin exactly as does the S wave, but instead of ingoing
because of the centrifugal force it tends to avoid the origin. In the presence of a force field the radial wave function
(XVII. 24}
Xi(r)=].Gi(r) satisfies
the equation
In this case the partial wave expansion of the wave function
* =
J
A,P,(cos
0) X i(r).
is
given by
(XVII.2.6)
z=o
A must be chosen in such a way that the wave function ^ shall wave and a scattered wave, so (XVII. 2.6) shall have incident an represent form the asymptotic The constants
i
t^e
ik r '
+ ~e
ikr
f(6).
We
must In a scattering process the wave function is everywhere finite. therefore choose that solution xzO*) of (XVII. 2.5) which is finite at the origin. For large r the form we have
last
two terms
of
Gi
(XVII. 2.5) drop
SB
l
sin (hr
+
e).
out,
and
for the asymptotic
(XVII. 2. 7)
The Elementary Theory of Scattering
456
To comprehend
[Chap. XVII]
this better, let us set
=
Qi
We obtain -
ih
where ui
is
F
a slowly varying function* of
^A dr 2
and
it
can be neglected.
We then
r;
therefore for large r
dr
get
2m
+
1(1
.
T7
jpV+ Coulomb
field,
>-
<*>
.
Thus, for fields
xi that is
if
'Jdr.
and only
which
fall
if
V
tends to zero
to zero faster than the
Gi has the asymptotic form .Bjsin (kr
For
1)1 J
r,
side tends to a constant
For large r the right than 1/r as r
faster
we have
+
e).
finite at the origin, the particular solution will therefore
have
the form
^ sin (kr
- |k +
6 Z),
(XVII.B.9)
where Ci is an arbitrary constant and 5z is a constant that depends on k and on the potential F. We added p/r to ensure the vanishing of 5i for V = 0. The arbitrary constant Ci can be fixed by defining xi(r) as that bounded solution of the equation that has the asymptotic
^ sin (kr
-
iZTr
+
form
8,)-
(XVILS.10)
wave can be obtained by subtracting the wave (IV.3M2) from the total wave function. The constants A in (XVII. 2.6) must be chosen in such a way that the result of subtraction of the incident plane wave does actually represent a scattered outgoing wave, without the term (l/r)e~ ikr in the asymptotic expansion. Thus for all I we must have
The
expression for the scattered
incident plane i
AM where Ci
is
(21
+
l
l)i ji(kr)
=
d i e**
(XVII. 2.11)
some constant. Hence using the asymptotic forms
2kr *
N. F. Mott and H.
Oxford, 1949, p. 23.
S.
W.
Massey, The Theory of Atomic Collisions, Oxford Univ. Press,
where kp
Waves and Phase Shift
Partial
[XVII.2.]
=
kr
we must choose A
|/TT,
l)e
i
Hence the wave outgoing
wave
so the second
i
A = (21+
457
is
term vanishes. Thus
l
'i
.
function, containing an incident plane wave plus a scattered
in the asymptotic limit,
is
00
t = ]T
(2Z
+
l)*V
5
'xz(r)Pz(cos 6).
(XVII.2.12)
1=0
This gives the asymptotic form of the scattered wave (l/r)e ttr/(0), where
= The form
(2Z
of /(0) can be understood
by comparing it with a plane wave. For example, if there were no potential there would still be an outgoing wave, which is just the outgoing part of a plane wave. In the presence of a force field the test for the scattered wave is to see whether the outgoing packet has been modified. Hence the asymptotic form of the scattered wave is obtained by subtracting from the actual outgoing wave the outgoing wave that
would be present if there were no potential. From (XVII. 2.1$) the cross-section can now be written as
=
a*
+
(XVII. 2.14}
V,
where
Once the phase
a
=
b
=
+
^2(2Z
I)(cos25
z
-
l)P (cos0), r
~ S(2Z + l)(sin 2S )Pz(cos z
shifts 5z are
(XVII.2.15)
0).
known, formula (XVII.2 .14) gives an expression
for the angular-dependent differential cross-section.
The phase
shifts di
can be obtained by solving Schrodinger's equation. The arises, in part, from the
angular dependence of the differential cross-section interference of I
=
waves
of different
o only, then there
spherically symmetric. to cos 2
B.
If
With
I
=
for example,
we
1
alone, the cross-section
=A
+ B cos
is
is
proportional
is
0,
=
2
|A|
+
2
|5|
cos 2
+
(A*B
+ A5*)
cos
interference term arising from the interference of
typical
scattered waves with
differential cross-section is ,(0)
The
If,
both waves are present, then the amplitude /(0)
/ (0)
and the
Z.
no angular dependence, and the cross-section
is
wave
0.
(XVII.*. 18)
S and P waves
is
a
representation of the scattered particles. are dealing with scattering of particles only; we including all the Z, form a wave packet in such a way that
In the classical limit must, therefore, by
we
The Elementary Theory
of Scattering
[Chap. XVII]
they build up to a maximum at a definite value of 9. This corresponds to a classical orbit in which particles come in with a fixed collision parameter and scatter through a definite angle; this is also the case for in both classical and quantum mechanical treatment.
The cases
series in
where
scattering.
it
(XVIL&.14)
The
wave) that ;
is,
o-
of
an atom
unit time, from a
one electron crosses a unit area per unit time
=
27r
F
cr(0)
Coulomb
for the scattering of electrons
number of electrons scattered beam of unit intensity (plane
defined as the total
is
by the atom, per
elastically
into a closed expression, except for
scattering
and there are not many
generally convergent,
total cross-section
a given velocity
of
is
summed
can be
Coulomb
sin 9 dd
=
~~
]T
(21
+
1) sin
2
$,.
:
(XVII. 2.17)
This result means that in the total cross-section the various partial waves
do not
interfere. It is only in determining the angular distribution through the study of the differential cross-section that they interfere. For Coulomb the scatterer can, scattering because of the infinite range of Coulomb force
at a given time, scatter all the particles in a resulting total cross-section
beam
of unit intensity so the
the nuclear charge is However, the screened by atomic electrons,, then the particle number scattered can be inhibited by cutting off the long Coulomb range. In this case the total crosssection
is finite.
infinite.
is
if
For, from
~
.
sln
"ar
e
'
.
we
obtain
and
wr * where or 1
is
,
the screening radius. Putting Po
we
write
(XVII. 2. 18)
=
as
ha,
22
/
V
/Trrrrr^N
(XVII.2.19)
where p Q
may
be regarded as the
momentum
at the screening distance.
XVIL2.A. Problems 1.
The
of the
potential energy of a
form
beam
of incident electrons is
assumed
to be
Partial
[XVIL2.]
Waves and Phase Shift V(r)
where a
-^r
-
(1
the screening distance and
is
e-Oer",
=
K
in the absence of screening
(XVII. 2. 20)
mc/h. Show that the differential
Born approximation
cross-section in the first
which
=
459
for scattering
is
given by
becomes
[
Comment on
the choice of the constant K as mc/h and the result that
(XVII. 8.2%) is independent of h. 2.
Protons of low energy, as compared to the proton rest mass, are scattered
from a nucleus.
If
Coulomb
interaction
is
neglected, the elastic scattering
of the protons can be assumed to be due entirely to nuclear interaction. The Schrodinger equation determining the scattering process is
=
(XVII. 2.2$)
0,
where
7=
for r
and
-[7 B (E)+;i for r > R, and R is the R = 1.31 X 10- A
7 =
13
1/3
cm.
that the wave function containing the scattering
Show from (XV1 1. 2. 23) amplitude
nuclear radius,
given by
is
^ (I
+
(I
+ $)i Pi(coB a) [gi(k
+ hi(kir)],
for r
<
R,
+ r)ihi(k r)],
for r
>
R,
z
4)i azPz(cos
of)
[gi(kir)
l
r)
Q
U=o where
772
=
exp
Bessel functions.
(2idi),
hi
= ji
+ in
t}
g
t
=
ji
ini,
and j
t,
HI are spherical
Show that the asymptotic form of the wave function for large
r is 1
T
where sa),
The Elementary Theory
460
3.
The
finiteness of the
wave function
normalization imply that the coefficients
ai
of Scattering
[Chap. XVII]
problem 2 at the origin and its and f\i can be determined by fitting
of
the internal and external wave functions at r
=
R. The radial wave functions
to be fitted are
where
Rn
shifts 81
and RQI refer to respective radial parts and the coefficients 0,1 are related by
i
= -
01
r
of ^.
Show
that the phase
f>
x,
.,
,
;
Zl L2oMSr)fcj-l(So)
where
2
= W?
=
and 0j fcr-R. Compare the scattering and complex potentials.
cross-section for
& and
P waves with real
XVII.3. Scattering of Spinless (Identical) Particles and Mott Scattering As an example for the scattering of spinless particles we consider the scattering of a particles. We shall assume that the energy of the particles is low enough to neglect nuclear scattering. We shall study only the Coulomb scattering of the particles. Mott and Massey illustrate the scattering of spinless particles
apertures.
with a thought experiment, using two parallel screens with function before collision is just the product of the wave
The wave
functions of individual particles:
Mri,
The wave function
r2
,
t)
=
ur(ri, t)uir(r 2)
after collision, ^(r 1? r 2
t).
the probability amplitude for finding the "first" a particle in the volume element dVr at time t at the point TI and the "second" a particle in the volume element dVn at the same ,
i), is
time at the point r2 With this interpretation of the two-particle wave function that is, the knowledge that the "first" a particle arrived from the aperture .
I and the "second" a particle arrived from the aperture // one can, in accordance with the discussion in Section XII.2, calculate the probability of finding the first or the second one of the particles at at time t as the sum of the probabilities
(ri,
dVi) and (r 2
,
dVn
)
:
[|lKn,
2
+
Mr,,
2
dVx dVn
(XVII. 3.1) method of interpretation of the wave function does not treat waves (since the knowledge of their distinguishability as "first"
Obviously
this
a particles
as
r,, *)|
ri, *)l ]
.
Scattering of Spinless Particles and Mott Scattering
[XVII.3.]
and "second no
77
wave behavior
particle destroys the
a
of
particles),
The
interference of probability amplitudes can occur.
461
and hence
correct procedure
to take into consideration the exchange interaction of identical particles. if #(ri, r 2 f) is the wave function of two a particles (spin zero), the of the is to particles interchange equivalent multiplying the wave function is
Thus,
by
,
wiiere
1)',
(
I
is
the orbital angular
momentum
of the relative
motion
two particles. But a particles obey Bose-Einstein statistics, and therefore under interchange of particles the wave function cannot change its sign. Hence a system of two identical particles with zero spin can assume only of the
even angular momentum.
The correct wave function of a two-particle system with zero spin is given by iKn, r2
The wave
,
*)
=
Afa/fa, i)un (n,
t)
+ uifa, f)un
(ri,
0].
(XVII. 3.2}
function remains symmetrical at all later times. two beams do not overlap, and therefore at time
Initially the
must have since
uu
^(riHiCrO =
at
=
t
refers only to the points
=
we
(XVIL3.3)
0,
on the
right of aperture II.
wave function (XVII. 3. g), using (XVII. 8.3),
the
initially
t
Thus
yields the prob-
ability 2 |*o(n, r2)|
where
2
l^ol 2
2
|A|
k(riKr(r2)|
2
+
|A|k(rOtto(n)l
a ,
(XVIL3.4)
near / and r2 near II, or vice versa. This means not the probability of finding the particle observed at I in the vol-
is
that
|^o|
ume
element
is
=
zero unless r x
(ri,
is
dVi) and the particle observed at II in the volume element
dVn}j since the latter probability is zero if r 2 is near /. The quantity z that any one of the a \fa\ dVi dViz is to be interpreted as the probability If the initial wave in the volume found one of elements. can be any particles (r 2 ,
function
is i/fy
then the wave function after time
*= The
=
t//( ri ,
probability that a particle
OK
r 2 )| 2
+
hKr,, n)!
2
r,, t)
is
+ ^(n,
at
+ iKr,, (ri,
r 2 )^*(r 2
t
n,
is
(XVII. 3.5)
t).
dVi) and the other at
,
n)
+ *(r
s,
(r 2 ,
n)**(ri, r 2)]
dVn)
dVj
is
dVa
,
(XVII.3.6)
The use
symmetrical wave suppressed. to a to not allow us does function finding a particular a assign probability if we try to see which particle in the volume element dVi or dVn. However,
where the time coordinate
t
is
of the
is observed, the interaction arising from the act of observation will interference terms in (XVII.S.6) by some unknown and unthe change controllable phase factors [see (XII.2.3)]. For example, by using slow
a particle
[(2e)*/hv
^>
1]
a.
particles, it is in principle possible to
perform an experiment
The Elementary Theory
462
of Scattering
where the paths
[Chap. XVII]
of the
wave packets
describing the a particles do not
overlap at any point. In this case the uncontrollable phases, as in (XII.2.3), will cause the cross-terms in
(XVILS.e) Consider
FIG XVII .
.
1
Symmetrical scattering of
.
tides falling on a
An a
spinless particles.
to vanish.
now
particle
flected
the slit
beam
AB
of
(Fig. 17. 1).
may have
been de-
through an angle 6 from
from AO. The wave equation for the two particles of mass system has the form center
or an angle
8
IT
= where
=
\L
Jm, r
r 2?
R=
+r
(XVILS.7)
0,
E=
BO
in the
2
=
2
7(r)
=
porequired solution, for a large relative coordinate, must of a plane incident wave and a scattered spherical wave:
tential energy.
be the sum
=
a par-
ri
J(ri
2 ),
J/ztf
iwii?
,
The
-e ikrf(0). were distinguishable, then the scattering would be described it and would be proportional to the probability that the line joining by |/(0)| the particles would be deflected by an angle 6. In this case the number of particles scattered along OP would be proportional to If the particles 2
,
This is incorrect, for we must use the symmetrical wave function (XVII. 3.5) and not the symmetrical probability. In the center of mass system the wave function can be expressed as a product of the center of mass motion and relative motion,
*(n,
r,, t)
where, the center of mass being at
-
(XVII.3.8)
*c(K)*(r, 0,
rest, ^c(JR) is just
a constant factor and $
equation (XVII. 8.7). The interchange of the particles is equivalent r. In polar coordinates, the reflection of coordinates implies replacing r by
satisfies Ibo
Hence the asymptotic behavior of the scattering beams .moving with equal and opposite velocities toward aperture / and II must have the form replacing 8
by
TT
6.
ikz
If each wave for the two beams and Vn, then the incident wave
r is
-
0)].
(XVII. 8.9)
normalized to unity in the volumes
Scattering of Spinless Particles and Mott Scattering
[XYII.3.]
e is
g-z =
2 cos kz
f
fcsj
ikz
_}_
=
2 cos
-
k(zi
463
z 2)
such that 12
2
cos
dVn =
dFj
2,
representing one particle per unit area in each beam. From (XVII. 8. 9) it follows that the effective cross-section for a collision in which either particle is
dQ
deflected into the solid angle (7(0)
If
one of the particles
^(9) and
is
+ /(TT -
|/(0)
2
dO.
0)1
(XVI I. S JO)
at rest, then the differential cross-sections
is initially
and center
in the laboratory
are related
=
da
mass systems,
of
respectively,
by
9 d9 =
d0,
where
=
2 sin - cos 2
2,1. =
.
.
^ tan 6 ,
-
sm0
6
+ cos
=
tan 2
;
Hence
2 cos--
0.?
6 = }0
and
=
CTL(O)
Hence the
dO =
|/(29)
For the Coulomb scattering and obtain
=
lf
}
+ /(TT
of slow
(cosec
a
6
4
is
given
2
26) 4 cos
we
by
dQ.
[
particles,
+ sec
4
-
(XVII..8.11}
0.
cross-section for differential scatteruig
d(e)
dcM(20)4 cos
(Z7JJ.SJ*)
use the expression for /(9)
+ 2A cosec
2
6
sec 2 9)4 cos 9,
(XVII.8.18)
where in
A =
which a
= Z
2
cos [a log (tan 2 9],
e*/hv.
We observe that in Coulomb scattering of identical particles the differential cross-section
is
not independent of
Bose-Einstein statistic
is
This
h.
quantum
only
expected, since the concept of mechanical. It is purely a nonclassical is
effect.
The
ratio of
quantum and
_ ~"
cosec
4
classical cross-sections
9
9 + 2A cosec 2 9 cosec 9 + sec 4 9
+
sec 4
4
=
1
+
2A
sin 2
9
1-2 sin
cos 2 2
9
9
cos 2
9
sec 2
9
The Elementary Theory
464 at
=
9
[Chap. XVII]
45 yields
=
aq so in a
of Scattering
quantum treatment twice
as
(XVII. 3.14}
2
many particles will be
scattered as under
classical theory.
XVIL3.A.
We The
Scattering of Particles with Spin
must use an antisymmetrical wave function by
|ii
to describe a collision.
state before the collision can be defined l^o)
=
Ml)>k2)>|a(l)>|b(2)>
where
and
effect
due to the
-
K2)>Kl))ia(2)>j&(l)>,
(XVIL8.15)
\u} \v) are unit vectors in the two-dimensional spin spaces, referring to the spin states of the two identical particles, and |a) and \b} refer to states other than spin. The state after the collision, neglecting the small relativistic
is
possibility that spins
may change
their average directions,
given by |*>
Kl))K2))i#(12)>
-
M2)>K1)>|*(21)>,
(XVILS.16)
the space part of the wave function. It will be convenient to rearrange (XVII.3.16) in such a way that symmetric and antisymmetric spin and space wave functions occur explicitly and write
where |#(12)}
is
A
(XVILS.17)
where
Let the average directions of the two spins be represented by the unit vectors u and v with polar coordinates (0,
From
(1 X.S.I 5)
we may
write
|(1)>
where
|1)
and
(*\K\*)
1'}
=
1(1
=
cos
1
1
1}
+
are eigenstates of a\ z and
-
cos e)(* fl |l* s )
tr^.
+ 1(3 +
(XVILS.18)
Hence cos e)<*
(XVII.8.10)
where the operator
K is defined by
Scattering of Spinless Particles and Mott Scattering
[XVII.3.]
K= and where we use the
(n
cos
-+
with 9
=
+
2
cos
cos
0'
sin
- sin
+ sin
sin
cos
0'
angle between the average directions cos
qj)
(XVII.S.S0)
/?'
fi
fi*
fi
|[1
-
qi)5(ra
relation
cos
=
-
465
6 = u-v =
The expectation value
cos 6 cos
0'
+
<')]
(
of the spins,
=
i(l
+
cos 6),
and
r
sin 6 cos
sin
(<
<')
K
with respect to the two-particle state |^} is the probability that one particle is in the volume element dVi at TI and the other in the volume element dVn at r2 of
:
=
1(1
1(3
+ |(3
-
cos 9)
J
+ cos 0) + cos &)PA
-
1(1
cos
(XVILSJ81)
,
where
J (XVI1. 8. and If 9 is not known that is, if the two colliding beams are unpolarized then we must average (XVII. 3. Iff) over all the cos 9 and obtain
dQ = fj (*\K\*)
The
differential cross-section in the center of
where
= =
A (B)
If
1 [P a
|/(2ff)
|/(M)
+ /(TT - /(TT -
+ 3PJ. mass system
2
20)| 2
20)|
4 cos
0,
(XVII.8.84) is
,
^
4 cos
JT
.
^
0.
the spins are in the same direction (triplet spin state scattering) then
cos
0=1, and for singlet spin state cos
solid angle
element dO
is,
=
1.
The number
scattered into
a
in the triplet state,
(*\K\*)T
=
(XVILS.27)
PA,
and, in the singlet state,
<*!*!*>.
We
=
PA
+ Ps) =
i [<*(*)
+
^(tf)].
can apply the above results to proton-proton
(XVII. S.2S)
(or electron-electron)
The Elementary Theory
466
of Scattering
[Chap. XVII]
Coulomb scattering, protons suffer a nuclear scattering. For a pure Coulomb scattering of protons Mott's formulas (XVII.8.%4) for singlet and triplet spin states gives the differential crosssections as(Q) and 0-^(0), defined by (XVII.3.26), and the asymptotic form of the Coulomb wave function* yields
scattering where, in addition to
e4
=
4 {cosec 6
-j
+ sec
-
4
2 cos [a log (tan 2 0)] cosec
sec 2 0},
(XVII.3.29)
where we use (XVIL3.%%) and -Icosec
4
+ sec
4
-^ /cosec
4
+ sec
4
+ 2 cos f ~ log (tan
2
-
2
TU
(X7I7.3.30)
0)1V
(XVILS.S1)
0)
ijj;
PA =
2 cos
[
However, the angular distribution
log (tan
arising
from the Coulomb repulsion
of protons will determine the scattering so long as the colliding protons do not can estimate the closest distance at which see each other's nuclear field.
We
the nuclear force will take over from the in a head-on collision (with
I
=
0),
the closest approach can be defined Tc ~~
by
_e?_mc?
me 2
Coulomb
E
E=
= ~
e*
me 2
e*/r
.
by noting that
E
of the protons,
Thus, for
[|mev]
E
to be of the order of the range of nuclear forces, of \
repulsion,
with a kinetic energy
E
must be
of the order
mev.
Finally,
(a)
we must
From
the
observe the following points. definitions
(XVIL8.eS)
and
(XV11.8. 88),
(XVIL8.e6), we see that an interference term/(20)/*(7r
-
20)
or
from
+ f*(26)f(* - 26)
occurs in the cross-section. This term characterizes the exchange interaction, In classical mechanics the particles are distinguishable and the resulting
+
Ps)' compare with Figs. XII.5 and XII. 6. 2 the in the (b) I) spin scattering process, then there exist (2/S different spin states; see problems 1, 2, 4 of Section IV.3.B. Of these, S(2S + 1) 1) to odd spin for half1) (2S correspond to states with even spin and (S
probability If
S
is
just f (?A
}
+
is
+
+
integral S; the converse holds for integral S.
Using problem of the
two
1
XIV.6.B, show that the scattering probabilities even and odd spin states are S(2S l)/(2S + I)
of Section
particles in
+
2
*
For the general form of a Coulomb wave function consult Mott and Massey, Theory Atomic Collisions, Oxford Univ. Press, Oxford, 1949.
of
rXVII.3.]
and (S
+
Scattering of Spinless Particles and Mott Scattering 1)(2S
+
+
1)/(2S
2
I)
,
respectively.
Hence the
cross-section can
be expressed as
for half-integral
S and
as
=
*(0)
for integral S.
The two
=
2
Js~
s(e)
+ 25^1 ^(6)
limiting cases in
/^
2 2 ) (l
+
3 cos 2
(XVII. 2. 27} 4
and
(XVII.8.33) are (a)
e*/fto
1,
in
1, (b) e^/fw 0) /sin 0, which gives
the classical formula.
CHAPTER XVIII
THE FORMAL THEORY OF SCATTERING
XVHI.1. The 5 Matrix In a scattering process one
is
interested in a transition
from a noninteracting
two colliding systems. In other from a state where particles are is a transition the words, scattering process approaching each other to a state where particles are receding from each other. state into another noninteracting state of
An
elegant formulation of the scattering theory has been given
by Lippman
and Schwinger and a more general approach is contained in a paper by Gell-Mann and Goldberger.* The total Hamiltonian H = T + H' is assumed to have a continuous spectrum of eigenvalues. For more general problems a discrete spectrum is superimposed over the continuum spectrum. If \ft) describes the state at time t and \It) the corresponding interaction picture state vector, then we have the equations
where If, t)
H
r (i)
so at
t
=
= e- /wa|7, = ef.WB.H'e-MXB;
we have If,
In the absence
(XVIII. 1 .8)
t ),
0)
of interaction the
=
(XVIII. 1 4)
\I, 0).
Schrodinger equation can be solved in the
form |fo, t)
*
\E,
f)
=
e-tf^'lJS)
= e -/va\E)
A. Lippmann and J. Schwinger, Phys. Rev., 79 (1950), 469. L. Goldberger and M. Gell-Mann, Phya. Rev., 91 (1953), 398.
M. 468
=
9
(XVIII. 1.5}
The S Matrix
[XVIILI.]
H
is an eigenstate of Q with eigenvalue E. We may use the eigenstate at the infinite past to set up a #o representation. The transition from an initial state \EQ ) to a final state \EP ) is caused by the interaction H'. From the rate of transition we can determine the differential cross-section. The
where \E} \E) of
of the statement of "an initial state \EQ ) with energy was discussed in the formulation of the time-dependent perturbation theory and it will not be elaborated any further at this point.
meaning
E^
Consider the unitary operator
=
U(t,
such that
time
it
can operate on a state
/
JZ
|
>
[7,0
dl
(XVIILI J)
<7,*o[,
at time
to produce the state
to
|7, t)
at
t,
(X7777J.7) can be taken to be the energy eigen-value of the ),
where I labels the
state. (It
system.)
The unitary
operator U(t, U(t,
=
ft,)
ft>)
-
1
the integral equation [see
satisfies
OHi(O #(*',
*+(,
j
which takes account of
ft,)
<
(1.7.5)],
(X7777J.fi)
A
>
of the direction of flow of time (t formal solution ft>). as discussed for the time-dependent orthogonal transforma-
(XVI ILLS),
tions [see (L7.9)],
is
given by a time-ordered exponential U(t,
where the operator
P
is
t,)
= P exp
'
[ -|
jf
Hr (O
that defined in Section
(X77I7 J.0)
df],
1.7.
The
initially noninteracting parts of the system will be represented state vector [7, -<x> ). The state at a time t is to be obtained from [I, t)
by operating with operator
J7(J,
i
)
U(t,
ft>)
on
will contain
|7,
-) a
some
t
Z
=
-. In the limit
oscillatory terms.
A
-
We
-co)
=
special
form
of the unitary operator
i
7
| f'^ ^(t
9
.
OffiWC* -*) ,
U is
ft,
A discussion
=
of
it is
and from
df-
the so-called
an infinite past to states at an can be defined from (XVIILI. 9) as
relates states at
->)
-co the
Goldberger, but
shall simply set
U(t,
=
These terms must vanish
to insure the existence of the limit of U(t, t Q ) for fo >this point, in great detail, is given by Gell-Mann and
not needed for our present aims. (Z7777J.8) we obtain
|I,
tQ
by a
(XVIILL10)
S
matrix, which
infinite future.
Formally,
it
,
S=
17(00,
-oo)
= P exp
f
r~-|
1^ Hr(t
)
dt'\
(XVIILI. 11)
The Formal Theory
470
Another symbolic
a spectrum
definition for
8 =
|I,
f
of Scattering
oo)
dl
[Chap. XVIII]
continuum eigenstates
of
-oo I,
(I,
is
(XVIII. 1. IS)
where
|I, oo) represents the final state corresponding to interaction and eventual separation of the systems. The state |I, t) at any time t is obtained from the state at infinite past |I, oo): oo) with U(t, oo), by operating on (I,
II, t)
=
(XVIII.1.1S)
U(t, -oo)|J, -oo>.
Hence |I,
=
oo)
=
U(t, oo)
(XVIII. 1.14)
S\I, -oo).
Furthermore, the unitary operator U(t, from the final state |7, oo), is defined by
j
which generates the state
oo),
|J, t)
dl
(/,
oo
|I, t)
(XVIII.1.1S)
|.
Thus |I, t)
From
these definitions U(t,
it
=
U(t,
oo
) [I,
oo
(XVIII. 1.16)
).
follows that
OS = j
|I, t)
dl
(I,
oo
oo) dl' (I',
|l',
,t, -oo
-oo
|
.
(For bound states the vectors |J, oo) do not form a complete oo |I, ) are orthogonal to all bound states.) Hence U(t, oo If
we make
t
*-
oo
and noting that !7(-oo,
The above procedure
=
)S
is,
oo
)
=
S-i
oo)
oo,
=
XVIII. 1. A. The Calculation
=
l,
^
of course, equivalent to
since the
(XVIILL17)
U(t, -oo). 17(
set,
we obtain (XVIII. 1.18)
a time-reversal operation.
of Scattering
Cross-Section In the representation constructed from the eigenfunctions of Ho we can (or representatives) of the S matrix and calculate the
form matrix elements
However, we must first observe that the final state must not be |J, after the scattering ) regarded as an exact eigenstate of Q the separation of the two interacting parts implies a localizability of the two transition probabilities. oo
H
]
free parts of the system. This localization can be described in terms of wave packets formed from a superposition of momentum states. In the center of mass system the two parts of the system can be regarded as a plane so
wave;
the probability of finding the center of mass in any specified volume,
when
[XVIII.l.]
The S Matrix
observation of
its
position
471
is
ume. But a superposition
independent of the location of the volenables us to form a wave packet
is
made,
of
momenta
mass
for the description of the center of
of the
two parts
of the system.
This
without being an exact eigenstate of HQ We can, can represent the however, find a procedure equivalent to a wave packet description of the final state,
final state
.
by using an eigenfunction
of interaction, arising
\Ep) of
HQ and by simulating the
from the separation
of the
cessation
component parts
of the
>co. system, by an adiabatic decrease in the interaction strength as t This can be achieved by using a function exp \_(e/h)\t\] where c is a small
number.
positive
Now
let \E<>)
=
eigenstate of HQ.
described
From
oo
|7,
be the
)
The change
initial state of
the system which is also an by the interaction can be
in the state caused
by the operator
T=
S -
(XVIII.1.19)
1.
S we have
the unitary property of
T T = "(T + f
(XVIII. 1.20)
T).
The
probability that the system will be found in a particular final state differing from the initial one is
From (XVIII. 1.10),
TFO = (Er \T\Eo) = o we write
setting
t
S =
-
1
(XVIII. 1.21)
3
^ J_ m
dtHr (t)U(t,
-oo).
(XVIII. 1
dtH(t)U(t,
-o)|E >.
(XVIII.1.88)
Hence
TFO = To be
r
-~-(Ep\
consistent with the adiabatic switch-off of the interaction
insert the factor exp
[
(e/h)\t\]
we must
and write
TFO =
(XVIII.1.84)
-^(Ep\H'\Ep+),
where
=
\E P +)
f"M
dt
exp
(EP |^|
Ho)] exp
(-|
1*1)^,
-oo)|JE?o>-
(XVIII. 1. iff)
In a similar way we have
=
(Ep\T*\Eo)
(XVIII. 1.26)
F I (E \H'\Ep-),
where IJ&JP-)
=
J*^
dt
exp
f|
(jBF
Ho)]
exp
(-|
U(t,
>)\E Q ).
||^
-
.,
(XVIII. 1.27)
The Formal Theory
472
To
derive
an
of Scattering
integral equation for \Ep+)
we
where T \Ef -}
=
=
t'|.
|<
df
_"
-
(EF
So)
-
#o)
-
(XVII LI .10). (XVIII. 1.25), we obtain
use equation
Thus, introducing (XVIII. 1.10) into the right side of
* exp
[Chap. XVIII]
|
Similarly,
-
(Er
exp
|
Carrying out the time integrations we obtain the integral equations '
We may introduce and reduce the
the states |=F)
(XVIII. 1 .28}
by
(XVIII. 1.88) to
integral equations
These same equations could be obtained by solving the Schrodinger equation (E
- H )\[ ) = Q
Q
H'\h)
form
in the
w
Hi
where the total Hamiltonian
The
integral equations
H
is
-r It
.
,
,
HQ
time-independent. provide a time-independent formula-
(XV11LI. 89)
tion of the scattering theory.
The small
positive
and negative imaginary parts
in the denominator serve to select outgoing or incoming scattered waves, respectively.
In terms
of the state vectors
expressed as
The
|+) and
},
the representatives of
T can be
TFO =
transition probability is given
WFO = From
|
by
47r 2 [5(^
-
EW\(E,\H'\+)\*.
(XVIII.1.31)
the definition
- K) =
Tta
_
exp
t(EF
-
Eo)
|*|
dt,
The S Matrix
[XVIII.I.]
473
we obtain [8(E F
-
~
=
2
)]
-
d(EF
Eo)
(^ dt.
Hence
=
IF
Y
-
5(J5JF
Eo)TOff'l+}|*
J^ dt.
(XVIII.1.32}
This formula shows that the only contributions to the transitions come from the equal energies in the
initial
and
final states.
The
the total time of effective interaction.
time
is
The integral ("tit represents
transition probability per unit
then given by
= dt
f
1
*(EF
-
Eo)\(EF \H>\+)\*.
Lippmann and Schwinger gave a second
(XVIILL33)
derivation of (XVIII. 1.33}
by
evaluating
W
'*
=
H^ltfft -)|J3o>| 8 ].
(XVIII.LS4)
This expresses the increase per unit time of the probability that the system starting in a state |E ) will be found at time tin. a state \Ep). Thus
W
'*
=
(EP \U(t, -o
=
F \U(t,
j-(E
-^Hi(t)\EQ)(EQ \U^(t
Using the definition TF^o
=
A JT
of
JEfj(i)
we
t
-GO)|EJP)
+ complex conjugate.
obtain
di/ (
/^
X (EFlH'eWM&'-^Utf, From (XVIII. 1.25} and
-oo)|E
the definition of |+)
lim e-^O
Hence
+ complex conjugate.
we obtain
dt 6 tf/
=
>
The Formal Theory
474 This
is
of Scattering
the state vector in Schrodinger's picture.
[Chap. XVIII]
The above results can be com-
bined into "
WFO =
+
f
dt ePtf-n/aiear-*)
\(EF \H'\+}\*
complex conjugate.
W
We multiply FO by the density of the final states p F and integrate over the final energies to obtain the transition probability per unit time in the form
WFO - y \(EF \H'\+)\*pP The
transition probabilities can be obtained
and second-order
first-
(XVIIL1.35)
.
iterating the integral equation for |+) and substituting in
by
(XVII1. 1.35).
Thus, neglecting higher-order transitions in H' we have ',
WFO
= WPO +
(XVIII. 1.36)
Wro, 2
1
where
and
=
\(Ep
V^ 4^
2lT ft
27T
\-^ /_.
a
These
results
WT* ^ Fiw "^ Irr/ITT
/
|
HprHio
w
i
,
o
w
i
1
j|
+ it - HO
#,
Tr/
|U }|2p,
(xviiLuyf)
.
were also obtained previously in the time-dependent perturba-
tion theory.
the relative velocity of the colliding systems, then the differential * \EF ) or the effective area that must cross-section for the transition |J5J ) If v is
be hit by an incident particle in order to be scattered in a unit solid angle is equal to the transition of the final relative momentum
about the direction rate divided
by
v.
Hence the
final result is
*FO
XVIII.2. The
=
T; \(E,\H'\+)\*Pr
The S Matrix and the Phase
relation of the
S
matrix to the phase
most conveniently, be developed in terms K defined by [see (1.7.10)1
8 =
(XVIILL38)
.
Shift
shift in
of
rrlf
a scattering process can,
a Hermitian reaction operator
The S Matrix and the Phase
[XVIII.2.J
We shall
Shift
A by PC = -tan A
introduce a phase shift operator
and obtain the S matrix
in the
475
the relation
(XVHI.2.2)
form
S =
exp
(2iA),
=
2i sin
(XYIII.2.3)
so
T= SThe
A
relation of the phase shift operator
tering process can be found
given representation. Let us assume that U(t,
M
where
is
U(o, -oo ) Hence
a constant
J(l
(XVIII. 2.5}
_
Y
i
and using
;
7(-oo)Af.
v^/
7(-)' 7(oo) = 1
\iK and F(
+00)
=
i7(0(l
+ ^f"
oo)
=
1
+
\iK.
7(0
= 7(-) -
V(f)
=
ij(t
'(
expression for
K
ijT(oo)
representatives of
(XVIII.2.8)
).
oo)
and
Z7(< 3
+00) become
V(*>)
HMWd') dt',
+
"('
where the step function
1
Thus
adding these two equations and using 7(
K=
in a
+ S). Then
integral equations for V(t).
The
700 AT,
these results the integral equations for U(t,
The
K
representatives of
t
U(t,
With
=
-oo )
which suggests the substitutions
M
to the actual phase shift in a scat-
can be written as
>)
*a
Hence
(XTVI7J4)
.
= oo and = oo operator. Putting = = 7(oo)j|f andl = -oo) 1, we obtain S
=, C7(-oo,
=
A e{A
by studying the
U(t,
By
I
t')
is
)
+ 7()
- 0^r(OV(0
defined
<
=
2,
we
get
(XVIII. 2.9}
by
*-'n) _f+l
ifO*', ifi
-i_l
is
_7(_
o)]
=
K are given by
00
i/"
Fi(<)F()
dt.
(XVIII. 2.10)
The Formal Theory
476
KPO = (Ep\K\E
\)
=
f
I (E
x
of Scattering
[Chap. XVIII]
\
=
e -W*>B.V(f)\Eo)
(EF\H'\E'},
(XVIII.,2. 11)
F(f) i#
(XVIII. 2.12)
where dt
The equation determining
-
~ (ft
exp
-
Fo)
|t|
the state |E'} can be obtained
>.
by using the
integral
equation (XVIII.2.9) in (XVIII.2.12). It yields
[f
(ft
-
Ho)
{
\t\]\Eo)
dt' dt-^ f'm f'm dt ,(t
(B
exp
-
exp
^P
(E
-Ito-j= Making the transformations [f
\t\]
+
f
^
exp
[
(^/K)\t\]j
t
/*
'
7
==
i
X |*l]
^' exp
(f *)
and
|
-
(<
X
-
ffo)
-
,
(-f
F
and suppressing the
factor
we obtain exp
X
ff
;
tit
_
exp
From
p we
- -
d /-- ^
(F^TO)
'
w exp [Sf (S ~ 54
obtain [/')
=
27rft5(
F
-
Eo) \E
)
+P
ff (^TH ) W-
(XVIII. 2.1
O
By writing we
-
(XVIII. 8. 1
tf )|I>,
find that |J)
=
|tf
>
+ P ^15;
Thus from (XVIII. 9.1$) and (XVIII..8.14) we 27r7i5(# F
-
tf
)1 />
which can be written as
==
/^ * exp [| (tf,
(XVIII. 2.1
H'|I>.
get
-
So)
|
V(t)\E l*l]
),
[XVHI.2.]
The S Matrix and the Phase
dt
Shift
(Er
exp
Hence, the time-independent state vector
477
-
H.)
-
describes a stationary state
\I)
according to the relation
(XVIII. 2.16)
.
XVIIL2.A. Eigenvalues The
of the Reaction Operator
representatives of the reaction operator
KPO = the operator form of which
-E
2irS(Er
K can be expressed as f
)(EF \H
(XVIII.2.17)
\I),
is
where
K=
27rAK,
A=
\E }(Er
and
=
(EjF\H'\T)
(XVIII.2J8)
(XVIII .2.19)
\
(Er\K\E
(XVIII.2.20)
).
T is
In a similar way the operator form of the equation for
T= where r also
is
(XVIII JiM)
-2irtAT,
defined for states of equal energies
(EF \T\E
)
=
and
is
such that
(XVIII. 2.22)
(EF \H'\+).
Now we
use the relations (XVIII. &t) and (XV I II. 2. 21) to obtain
= -I A-
1
tan
(XVII1. 2.3),
(VIII. 2. 18) and
A
(XVIII. 2.23)
7T
and
Let us introduce an operator
by
5
tan
A = A tan 5,
so '
sin
Ael A
= A sin 8 e.
Hence
T-
-27riAr
=
2ii sin
5 ew
(XVII1. 8. 8ft
and 2
-T \(Ep\ irn,
sin 5 e*\E )\*8(Ep
JSf
)
/
*.
The Formal Theory
478
The
transition probability per unit time
Wpo =
(
EP\
sin d
of Scattering
is
eiS
\
E o)\* (^ - E O)-
Using the operator property (XVIII. 1.80) write the relation between the matrix elements 47T
2 ]
Er)d(Er - E O )T*IF TIO =
8(EF
[Chap. XVIII]
(XVIII.$.26}
of the operator
T we
can
of r as
2irid(EF
- E O )\JFO -
r%F].
I
Summing over EF
Hence the
or
and
jE7
setting
total rate of transition
E =
-Z?F,
from the
we
get
initial state is
WFO = ~| Im(roo).
We now introduce the
eigenstates of the reaction operator
K\KA ) = The
vector
(jfi^i)
is
also
an eigenstate
KA)
=
KKA
=
r
(XVIII. 8.87)
TJ.|J?A)
of
K by
X^>.
(XVIII.2.28)
so that
r,
1
=
sin 5^ et5A JE4 )
7T
and )
7T
The eigenvalues XA and TA
KA
- tan
-
given
are, therefore,
=
T^L
A,
by sin 6^ eiSA
Introducing the representation in which probability (XVIII. 8. 86) can be expressed as
WFO = 4
s 1
TTll
where /FA
Ho
is
=
(EP \KA )
is
.
7T
7T
sin
^^fpAflo
the eigenf unction of
K
is
**(Ep
diagonal, the transition
- So),
(XVIIL8.89)
K in the representation in which
diagonal.
Formula (XVIIL8.87)
V *& Er
TfFo
yields the result
= 4- Im(E TTfl
\sm de*\E
)
=
~
2
<J5
|sin
8\E
)
TCll .
=^Li/^i
2sin2
(XVIIIJ8.SO)
^.
A.
which
is
the total probability per unit time for transitions from a particular
[XVIH.2.]
The S Matrix and sum
state. Finally, the
the Phase Shift
of the total transition probability per unit time over
same energy
all initial states of the
is
=
$4
=
.
and the functions fOA can be
eigenfunctions fOA = (EF o\KA ) are func2 2EM/^ ] and the representatives FO as func-
related to spherical harmonics. tions of the vector fcoF^o
sin 2 8 A
-
are the usual phase shifts,
5A
by
expressed
oA* sn Obviously, the
479
The
K
,
kF and fe are invariant under a simultaneous rotation of kF and k The vectors k and kp are the propagation vectors defining the initial and tions of
.
,
final states.
Following Lippmann and Schwinger, we put /c^ == CYim (k Q ) and A = I, m; the eigenvalues for depend only upon the order of the spherical harmonics
K
that
is, d
A
=
di.
From
the normalization condition
we have
where p dO is the number of states per unit energy range associated with the motion within the solid angle element cK2, or P
z = V ~ Vp dp ~
8-jrW
dE
87r%
v
where we take V to be a unit volume. Spherical harmonics normalized in a unit sphere require that
From (XVIII. 2.29), the direction ko
kF we have ,
for the probability per unit time that the particle
scattered into the solid angle
is
(Kl
from
around the direction
of
the expression
w=
I
Tfl
Z
sin
^ ^C\*Y lm (kF )Y lm (k^ P dfi.
(XVIIIJSM)
l,m
Dividing by the relative velocity v of the particles, which measures the flux of incident particles, and using the spherical harmonics addition theorem 07
I
^ ffl
we
=
-1
I
Y lm (kF )Y?m (k = ^^(cosfl), )
I
obtain the differential cross-section for scattering through an angle
dff(ff)
=
~ K
|
Y)
(21
+
1) sin di
e^'P (cos z
2
^)[
d&,
(XVIII &JS8)
l
where
The
6 is
the angle between
feo
and
kp>
total scattering cross-section, according to
6:
(XVII1. 8. 80),
is
The Formal Theory
480
=
9
lb
WIIV
Zm
sin2
we
note that
WlFi-Cfco)!
lt
1
=
of Scattering (2Z
K
+
1) sin*
[Chap. XVIII]
,,
I
where
S =
"
2lA
= T+
1 can be used to study the relation phase shift and the potential which is the cause of the phase shift. the phase shifts are real quantities. Because of the Hermitian property of
Finally,
of the
K
A careful study by
Bargmann
of the relation
between the phase
shift
and
scattering potential shows that phase shifts do not determine the energy of bound states. The extent of a possible determination of the scattering poten-
by the phase shifts 5i as functions of energy has been studied by Levinson and also by Bargmann.* Levinson has shown that two potentials tial
V(r)
which
fall off
momenta let
same phase
rapidly enough and have the
shifts 5 L for all
are identical, provided they do not give rise to
Vi and
7
2
be the two potentials and
$/,
bound
angular
states.
5" the corresponding phase
Thus
shifts.
If
for i
=
1, 2, is finite
[see
(XVIIL5}] and
if
for
some
I
8" implies that Vi(r) = Vi(r). For every bounded potential which has the property that for large r it is smaller than C/r 2 where C is a constant, the above inequality is satisfied,
where
i
=
1, 2,
then
8{
=
,
I. Thus if V\ and F2 are potentials of this kind and if then for a large enough IQ the inequality holds and we obtain This argument is not, of course, sufficient for the construction of a
for sufficiently large
8" for
==
5/
Vi
= F2
-
all
I,
potential from
its
the phase shifts
81
exp
corresponding phase shifts. In terms of the S matrix, are related to the eigenvalue of the $ matrix by S(K) =
[2iz(fc)]-
By
analytic continuation, the function S(K)
and, in particular for imaginary values
served
by Kramersf that the
zeros of
/S
(-&*). However, as
fc
=
may i/c,
be defined for complex fc, > 0. It has been ob-
for K
stationary states should be obtained from the
shown by Mat there
exist
some zeros
of S(k)
* V. Bargmann, Phys. Rev., 75 (1949), 301. N. Levinson, Phys. Rev., 75 (1949), 1445. t Quoted by C. Holler, Kgl Danske, Vid. Sels. Math-Fys. Medd., 24, No. 19 (1946). P. A. M. Dirac, Proc. Roy. Soc. London, 49 (1937). tS. T. Ma, Phys. Rev., 69 (1946), 668; 71 (1947), 195. See also D. ter Haar, Physica, 12 (1946J, 509.
The S Matrix and the Phase
[XVIIL2.3
Shift
481
which do not correspond to bound states.* For further discussion of see Section
this point
XYHI.6.B.
XVIIL2.B. Problems Consider the unitary operator
1.
U= where the states |+> and \E } are those of (XV711.1.29). By using the defi= I of unitarity, prove that U is not unitary for bound = nition
WU
UW
states.
Show
2.
S matrix
that the
is
unitary for bound states.
XVIII.2.C. Scattering from
An example
of scattering
Two
Potentials
under the combined influence
Coulomb and
of
is the low-energy P P scattering. Another interesting examthe process of "bremsstrahlung." In the case of electron bremsstrahlung the potentials are the Coulomb field of the nucleus and the interaction of the
nuclear fields is
ple
One
electrons with photons. exactly,
and the
of the potentials
the
Coulomb
can be treated
resulting states can be used to calculate the transition
probabilities caused
by the
interaction between the electron
and the
radiation
field.
Let us consider the integral equation
=
1+)
\Eo)
+E+
_
flo
(XVIII.2.SS)
H'\+},
where
H and
H
HQ
of
'
is
to be treated
1
= H{
+ Hi
by perturbation
(XVIII. 2.36) theory. Thus,
to another free state (plane wave)|-EV) with energy
Tro
= =
(Ef \T\E
)
-2*tS(JZ,
=
-2*ti(EP
-E
- E
)(-~\(H[
)(Ep\(H(
+ H)\E
Let us further introduce a state vector \F
|F-> where the *
if
\E
) is
the eigenstate
belonging to energy EO, then the probability amplitude for transition
|JF
)
=
\EF)
Ep
+
is
given by
TJ)|+>
)
'
by
+ E _^_ HQ Hi\F~)
are solutions of the problem without
See R. Jost, Helv. Phys. Acta., 20 (1947), 256.
(xvTTTevft' ^
).
ffi.
(XVIII JB.S8)
The
reasons for the
The Formal Theory of Scattering
482
wave boundary
choice of ingoing
[Chap. XVIII]
conditions can be understood,
we
if
solve
substitute in
(XVIII 3.88),
+ H$\+),
(Ef \(H{
and use (XVIII.8.3ff), which
leads to
+ H)\+) =
(EF \(Hi
(F-\H'2 \+)
+ (F- \Hl\Bo).
(XV III. $.39)
This interesting result was obtained by Gell-Mann and Goldberger. If Hz were actually zero, the state [+) in (XVIII. 2. 35) could be replaced by \F+),
and (XVIII.
.39)
+ HS)\+) =
(Er \(Hl
The second term found, even
if
H
wave boundary
E
from a state
could be written as
r
in
(XVIII.
were
.40) is
and
+ (F-\Hl\E
).
(XVIII JB.40)
the scattering amplitude that would be
term the
final state contains the ingoing the probability amplitude for transition under the action of H[, But in the case of
zero. In this
condition, )
(F-\m\F+)
it is
to a state \F
},
bremsstrahlung there is a photon in the final state, while the state arising from the state \E } under the influence of H[ cannot give rise to a photon,
by assumption H{
since
is
particle with the radiation
not related to the interaction of the charged This term must, in this case, vanish. The
field.
only term contributing to the transition probability
is,
therefore, the first
(XVIII. 8. 40).
term in
of the small imaginary part in the denoma with state +} outgoing wave boundary conditions becomes, under inators, the operation of Hermitian conjugation, a state } with ingoing wave boundIt
must be noted that because |
|
ary condition.
The
first
term in (XVIII.
conjugation, that
and the
.4ff)
is
invariant with respect to Hermitian
is,
final state is
an incoming state vector contrary to the usual practice
of choosing final states. However, the fact that the choice of |F final state is the correct one will be explained in the next section.
For later reference
}
for the
be useful to write the integral equation (XVIII.8.85) in different forms. We can introduce an outgoing wave state \F+) as a stationary solution of Schrodinger's equation with the total Hamiltonian
is
HO
+ H{.
then solved by
it
will
The Schrodinger
equation,
[XVIII.3.]
Scattering Problems and Ingoing Waves
where \F+) is an elgenstate of boundary condition is, of course,
H + HI
and the
483
original outgoing
wave
maintained.
still
XVIII.3. Final States in Scattering Problems
and Ingoing Waves In an interesting paper Breit and Bethe have discussed* the outgoing or ingoing wave nature of the final state in the calculation of transition prob-
This question will arise only if the states in question are not pure plane waves, but are modified by the addition of an ingoing or outgoing wave. In this case the transition probabilities are to be calculated with respect ,to plane waves modified by the action of a potential.
abilities.
In the space-time description of scattering (the wave picture) we use stationary-state wave functions, and the time dependence of the process is represented by forming a wave packet. In space-time description of scattering
we
start with a steady incident beam, part of which is deflected, and from the intensity of the scattered wave we compute the differential cross-section. In a scattering process the size of the incident packet is large compared to the scatterer, and it is a good approximation to represent the incident beam
by a plane wave exp (ikz). When the wave enters the force field a scattered >- oo wave is produced, and for the stationary-state wave functions (r )
we
write
$
= exp
(ikz)
+ x(r).
A wave packet can be produced by sending a steady beam of incident particles through a collimating slit. In scattering by a central r
>-
oo
field
the time-independent
wave
function for
has the form ft,
S exp
(ikz)
+ f(8) ~ exp (ikr).
(XVIIL3.1)
The last term represents a modification of the plane wave by an outgoing we form a wave packet which at time t = wave. For a time-dependent will be supposed to be moving along the s-axis toward the scattering center. \[/
The time-dependent wave
function
where
E For *
t
<
is
represented
~
by
(XVIII.3.2)
the wave packet has not yet seen the potential, and
G. Breit and H. A. Bethe, Phys. Reu., 93 (1954), 8S8.
it
behaves
The Formal Theory
484
of Scattering
[Chap. XVIII]
as a free packet in space. After the collision there are scattered waves, both ingoing and outgoing (Fig. 18.1). The ingoing waves arise from the interfer-
ence of the secondary waves (Huygen's principle) with the outgoing waves.
Outgoing spherical waves
Wave packet Collimating
Incoming spherical waves
slit
V.i
\
\
)i
Steady stream of particles
t
t>t z
FIG. XVIII. 1.
=
Ingoing and outgoing scattered waves.
The wave packet (XV'III. 3.1)
consists of
superposition of the incident plane
waves
two terms: one formed from the
(to the left of z
=
ZQ)
exp
(ikz),
and
the wave (ikr). For t < there are no scattered waves;
the other formed from the outgoing waves l/r exp
has not yet hit the scatterer, and therefore, that
is,
for
t
<
0,
f In the time region
t
<
Ck
exp (ikr
0, r is
|
E\
d*k
the same as z and kr
This shows that along the
-
=
(XVIII. 3.3)
0.
negative.
Thus
for
t
<
0,
-kz.
=
the phases in the the reason for the destructive inter-
z direction, just
packet change in opposite ways. This
is
=
is
before z
wave l/r exp (ikr) and an ingoing wave. For t > WiMon the signs of r and z are the same, and constructive interference occurs in the same places for both terms (XVIII. S.I). If the plane wave is
ference between outgoing
modified
by the addition
of (l/r)/(0) exp
(ikr), the relative phase relations and the corresponding process
discussed above would reverse themselves,
would not be the intended one. Let the total Hamiltonian be
H
= HQ
+
H' where H$ comprises the y
energies of the incident particle plus the scatterer. for
H
Q
is
kinetic
The Schrodinger equation
[XVIII.3.]
Scattering Problems and Ingoing Waves ih
we
I \E,
=
t}
H%
we
can represent the state IE, wave, for example, is incident, the state \E, If
wish,
485
(XYIIL34)
t>.
t}
by a wave
t}
is
packet. If a plane modified by an interaction
energy!?': ih
I
|r/ > *>
= (H
+H
'
(XVIII J^)
)|f/ > *>'
Let |E> be the time-independent solution of (ZFI/LS4), so
where the label S enumerates the continuum of possibilities for solutions at a fixed energy E. For example, the waves may have different propagation vectors and spin specifications.
The
observables
H
S,
,
and spin
s
must form a complete commuting
wave function
representing the state can be normalized by picture so the
f
(r\E, S, s)
dSd E (E,
S,
s'\ r ')
=
[JE?,
set,
S, s) in the Schrodinger
'8(r
-
(XVIII. S.6)
r').
In the interaction formulation, equation (XVIII.3.5) can be replaced by the integral equation
where [f,
t)
=
e
-
*wa|I,
|f, 0}
*),
=
|7,
0}
=
\E, S, s),
and It is solved, to the first order in
|I, *>
which |f, )
is
=
= |^
S,
>
H
',
-
by '
|
/
equivalent to \E, S,
s,t}-\
exp
[I
(f
Oflo]
fl'IB, S,
s,
df
(XVIII. 8.7}
Introducing the unit operator
we
obtain
*(r,
-S, t)
=
X
d'r'
exp
(*
-
*')^
tia(r, s)
s)H^(r',
s, ,f)
(XVIH.3.8)
The Formal Theory of Scattering
[Chap. XTIII]
where
=
*(r, S, V>o(r,
*>,
S,
t},
fas(r, s)
}
(r'\H'\r)
The completeness relation (XVIII. 3.6) can be satisfied by having
Neglecting the spin for the present, the distorted plane waves represented in the form
may
be
(XVIII.S.9) where, outside the potential, Tz
=
(Fi cos
Sz
+ Gi sin ^e^z,
fc-r
=
cos
0.
plus sign corresponds to outgoing waves, and the minus sign to ingoing waves. The functions Fi and Gi are the standard regular and irregular solu-
The
tions, respectively.
From
the addition theorem of spherical harmonics, equation (XVIII.8.9)
can be written as
z
Multiplying by Yv m '(k) and integrating over angle around the direction fc), we find that
Thus the polar coordinate
dOfc
(an element of the solid
solutions are related to %&
by means
of a unitary
transformation, the coefficients of which are F m (lc) rfQ&. The unit vectors k and P are defined by the polar angles. As a consequence of the unitary char*
z
acter of the transformation, the functions Yim (f)[Ti(kr)/kr]
The
form a complete
the factor exp (d=i8i) is immaterial because of the Unitary nature of the transformation. If we use an outgoing wave modification of the plane wave, then in the region of large kz the fas contains terms exp (ikr) coming from exp (ikz) and also terms exp (ikr) set; so does %z-
choice of sign for the
6 Z in
coming from the outgoing wave. In (XVIII.8.T) the second term represents the first-order effect of in the form of a superposition of fas- We have seen in the /S-matrix formulation
H
of the scattering theory that for large times
we
/
are led to conservation of
[XVIH.3.]
Scattering Problems and Ingoing Waves
487
energy. For example, an inelastically scattered electron (as In the case bremsstrahlung) wave in each energy region Is represented as a
of
superposition
of 4>S8 As pointed out above, in the direction of k each. ES contains two terms with the same phase exp (ikr), and they both contribute to the .
<j>
wave
packet of inelastically scattered particles. The contributions from exp (ikz) and l/r exp (ikr) at large r are of the same order, since from the partial wave analysis of exp (ikz)
l/r exp (ikr)
beam
arises
we know
that
it
contains mainly terms of the forms
and l/r exp (-ikr). The reduction of intensity in the primary from the interference of the outgoing wave parts of + exp
UA)/(0) exp (ikr) with the unscattered wave packet. Hence tion of an inelastically scattered wave packet, the
(ikz)
in the calcula-
outgoing wave part of fas cannot be neglected. This argument of Breit and Bethe shows clearly that the calculation of (XVIII.S.8) in terms of distorted plane waves with outgoing wave modification is possible but is not directly interpretable in terms of a differential cross-section.
on the other hand, we take fa s as plane waves distorted by "ingoing"
If,
wave
modification, that
is,
exp
as (ikz)
+ l/r exp
(-ikr)f(ff),
then in the direction of each & the phases of the ingoing waves are just opposite to those of the plane wave parts. For the ingoing wave part, one has destructive interference. Then the second term of (XVIII.8.8) gives a representation of the inelastically scattered
wave packet in terms of undistorted waves and of plane is, course, interpretable in terms of the number of particles. The above argument does not effect the initial state of the system. It is still,
asymptotically, a plane wave plus an outgoing wave. But the final be chosen in such a way that at large r it consists a of a distortion
state
must
plane wave of by an ingoing wave. All the above considerations in the choice of a final state are closely related to the invariance under time-reversal operation, which for a time-independent
wave function
consists of a
complex conjugation
operation.
XVIII.3.A. Problems Prove that the eigenvalues of the $ matrix are of the form e a<x where X an eigenvalue of the phase-shift operator A introduced by (XVIII. 2.2}, where X is a function of wave vector k. 2. Prove that the S matrix commutes with the Hamiltonian, momentum, and angular momentum operators. 1.
,
is
3.
Show
related
by
that the basic operators
K
and
i
of the scattering
theory are
The Formal Theory
488
r
=
K-
of Scattering
[Chap. XVIII]
i-
Hence ==
TFO
This 4.
is
KFO
called the radiation
From
damping equation.
(XV711. 1.29) show
equations
that the states |+> and
>
|
are
related according to is a time-reversal operator. Show also that asymptotically the wave function contains a plane wave. Discuss the significance of the state } in is the problem of bremsstrahlung where, after the photon produced, the charged particle and the nucleus may interact strongly through the long-
where 5
|
range Coulomb
By
5*
field.
Does
this affect the y-ray
spectrum?
using the transformation
show that the operator
satisfies
the integral equation
Q=1 +g + *e-gc gU Show
also that it is solved
by '
where (E
+ ie
l
H)~ can be
regarded as the Green's operator of Schro-
dinger's equation. 6.
Discuss the physical significance of the scalar product of the states
XVIIL4. The Principle of Detailed Balancing In classical statistical mechanics, a collision between any two members of an ensemble in an equilibrium state corresponds to a state where the two members can occupy a part of phase space with initial momenta p1 and p2 and leave that part of phase space, after collision, with momenta p{ and pThe reverse process of entering with momenta p{ and p^ and leaving with
momenta pi and p 2 (the time-reversed state) is also possible. In general, the total rate of transfer of members from either portions of phase space where momenta arep or p to other portions of phase space where momenta are
r
p and
theory
p
this is
f
equal to the transfer rate in the opposite sense. In classical regarded as a sufficient condition for the entropy to increase is
and permit the system to reach statistical equilibrium. In statistical mechanics the process is governed by what is called the "principle of detailed balancing."
The
[XVHI.4.]
Principle of Detailed Balancing
489
In quantum mechanics the principle of detailed balancing, for a system in equilibrium, refers to a direct balance between the rates of processes in opposing directions of time. This means that a transition from a state A to a state B can be balanced by a transition from the state B to the state A
without the need of an intermediate transition between
remember
A
and B.
We
must
at this point that the transitions in question are determined
by
transition probability amplitudes (see Chapter XII).
For
particles without spin the detailed balancing
ability that the collision process,
P2
Pl,
will occur is the
means that the prob-
-^p^pg,
same as the probability that the reverse
~Pl
~P2 - --
Pl,
~~
collision,
P2,
particles with spin we must recall that under time-reversal the spins are reversed. In this case the principle of detailed balancoperation
will occur.
For
ing requires equal probability of occurrence of the opposing processes, Pi,
p
2 , Si,
and
~P2
""Pl?
3
~~ s lj
S2
~S
'2
-
>-
pi',
~
P2,
Pl,
Si, $2
P2,
Si,
S2
.
A general quantum mechanical derivation of the detailed balancing theorem was given by Coester and
also
by Watanabe.* Here we give a
short deriva-
tion of the principle.
We wish to is
determine in what
way a
related to its reverse transition
from
from a state A to a state -B Under a time-reversal opera-
transition
B
to A.
tion the transition probability amplitude
(XVIII. 1.35), as follows from
(VI. 1.20) in problem 2 of Section VI.2.A, assumes the form
WFO =
(XVIIL4.1)
W^O-F,
where the right side represents the transition probability amplitude in reverse order. Thus, in accordance with (VI. 1.18) and (VI. 1.19), the transition
W-O-F contains the states with momenta and spins opposite to those contained in
WFOLet
da-fo
be the
cross-section for a collision
between two
particles
where
the relative velocity of the particles deviates into a solid angle dtip in a coordinate system where the center of mass is at rest. To incorporate conservation of energy (Eo
=
EF)
we may,
equivalently, write the cross-section in
the form
*F. Coester, Phys. Rev., 84 (1951), 1259. Watanabe, Phys. Rev., 84 (1951), 1008.
S.
The Formal Theory of Scattering
490
which must be equal to
[see
(XVIII. 1.88)]
~
which
2
for the reversed transition
is
(XVIII 4
given
by
-
~ UF
where
^^
Eo)
dE
(XVIII 4.3)
,
W7so
Ns0 and N F are spin degeneracies in states
and F,
S
differential cross-section
[Chap. XVIII]
(XVIII 43)
is
a function of
respectively.
pi,
p
2,
pi,
The p-2-
But (XVIII43) does not change under a space reflection, and it is therefore a function of pi, j> 2 p{, p 2 Since the sums over spins of the initial and final states in both (XV III 4^) a &d (XVIII 43) are the same, we obtain the .
,
detailed balancing theorem:
Ns0 PO =
v
The
An
^
vpN, f
detailed balance cannot be established without
interesting example
in classical statistical
(XVIII 44)
pp.
summations over spins. is Boltzmann's obcollisions between mole-
mechanics
servation that detailed balancing does not hold for cules with a nonspherical shape. For quantum theory,
when angular momenta, polarizations, or any other intrinsic properties are involved, the principle of detailed balancing will hold only to the first order. However, in general, to insure the increase of entropy of a
lack of detailed balancing will
system
it is
assumed that the
be made up by other processes.
XVIII.4.A. The Spin of the
TT
Meson
was suggested by Marshak and by Cheston* that the spin of a TT+ meson may be observed by producing it in a collision process of the type It
p
+p
v
D+
TT+
and also by the inverse process of T+ absorption in deuterium. In the above proton-proton reaction
we
right sides, respectively.
and F to the left and symbols mass system the velocity of is the reduced mass and p 2p/M, where f
shall assign the
Thus
in the center of
the proton is given by VQ = refers to the momentum of the proton.
M
The
center of mass, neglecting deuteron energy, is
the
tem. *
momentum
of TT+
and
is
VF
= (l/M^prj
where
pT
M D refers to the reduced mass of the ir-D sysV
From
R. E. Marshak, Phys. Rev., 82 (1951), 313. B. Cheston, Phys. Reo., 83 (1951), 1118.
W.
velocity in the final state in
Expansions of Scattering Amplitude
[XVIII.5.]
we get,
=
PO
= &M,
<>
dpr
P'-P'^'
approximately,
Since spin of the proton
N,o = If
o dp p-2l'
p
491
the spin of
IT is s,
is
+
=
(2s
+
Hence the
=
1)
is
4.
Tr+D) 2p
M
is
pM 4 ~
d
+D
2
ratio of cross-sections d(r(Tr
X 1 + 1) = 3(2s + 1) h. We may now use equation
1)(2
where the spin of the deuteron and obtain
^ da
AfxPr-
then
^SF
d
=
the spin degeneracy
-|ft,
2(2s
pF
+D
^
*-
pp)
p* MTi)
(XVIII.4.4)
PW
is
>-
pp)
+
in which the only unknown is the spin degeneracy (2s 1). For agreement with the experiments carried out at Berkeley* and Columbia! the choice for the ir + meson is s =
necessary.
XVIIL4.B. Problems 1.
Prove that
of the 2.
S matrix
for the scattering of spinless
S waves
the representations
are symmetrical.
Prove that
in the scattering of the initially polarized particles time-
reversal operation does not yield detailed balancing. 3. To get detailed balancing for the scattering of unpolarized particles, it is necessary to average over the initial spins and to sum over final spins. Does
one obtain detailed balancing
if
the scattered particles are polarized?
XVIII.5. Expansions of Scattering Amplitude
We
have seen that the
S
matrix elements represent the probability ampli)) at an infinite past to a state ]*"(
tudes for the transition from the state *
W.
F. Cartwright, C. Richman,
M. N. Whitehead, and H.
A. WHcox, Phys. Rev., 91
(1953), 677. t R. Durbin, H. Loar, and J. Steinberger, Phys. Rev., 83 (1951), 646, 84, 851L. D. L. Clark, A. Roberts, and R. Wilson, Phys. Rev., 83 (1951), 649L.
The Formal Theory of Scattering
492 |^(oo )} at
an
[Chap* XVIII]
infinite future.
may
According to this description the states |^(=boo)} consist either of elementary systems not interacting with one another
that
is,
sufficiently separated
from one another
or of their combinations
bound states. Therefore the problem of distinguishing between the initial and final states of noninteracting particles and their bound states is important. This rather complicated problem has been solved by Klein.* Its discussion
in
not included here, but the interested reader may find it very useful to read Klein's paper. further note that in Section XVII. 1 only the first Born approximation was used; contributions from higher orders were neglected. The first Born
is
We
approximation
only for higher energies and for weak potentials. are usually too complicated for practical purposes
is reliable
The higher approximations and
on scattering cross-section. Higher terms in the Born expansion have been investigated by Wu and also by Kallenf for various central potentials. In particular, in view of breakdown of a potential for rapid estimates
description for high-energy phenomena, the Born expansion is not expected to be of any use there; for a low-energy region, higher terms in the Born expansion deserve closer attention. detailed and rigorous treatment of this
A
and
also
an investigation
problem
of the general behavior of the solutions for the
scattering integral equation as a function of potential strength parameters
have been carried out by Jost and Pais.J Here we
shall follow an operator technique to reproduce some of the results of Jost and Pais. We shall begin with the integral equation (XVII. 1.7} of the scattering theory. Its formal solution can be written down in the form
i*}
where
the
=
(1
-
(?off'H*
(XVIIL5.1)
>,
the scattering system of particles. For the scattering process to take place, the operator must have a finite norm (see Section I.5.D). The condition for this can be obtained by extending the definition in Section 1.5. D, for finite-dimensional spaces, to continuously infinite-dimensional spaces. Thus, from [*b> represents
initial state of
[norm (Goff')?
we
find that
norm
(GU5P)
=
V(r)8(r
/
(r\(G*H'yGtH'\r) d'r,
must be smaller than or equal
L* where
=
-
r \V(r)\ dr
r'),
=
constant,
and 7(r) represents the potential energy.
*
A. Klein, Prog. Theor. Phys. 14 (1955), 580. T. Y. Wu, Phys. Rev., 73 (1948), 934. G. Kallen, Ark. Fysik, 2 (1950), 33. }
t
t
to
R. Jost and A. Pais, Phys. Rev., 82 (1951), 840.
Expansions of Scattering Amplitude
[XVIII.5.] If
we assume
the validity of an expansion in powers of the operator
=
K from
493
(XVIII. 5.1)
we obtain the
Goff'
(XVIII.5.2)
result
(XVIII.5.3) 71=1
where
=
Kn
the vector
where
Kn
From
(G^H')
|r)
(r,
n
and #1
rO
=
K. The scalar product of
(XVIII. 5. 8)
with
K n we have Kn = KiKn-l = KKn
the definition of the operator
i,
obey the equation
so its representatives
Kn
(r, r')
where
By
=
yields the equation
=
^(r, substituting from
7
J
K(r, r'O^n^iCr'
r')
=
(? (r
(XV III. 5.5)
-
,
r')
(X7IIL6.S)
d*r",
r07(rO-
(XVIII. B. 6)
(XVIII. 5. 4), we can separate out the
in
scattering amplitude /(0) as the coefl&cient of 1/rexp
Thus we obtain
(ifcor).
(JC7/II.5.7)
where the vector
is
fe
in the direction of the position vector r
and where we
=
|fc|. took^o(r) = e^' and |fe We can also work in terms of the
r
|
momentum
from (XVIII, 5. 3) the momentum wave function
eigenstates p) in the
y where 0(p)
=
{jpl^}
Kn (p,p
and ^,(p)
f
)
=
=
=
(pi^nlp'}
(pl^o)
=
f^T | K
j
and obtain
form
,
(XVIII.5.8)
and
(p\r)(r\Kn\r')(r'\p>)d*rd*r'
n (r,
r>)eW-**+''*')d*rd*r'.
(XVIIL5.9) If
we
choose
wave
that
is,
a delta function
then
(XVIII.5.8) becomes
(XVIIL5.W)
The Formal Theory
494
The
kernel of the integral equation in
=
K(p, p')
=
(p\K\p'}
/
of Scattering
momentum
[Chap. XVIII] is
space
(XVIII. 5.11)
It can further
be simplified either by direct integration in coordinate space or by using the definition (XV1 1. 14) of the Green's function and writing
1
=
E-H
W
2
p 2m
kl
-
fc
2
+ ie
Hence (XVIII4.11) becomes (xviii. 6.1 where &J
= 2mE/H 2 and
serves to
make
Je
=
2
2
jp
A
2 ;
the positive imaginary part i, as before, p well defined, so that the kernel
K
the integration over
corresponds to outgoing
waves
A further useful relation is
only.
given
f Kn(p, p)
by
d*p
=
I
Kn
(XVIII. 5.13)
(r, r) d*r,
which follows from
(p\Kn \p) d*p
=
(pjr^Cr,
r')
d
We also have ,
The
7
)
scattering amplitude can easily be obtained
(XVIII. 5. 14), as in Section
where J?
=
XVII. l.B,
in the
infinite series in
from (XVIII. 5. 10) and
f
n
x
5(j>
(XVIII. 5.14)
d*q.
form
-i I? / (po\H \p')K ^(p' ** E = n
an
P
jp).
i
">
We
J
}
p) d'p'
(XVIII. 5.15)
,
have thus expressed the scattering amplitude as
powers of the potential strength.
The expansions (XVIII. 5. 7) and (XVIII. 5.15)
for/(0) can readily be ob-
tained from the scattering amplitude operator, defined
where the position operator
q,
by
or
a is
defined
by
(V.I.I) with eigenvalue r
and eigenstate
|r).
The
state vectors
Expansions of Scattering Amplitude
[XVIII.5.]
and
'#>
operator
By
T
^V
(XV
are those appearing in the
I II. o. 16)
using the operator r, as
the differential cross-section can be defined,
/,
=
*(#)
XVIII. o.A Expansion of
exp
where
r
>
r
=
2
*
in Spherical
&
]
+
(XVIII.5.18)
.
(ikr) in spherical
l/R exp (ikr)
KH/>)!
Harmonics harmonics
Gi(r,
/)Pz(cos
the angle between the position vectors r and
6 is
=
Gi(r,r')
For
wave equation (XVII. 1.7). The
can also be written as
for large
The expansion
r
495
is
6),
r'
given* by
(XVIII. B. IS)
and, for
r'
>
r,
+ i/2(fcr)[(^l)V-z- 1/2 (fcrO + iJl+l/2 (kr>)]. -7=Jz V(rr')
the function
(? is
defined
by
interchanging r and /. Introducing
the expansions
in the
wave equation (XVII. 1.1$), we obtain
/
+
where the angle
2)
co
Gz(r
'
r/ )
between cos
P,(cos
a?
co)
=
r
p z( cos wJH'fr')^* d^CO^-Ccos 00 and
cos
r' is
cos
6'
2
=
sin
0' <$'
d*
7 ,
given by
+ sin 6 sin 7,^(6',
0'
cos (#
tf/)j
^'J^Ctf, *),
Hence, using the orthogonality property (IV.S.21) of spherical harmonics also (IV. 2. 27), -we obtain the integral equation
and
= * -
/z(r)
-
G. N. Watson, r/ieory o/
366.
" ^'fr.
J5e55eZ Functions,
OnrO^KrOr" dr*, Cambridge Univ.
(XVIII. 6*0)
Press, Cambridge, 1944,
The Formal Theory of Scattering
496
where
fi(r)
[see (IV.3.22)] is given
f
l
(r)
=
l
i (2l
[Chap. XVIII]
by
+
Equation (XVIIL5.20) for the radial wave functions was derived by Jost and Pais. It is an equation for partial waves and is therefore suitable only for scattering processes where only a few partial waves are needed for a good estimate of the cross-section. This type of equation is not suitable for central potentials with long tails (like a Coulomb scattering), where a complete set
waves is required for a correct calculation of the cross-section. For nuclear interactions (nucleon-nucleon, nucleon-nucleus scattering) where usually depending on the energy of the incident beam and the final state interaction the first few partial waves are sufficient to describe the scattering,
of partial
equation (XVIII. 5.20) may be found quite useful. Consider an S state (Z = 0) scattering and the corresponding integral equation
where, since r
>
r' 9
i/ 1/2 (fcr)]
and/o
=
sin kr/kr. Putting
=
we
get
(XVIII. 5.21) where
= 6^ -
e i*|r-r'|-tfcr
and the equation contains a boundary condition where The asymptotic form of <j>(r), for large r, is
(XVIII. = for r =
0.
(XVIII.B.gg)
where
sn is
the eigenvalue of the
T
(XVI II. 6. 94)
matrix corresponding to zero angular
momentum
and wave number fc. Calculation of the scattering cross-section requires either the use of Born expansion or one of the methods discussed in later sections.
[XVIIL5.]
Expansions of Scattering Amplitude
497
XVIII.5.B. Dispersion of S Waves
We shall consider a collision process where dispersion can play an
important This type of scattering resembles optical dispersion by a medium containing damped oscillators with various natural frequencies. Xucleon-nucleus role.
and meson-nucleus
collisions,
as
many-body phenomena,
A
contributions of dispersive scattering.
are affected
by the
simple case refers to the dispersive
S waves. We will discuss this in some detail. Let us assume that the potential can be prescribed
scattering of
up
for parameter a such that V(r) = (XVII1. 5 .21) for S waves becomes
HT- + If r
=
a,
then
g(r, r
f
)
***
becomes
H
r
a.
fffr,
{o
g(a, r')
sin
>
=
ka
to
an arbitrary
Therefore the integral equation
r')7(r')
2i sin
kr',
(XVIII. 5. 25)
so
.
(XVIII. 5. 26)
By
comparing
this
with
scattering amplitude
is
(XVIII. 5. 2$) and
(XVIII.ff.g4),
we
see that the
given by
-*.
(XVIII. 5.27)
We
note that the wave function outside the potential (that is, for r > a) is given by the right side of (XVI II. 5. gS) and equals the wave function given by (XVIII. 5.25) at the point r = a.
Let us assume the existence of the eigenfunctions u(r) defined over the ^ r <; a and also the corresponding eigenvalues k s as solutions of the homogeneous integral equation interval
.(r,
where &
will, in general,
^7(0^(0
rf/,
be complex and a function of
fc,
(XVIII. B. 98)
and where g8 (r,
r
r )
r
obtained from g(r r ) by replacing k by k s The theory of dispersive scattering of S waves was first worked out by Kapur and Peierls.* Here we shall follow the method of the integral equation which contains all the boundary conditions of the problem. To obtain the is
.
}
differential equations corresponding to the integral equations (XVIII.5.25)
and (XVIII.ff.g8), we introduce the function cient by *
S.
Kapur and R.
Peierls, Proc.
e(r, r')
and
its differential coeffi-
Roy. Soc. London, A, 166 (1938), 277,
The Formal Theory
498
=
1
for r
I
f or r
-
2*(r
dr
of Scattering
> <
[Chap. XVIII]
r',
r',
rO,
and obtain fl(r / \
j
T '
f
} /
=
fffir'
r')e(r--r')
f>ik[(r
*
-
S(r
Hence the corresponding
+ From
r')
-
2ifc
differential equations are
-
As seen from the
r]
[
|r
^)] *
=
|r
^W ]
=
integral equations, both
the integral equations
we
.
o,
(xvm.5.29)
o.
(xvm. B.so)
$ and us vanish at the point
also obtain the
boundary conditions
r
=
0.
:
.., Jo.
= e->
We
shall
+ ik^a).
(XVIII. 5. 32)
assume that the eigenfunctions us (r) constitute a complete ^ r ^ a and expand <j>(r) in the usual way,
set
in the interval
where the boundary condition 0(0) =0 is satisfied. By multiplying equations (XVIII. 5. 29} and (XVIII.5.30) by u 8 and 0, respectively, and subtracting the resulting expressions,
we obtain
d f
du s
d
^L"**"*"* Integrating both sides in
g
r
g
a,
we
get
JM
=
(*J
dr
=
- W)C N a
a,
(XVI 1 1. 5. 34)
where 1
dr,
and
we
/4
^
obtain
fcj.
f |.
*
1,
f" tt,i dr
=
0,
Using the boundary conditions (XVIII. 5.31} and (XVIII.B.SS),
Fredholm's Method to Scattering Theory
[XVHL6.]
C =
499
(XVIII.535)
A^'ff-T*)'
Hence
Thus the
total cross-section,
as follows
from (XV111. 5. 36) and "
=
N
jLj
ls>
(XVII1. 5. 27), "~
(w
^~F
1
-p
\
is
given by
2 sin ka e~ ika
\
,
where we take
=
f
(s.
fc
I ff.)
=
f
wi -
js,
v.a. (Z7I7L5.SS)
The use
of the
boundary condition (XVIII.5.31)
T s = |WS
The
cross-section
is
the
sum
of
yields
2
(XVIIL5.S9)
. |
two terms. The sum
in the first
term
is
the
characteristic of dispersion theory for a set of oscillators with energy levels
E
and
of natural width
F s The second term
in (XVIII.5.87) represents extension of the potential. If the "potential scattering natural width of the oscillators vanishes, then the scattering cross-section 3
77
is
.
due to the
just
ff= so a.
finite
4x
sin 2
ka '
w
represents the amplitude scattered from an impenetrable sphere of radius For an incident wave with wavelength much larger than the radius a,
it
the potential scattering cross-section reduces to
which
is
just the geometrical cross-section.
XVIII.6. Application of Fredholm's to Scattering Theory
Method
Fredholm's method* for the solution of integral equations can be applied to an investigation on the validity of a
problems and *
W.
it
Born approximation
in scattering
can also serve for a deeper understanding of scattering proc-
V. Lovitt, Linear Integral Equations, Dover,
New
York, 1950.
The Formal Theory
500 esses.
In this section we
shall develop
of Scattering
[Chap. XVIII]
Fredholm's method in an operator
formalism, suitable for application to nonrelativistic as well as to relativistic
problems. Consider a one-dimensional integral equation,
= /(a?) +
u(x)
f K(x, x')u(x
X
Ja
l
)
dx'.
(XVIII. 6.1)
In Fredholm's method the solutions of this integral equation are expressed as a ratio of two infinite series and, for a well-behaving kernel K(x, x'), these solutions are valid for
values of
all
integral over the interval I ^
by
(a, V)
_.
X. The method consists of replacing the the limit of a finite sum. Thus, putting
*>- a
>
Xn
^
=
_L n a _j_ ^ I.
Ti
for
n =
-
-
1, 2,
we divide the interval (a, 6) into n now be replaced by the approximate ,
(XVIII.6.1) can
u(x)
-
x
x{ X2,
'
}
,
x^
and obtain n
is
valid also
(XVIII.6.2) if
we choose
linear equations,
-
\u)
expression
= /(),
K(x, x$u(x'd
valid for every value of x. Equation (XVIII.6.2)
=
equal parts. Equation
\hK\u)
=
(XVIII.6.3)
|/},
where fte) !/)
=
and 1
K(xi,
0:1),
K(Xi, X n )
K(XI,
K
The
(XVIIL64)
existence of solutions for (XVIII.6.8) require that the determinant
A = be
finite.
The
linear
det (I
\hK)
(XVIII. 6.5)
system of equations are solved by
u * *
where A# denotes the of A.
-
first
minor
1
of the
A ^"'
(XVIII. 6. 6)
element in the rth row and jih column
Frediiolm's
[XVHI.6.]
Method
to Scattering Theory
501
The
solution of the Integral equation is to be obtained by letting n increase indefinitely. Therefore we need to know the limits of A and Ay as n tends to infinity.
From (XVIII..6.5} we have
+ Hence, for n in the
,
n n n (-l} h \
we may,
(XVIIL6.7)
formally, write Fredholm's determinant D(X)
form
X
(XVIII.6.8) _
By
K(x n
ai) ,
X(x n
,
ar 2 )
1
- \K =
,
,
using theorem 6 of Section I.5.A
Thus, from it
,
we can
exp [log
(1
K(x nj x n ] write D(X) in a closed form.
- \K)l
follows that
D(X)
K
where now
is
=
an operator
exp
{tr [log (1
- \)]},
(XVIII. 6. 9)
in infinite-dimensional space
and
its
trace
is
defined as
The
trace operation
=
tr
K=
is
defined over the limits of the integration in the integral
I
(1\K\1) dl
f
K(l,
1) dl.
A
point with coordinates x i} x%, x% in three-dimensional (or fourequation. dimensional) space is represented by a numeral. In this way we are extending
Fredholm's one-dimensional determinant to the case of higher-dimensional spaces. The vectors \1), \2), and so on, are to be regarded as the ket vectors of the infinite-dimensional space of the type introduced in the previous chapters.
Thus tr
X
2
=
f
(1\K\S)(S\K\1} dl d2
=
ff K(l, $)K(%,
where, for example, in three-dimensional space
K(l, *)
= K(r
l}
r2 ),
dl d
1} dl dS,
we have
=
d'n
d*r*.
The Formal Theory
502
We
of Scattering
expand the operator log
can, formally,
-
log (1
= -
XX)
[Chap. XVIII]
\K) as
(1
]
n
and obtain
-
tr [log (1
Hence
it
XX)]
is
= -A
K(l,
X
-
dl dB
easy to
1) dl
X3
X(J,
JX
*)X(,
the
of
expansion
1)
1) dl
X(J, 0)X(0, S)X(S,
J
that
verify
-
+
dS d8
(XVIII.6.9)
.
yields
(XV11 1. 6. 8). In a similar way one can show that the limit of A^ as n lim
where
D
C1)
is
=
A,,-
D(l,
2, X)
>- co
is
given by
=
an operator for Fredholm's
minor D(l,
first
8] X),
which
is
given by
',#)
'd3'
K(S', S
3',%'),
+
f
)
(XVIII.6.10)
From
(XVIII. 6. 8) X
and (XVIII.6.10) we may
= ^~f^ d/X
= -
f D(l, 1} X) dl
= ^^ d\
-
tr [(1
we
for the first minor, so the ratio of
~7rr
=
X(l
D<.
-
D
(1)
XK)-
1
/D(X)
X=
is
(XVIII. 6.11)
get another relation,
(XVI 11. 6. 12)
XX^XJjDCX).
(XVIII.6.11) and (XVIII. 6. 12)
nd)
tr
J
Furthermore, by differentiating (XVIII.6.9)
On comparing
easily obtain the relation
we
get the operator equations
given by
XK(1
-
XK)"
1 .
(XVIIL6.14)
JJ(\)
We
can now use (XVIII.6.18) and obtain Fredholm's first and second fundamental relations. From (XVIII.6.18) we have the operator equations
Hence, taking the representatives, ,
g; X)
==
\K(1, *)D(X)
+
we X
get X:(S,
*)Dy,
5; X)
d3
Fredholm's Method to Scattering Theory
[XVIII.6.]
503
and jD(Jf,
=
0; X)
\K(1,
X
K(l, 3}D(3, 2;
J"
(XVIII. 6.16)
X) dS,
and second fundamental relations, respectively. consider the operator form of a linear inhomogeneous integral equation
which are Fredholm's
Now
+
)D(X)
first
=
\u)
|
+ \K\u),
Wo >
(XVIII.6.17)
which can be written as
=
\u)
\K on
Operating by
both
we
sides,
-
(1
get
=
\K\u)
and from (XVIII.6.17), we write
Thus, from (XVIIL6.14), is,
follows that the integral equation (XVIII.6.17)
it
at least formally, solved
by
=
\u}
^ nu)
+
|uo>
K>.
(XVIII.6.19-)
Taking the representatives we obtain the expression u(l)
=
~
+
u(l)
f
(XVIIL6M)
D(l, 2; X)iio$)
as the solution of the actual integral equation, where u(l)
=
(J|
M),
Finally, let us consider the
it
=
of (XVIII.6.%1)
of X
=
XQ,
in is
(I
|
A
=
X
X(i,
|
(XV111. 6. 20) we
=
given by u
0.
= <J|D|>.
D(i, *; X)
homogeneous
u(l)
On setting
=
/CO
integral equation
)u() d.
(XVIIL6JB1)
if
D(X)
=
for a special value
then the equation u(l)
=
X
f
K(l,
)u(8) d2
has a nonzero solution. This can be seen by putting X in (XVIII.6.14)
7
g.f Xo )
=
X
f
K(l, 3)D(3, 8; X
we assume that equation (XVIII..6.
and therefore
for
a fixed value 2 D(l,
which
is
=
X and D(X
)
=
and obtaining D(1
If
the only solution
see that for D(X) 5^
However,
just equation
#o;
X
)
=
= X
2
f
,
)
)
dS.
(XVIIL6.aH)
holds for every value of the point 2
then we get
K(l, 8)D(8,
ft,
X
)
(B,
(XVIII. 6. 21) with w(l) replaced by
(XVIIL6.8S) D(Jf,
J^
;
Xo).
Thus
The Formal Theory
504
=
u(l)
D(l, #
of Scattering ;
X
[Chap. XVIII] (XVIII.6.24)
)
a solution of the homogeneous integral equation corresponding to X = X for a fixed point 2 = 2$. Here we assume that D(l, 2] Xo) does not vanish is
for the choice
2
=
%Q
.
Further discussion, especially mathematical justifications of the above summary of Fredholm's method, will not be included here. The interested reader
can find satisfactory answers and other
details of integral equations in spe-
cialized texts.
XVIII .6. A. Expansion
of the Scattering
Amplitude as a Fredholm
Series
The singularity in the kernel of the integral equation (X VI11.1.7) does not allow application of Fredholm's method. However, if we iterate it once, then a Fredholm solution for the iterated form of the integral equation does exist.* Thus the
integral equation to be solved in terms of a
I*)
=
(i
+ GoHOhPo} +
x 2 JK|^>
=
|*o>
K
=
+
Fredholm
x 2#|#),
series is
(xvin.
where
= and X
is
(1
+ 0oH')I*b),
GoH'GoH'
(XV711
.
6'.SB)
the basic parameter of the scattering process (for example potential From the definition (XVII. 1.12) of the Green's function, it follows
strength).
that the representative of the kernel operator
K(r,
r')
=
is
given
by
which has no singularity at is
K
r
=
r7
.
The asymptotic form
of K(r, r') for large r
given by
^or ^ - j^i W) ^/
r
9!>r?
K(r, rO
-*^G
(r'
-
r-)F(r
where the propagation vector fe after scattering is fc = fc ^. We may now write the Fredholm solution of (XVIII.6.86) in the form
(XVIIL8.89) where Fredholm's determinant jD(X 2 fco) and the operator of the first minor D C1) are to be formed with respect to the kernel defined by (XVIII. 6. 26). ,
K
*
N. N. Khuri, PKyt.
Rev.,
107 (1957), 1148.
Fredholm's Method to Scattering Theory
[XYIII.6.]
505
Hence, by taking the representatives, we obtain the wave function #(r):
= ^( r)
^(r)
+
D( r
,
r';
X2
fo>)*o(r')
dV,
(XVIII. 6. SO)
F(rO
dV,
(XVIIL8JS1)
,
where *(r) *>(r)
^o(r)
= = =
*
>
=
^o(r)
+ f G (r Q
r')
ik*' r
e
-
and Pais have shown that the series solution (XV 1 1 1. 6. SO) for the wave some restrictions on the potential, converges uniformly and absolutely. Moreover, ^(r) has no singularities for real X and real k (except
Jost
function, with
=
0). possibly at k The scattered wave
*.(r)
=
=
*(r)
f
-
(? (r
*
-
is
(r)
r')F(r')^o(r')
dV +
^^
f
fa)
2 D(r, /, X *o)*o(rO ,
dV.
(XVIII.6.8g)
Hence the
scattering amplitude as a function of &
and momentum
=
fco|, [see (XVI I.I. 84), where the scattering amplitude |fe q as a function of \k fc'|], is obtained as
/(fco, a)
To
find the asymptotic limit
relation (XVIII.6.16)
=
we
lim [re-^.(r)]. shall use
is
transfer
expressed
(XVIII.6.SS)
Fredholm's second fundamental
and the asymptotic form (XVIII.6.28)
of K(r,
r').
Thus r, r';
X2
,
fc
)
X/ Substituting in
e- ife -"G (r"
-
(XVIII. 6.3$) and
X
+&>)
ri )V( ri)V(r")D(r", r';
(XVIII.6.S3),
dV
we
>';
X1
,
,
fc
)
get
^J^ /
X F(n)F(rW",
X2
fco)*o(r')
.
"G,(^
-
n)
dV dn d'r",
where /"
\q\
=
ei (-w-'-F(r) d'r,
jfco
- M| =
/b
V [2(1 - cos 0)1
(XVIII.6.SS)
The Formal Theory of Scattering
506
[Chap. XVIII]
This completes the formal expansion of /(fco, #). It has been proved by Khuri that for a large class of potentials and for a complex fc the scattering amplitude is K.
an analytic function of fc0? regular in the complex k Q plane where Imfc = > and also uniformly bounded in the region K ^ 0. On the real axis,
/(fco, q)
has branch points at k Q
For bound
i?; that
is,
6
=
71-.
states the discussion of the
to the result that
The
=
all
zeros of
D(X
2
fc ,
homogeneous integral equation leads ) lie on the positive imaginary axis.
corresponding eigenfunctions are the bound-state
wave
functions.
XVIIL6.B. Remarks on Dispersion Relations For a
nonrelativistic study of dispersion relations
on dispersion
ences to the extensive literature
and
also for
relations
we
more
refer-
refer
the
reader to the paper by Blankenbecler, Goldberger, Khuri, and Treiman.* The method of dispersion relations avoids any form of series expansion, but is based on the analytic properties of the representatives of the S matrix as a is found that a relation between real and imaginary parts of the $-matrix elements (dispersion relation) contains the information required for the description of a physical process. We must therefore study the analytic properties of the S matrix elements as a function of energy. The basic mathe-
whole. It
matical tool for dispersion relations is the Cauchy integral formula for an analytic function /(z). Thus from a typical energy denominator,
we may,
for
an analytic function g(E), write 1
Pn
dE f flW W-E' '
(YVTTTfi>fi\ (XVIII.
6.S6)
/
Hence, taking the real part, we obtain a relation between the real and imaginary parts of g(E)
:
Re The
g(E)
function g(E), denned
=
1
P
j
^Tp
dE'.
(XVIII. 6.37}
by
f(k,q)+~t
E
shown by Khuri, is analytic everywhere in the plane except for a branch cut on the real positive axis and poles on the negative real axis. Thus if Ri(q) as
* R. Blankenbecler, M. L. Goldberger, N. N. Khuri, and 10 (1960), 62.
S.
B. Treiman, Ann. Phys.,
Fredliolm's
[XYIII.6.]
Method
bound
are the residues of g(E) at the
formula,
JL ?rt
and obtain,
Because
for
Go-H"'
F f Jr
Ei
z'
<
^ -
dz
r
to Scattering
states Ei, then
we apply Cauehy's
= SF &
if
z lies insid e r,
\o
if
z lies outside r,
2
507
Theory
0,
and GJH'GvH' are
the residues Rt(q) above are
real for
on the positive imaginary
fc
axis,
Noting the negative imaginary part in energy E, formula (XVIII. 6.88) yields the dispersion relation real.
(XVIII.6.89)
where
N
is
the number of bound states for potentials satisfying the condition* I
r\V(r)\ dr
=
constant.
Investigations on the forward scattering amplitude / have shownf that its imaginary part is proportional to the total cross-section. This is called the "optical theorem." For a meson-mi cleon system it is given by
Im/ =
^
o-
(XVIII. 6 40)
and -- |f|2
where *
p
is
the
momentum
of the
meson.
V. Bargmann, Proc. Nat. Acad. Sci. U.S.A., 38 (1952), 961. R. Karplus and M. A. Ruderman, Phys. Rev., 98 (1955), 771. M. Goldberger, H. Miyazawa, and R. Oehme, Phys. Rev., 100 (1955), 986. A. Salam, Nuovo Cimento, 3 (1956), 424. A. Salam and W. Gilbert, NUOJO Cimento, 3 (1956), 607. C. Polkinghorn, Nuovo Cimento, 4 (1956), 216. t
AUTHOR INDEX
Eden, R. J., 437 Edmonds, A. R., 106
Aharonov, Y., 345 Akhiezer, A. I., 266 Alliluev, S. P., 363 Ambler, E., 279 Archibald,
W.
Bargmann, Bass,
J.,
J.,
Einstein, A., 2, 4, 59, 135, 339, 340, 345, 414,
415, 419, 446
Eisenbud, L., 191 Emde, F., 267
63
V., 363, 480,
507
439
Berestetsky, V. B., 266, 380 Bethe, H. A., 273, 362, 373, 392, 424, 425, 483 Blankenbecler, R., 506 Blatt, J., 263, 267 Bohm, D., 345 Bohr, N., 59, 91, 141, 147, 339, 345 Boltzmann, L., 135 Bose, S. W., 172 Breit, G., 302, 408, 483 Brown, S., 381 Bruckner, K. A., 437 Burgoyne, N., 326
W.
F., 491 443 Cheston, W. B., 490 Clark, D. L., 491 Coester, F., 350, 489 Condon, E. IL, 110, 120 Corben, H. C-, 380 Cowling, T. G., 443
Cartwright,
Chapman,
S.,
Darwin, C. G., 359, 368 DeBenedetti, S., 380 de Broglie, L., 58, 345 deGroot, S. R., 176 DeHoffmann, F., 273 Deutsch, M., 381 Dirac, P. A. M., 85, 90, 91, 156, 194, 350, 359, 375, 377, 380, 480
Durbin, R., 491 Dyson, F. J., 31, 375
Fano, U-, 175 Fazzini, T., 187 Fermi, E., 410 Ferrell, R. A., 380 Feynman, R. P., 342, 347, 348, 350, 351, 353, 356, 375 Fidecaro, G., 187
M., 327 Fock, V., 194, 363, 435 Foldy, L. L., 302 Francis, N. C., 437 Fulton, T., 381 Fierz,
Garwin, R. L., 279 Gell-Mann, M., 185, 382, 468, 469 Gilbert, W., 507 Goldberger, M. L., 468, 469, 506, 507 Goldstein, H., 1 Good, R. H., 63
Goto,
K,
350
Hartree, D. R., 435
Hayward, R. W., 279 Heisenberg, W., 94 Heitler, Hill,
W., 408, 417
E. L., 363
Hoppes, D. D., 279 Hudson, R. P., 279 Irving,
J.,
439
Jahnke, E., 267 Jauch, J. M., 350, 363, 375 Johnson, M. H., 368, 371
509
Author Index
510 Paul, H., 187
Jordan, P., 307 Jost, R., 481,
492
W., 121, 172, 175, 179, 219, 255, 277, 322, 326, 330, 363
Pauli,
Kallen, G., 492
Kapur,
S.,
497
Peierles, R.,
Peres, A., 345
497
380
Karplus, R., 381, 443, 507 Kemble, E. C., 425
Pirenne,
Kennard, E. EL, 368 Khun, N. N., 504, 506
Podolski, B., 345
443 Klein, A., 381, 492 Klein, O-, 307 Kiimentovich, Y. L., 439 Kramers, H. A., 277, 480
Powell,
Kirkwood,
J. G.,
J.,
Planck, Max, 58, 134, 414 J. G., 356,
Polkinghorne,
507
380
J.,
Present, R. D., 425 Prigogine,
I.,
443
Kursunoglu, B., 208, 300, 302
Rainich, G. Y., 363 Retherford, R. C., 374, 376
Lamb, W.
Roberts, A., 491 Rohrlich, F., 375
Landau,
E., 374, 376
L., 276, 368,
380
Rose,
M.
E.,
110
Laperte, 0., 363 Lederman, L. M., 279
Rosenfeld, A. H., 185, 382, 410
Lee, T. D., 276, 279
Ruderman, M.
Levinson, N., 480 Lipmann, B. A., 368, 371, 469, 473, 479 Loar, H., 491
Saenz, A. W., 363 Salam, A., 276, 301, 356, 507
Lomont, Lovitt,
297
J. S., 42,
W.
V., 499
Ross,
J.,
443 A.,
507
Salpeter, E. E., 362, 373, Schilpp, P. A., 339
392
Schluter, R. A., 410
Luders, G., 277, 326
Schrddinger, E., 177
Mclntosh, H.
V.,
363
J., 156, 157, 277, 301, 307, 309, 375, 443, 468, 473, 479
Schwinger,
Shortley, G. H., 110
S. T., 480 Magnus, W., 366 Marshak, R. E., 490
Ma,
Silin,
V. P., 439
Singer, P., 345
Martin, P., 381 Massey, S. W., 147, 456, 466 Mathews, P. T., 356 Mayal, J., 439 Moller, C., 480 Moses, H. E., 297 Mott, N. F., 147, 456, 465, 466 Murnaghan, F. D., 233
437 491 Symanzik, K., 350 Szego, G., 108, 249 Slater, J. C.,
Steinberger,
J.,
ter Haar, D., 480 Thining, W., 350 Tobacman, W., 350 Tolhoek, H. A., 176, 269
Y., 350 Nordheim, L. W., 192
Tollestruf, A. V., 187
Oberhettinger, F., 366
Uhlenbeck, G. E., 368
Oppenheimer, R. Ore, A., 380 Orear, J., 410
Van Hove,
Nambu,
J.,
Treiman,
S. B.,
506
63
350 345
L.,
Vigier, J. P.,
Von Neumann,
J., 17,
Page, L., 368 Pais, A., 382,
Pasternack,
492 425
S.,
Ward, J. C, 301 Watanabe, S., 489
18
Author Index
511
Watson, G. N., 495 Weinrich, M., 279
Wilson, R., 491
Weinstein, R., 381 Weisskopf, V. F., 263, 267
Wu, C. Wu, T.
Welton, T. A., 427 Wentzel, G. 310 Weyl, EL, 30 }
Whitehead, M. N., 491 Wick, G., 350 Wigner, E. P., 23, 55, 56, 191, 277, 307, 384, 408, 439 Wilcox, H. A., 491
Wouthuysen,
S. A.,
302
279 Y., 492 S.,
Yang, C. N., 276, 279 Yost, F. L., 192 Young, X., 368
B., 326 Zwanzig, R., 439
Zumino,
SUBJECT INDEX
Abelian group, 6, 34 Absorption: 57; of radiation, 415 Accidental "degeneracy, 363, 368 Action at a distance, 93 Action function, 1, 27, 78, 159 Action operator, 160
Black-body radiation, 59 Bohr magneton, 150 Bohr's atomic theory, 59
Boltzmann distribution, 177, 178, Boltzmann equation, 442, 444 Born approximation, 452, 499 Born expansion, 492
Act
of observation, 94, 95, 96 Adjoint, 217
Angular momentum:
3, 11, 74, 92, 97, 98,
110, 146, 148, 166, 184; addition theory for, 111, 116;
248;
four-dimensional Euclidean,
orbital,
75,
97,
105,
109,
113;
Schrodinger representation of, 71 Annihilation operators, 315, 331 Anomalous Zeeman effect, 374, 377 Antibaryons, 185
Anticommutation 236, 237, 318
relations,
40,
199,
Bose-Einstein
statistics, 172, 179, 192, 270, 315, 329, 439, 461, 463
Boson, 197
235,
Antilinear operators, 21, 22, 222 Antineutrino: 186, 321, 324; right-handed, 277, 278
Antineutron, 184 Antinucleons, 185 Antiorthogonal, 211 Antiparticle, 188, 320 Antiparticle states, 289
Antiproton, 184
Antisymmetric, 11 Antisymmetric operator, 16, 24, 26 Antisymmetric state: 115, 171, 271, 439; singlet, 190 Antisymmetrization operator, 200 Antiunitary operator, 22, 23, 84, 168, 296 Asymptotic limit, 453, 457, 505 Average square deviation, 143 Average value, 137 Axial vector, 11, 14, 27, 218, 273
Baryon charge, 185, 186 Baryon number, 185 279
363, 364, 380, 392,
480, 481, 506, 507
Breakdown
Antihyperons, 185 Antileptons, 186
/3-decay, 186, 189,
Bose-Einstein condensation, 179 Bose-Einstein distribution, 176, 178, 325 Bose-Einstein oscillators, 199 Bose-Einstein quantization, 329
Bound plasma, 437 Bound state, 359, 361,
Anti-Hermitian operator, 16, 92, 132, 144, 289
Bare mass, 426
415, 416
of parity conservation, 299
Breit-Wigner formula, 411 Bremsstrahlung, 481 Brownian motion, 135 Canonically conjugate variables, 74, 94, 309, 310
Canonical transformations, Cause, 94
3, 4,
73
Charge conjugation, 188, 277, 294, 324 Charge exchange operator, 189 Charge reversal operator, 296 Circular polarization, 373 Classical description, 141 Classical path, 82
Clebsch-Gordon 264, 282
coefficients, 111, 112, 115,
Collision matrix, 170
Commutation relations,
40, 71, 98, 106, 118,
128, 133, 159, 189, 193, 196, 237, 251, 254,
310, 312
Complete commuting
set of observables, 91,
387 Complete orthonormal bases, 8 Complete orthonormal set, 18, 255, 293 99, 125, 126, 138,
Complete set, 18, 201, 264, 498
84, 102, 107, 110, 131, 195,
Complex orthogonal matrix, 212, 243 Complex orthogonal transformations, 210 513
Subject Index
514 Complex
sphere, 230
Euclidean group, 48, 256
Compound nucleus, 147, 405, 411 Compound state, 406, 409 Compton effect, 340
Even parity levels, 384 Even permutation, 13, 210, 437 Even states, 272
Connected group, 25 Continuous group, 36 Contravariant, 46
Exchange forces, 189 Exchange operator, 271 Exchange process, 171 Exchange transformations, 44
Coordinate transformations, 3 Correspondence principle, 91, 141, 432 Coulomb interaction, 182 Coulomb repulsion, 149 Covariant, 46
Excited state, 411 Exclusion principle, 172, 179, 281, 318, 328, 331
Expectation value, 69, 72, 144, 330, 331, 465
Cross-section: 447; differential, 457, 465,
469, 483, 495; effective, 463 Creation operators, 315, 331
Degeneracy, 171, 363, 387 D-function, 317 S-function, 85, 87, 88, 128, 130, 454
87 Density matrix, 173, 176, 269, 440, 443 Density operator, 174 Determinism, 94 Dipole radiation, 381, 418, 422 Dipole transitions, 423 Dirac equation, 287, 292 Dirac matrices, 235, 246, 287 Dirac neutrino, 325 Dirac's hole theory, 319, 328 Dispersal time, 408 Dispersion law, 67 Dispersion relation, 506 Dispersive scattering, 497 S +-function,
Fermi-Dirac distribution, 176, 325, 331 Fermi-Dirac oscillators, 199 Fermi-Dirac quantization, 329 Fermi-Dirac statistics, 179, 199, 318, 328 Fermions, 200 Feynman's first postulate, 349 Feynman's second postulate, 349
Born approximation, 451, 454 Fock representation, 193, 194 First
Forced degeneracy, 363 Four-component spinor, 287 Fredholm series, 504 Fredholm solution, 504 Fredholm's determinant, 501, 504 Fredholm's first fundamental relation, 502 Fredholm's first minor, 502 Fredholm's method, 499, 500, 504 Fredholm's second fundamental relation, 502, 505 Frequency distribution, 446
Distribution function: 135, 173, 175, 437; free harmonic oscillator, 445 Double commutator, 118 Double slit experiment, 342
Dual, 13
Dual
space, 5
Galilean transformation,
the
Eigenvalue equation, 33 Einstein coefficients, 415 Electric dipole moment, 430 Electric quadrupole, 423
Electric quadrupole
moment, 430
3,
4
Gauge group, 227 Gauge invariance, 297 Gauge invariant, 297, 299, 300 Gauge transformations: 3, 35, first
82, 300; of
kind, 182
Gegenbauer function, 365, 366 General relativity, 93, 332 Gibbs ensemble, 172 Green's function, 153, 162, 450, 494, 504 Green's operator, 449, 488 5, 10, 38; character of the, 42; complex orthogonal, 208; discrete symmetry,
Electromagnetic mass, 425 Electron-neutrino correlation, 279 Emission, 57
Group:
Energy eigenvalues, 361 Energy width, 409
Group Group
277 invariant, 121, 122, 251, 252, 253 velocity, 146, 150, 306
Entropy, 177, 488, 490
Equal a priori probability, 177, 179 Equal statistical weight, 136
Hamilton's equations of motion, 91, 135, 141
2, 67, 73,
Subject Index Hamiltonian:
515
1, 2, 72, 76,
133, 135, 138, 193,
269, 440; effective, 311; ordered, 162 Hamiltonian density, 310, 312, 318
Hamilton-Jacobi equation, 79, 82, 160, 162, 300, 336 Hamilton's principle, 1, 4, 82, 308 Handedness, 276
Harmonic oscillator: 76, 193, 203, 444; normal state of, 204; one-dimensional, 76, Heisenberg operator, 141 Heisenberg state vector, 142 Heisenberg's equations of motion, 73, 141, 143, 161, 207, 281, 337, 369 Heisenberg's representation, 140 Heisenberg's uncertainty principle, 82, 91, Helicity, 280 Hermite polynomials, 202 Hermitian conjugate operation, Hermitian operator, 16, 17, 18, Hidden symmetry, 363
408
355,
Isotopic multiple!, 191 I-spin (Isotopic spin): 97, 137, 180, 184, 185, 188, 382; conservation of, 182; orbital, 182, 186
Isotopic spin space, 181, 188
Jacobi polynomials, 108
Kaon, 185, 186 Klein-Gordon equation, 163, 301 Kronecker product, 233, 235 Lagrangian, 1, 2, 28, 80, 156, 307, 349 Lagrangian density, 307, 311 Laguerre function, 362
Lamb 25,
168
23, 69, 157
Hilbert space, 29, 83, 84, 130
Homomorphic, 208 Homomorphism, 41 Huygen principle, 340, Hydrogen atom, 357
Isomorphic, 124, 370
J.W.KB. method, 335
161, 368; three-dimensional, 77
94, 95, 96, 145, 346, 400, 405,
Irreducible, 42, 46, 102
484
Hyperfine structure, 110, 375 Hypergeometric equation, 366 Hypersphere, 366
Improper rotation, 10 Improper transformation matrices, 12 Incident plane wave, 456, 457 Incoherent sxiperposition, 269, 344 Incoming scattered wave, 472 Indcterministic interpretation, 345 Induced emission, 415, 420 Inelastic scattering, 405, 411 Inertial group, 93
Infinite-dimensional spaces, 82 Infinitesimal rotation, 26, 27, 98, 242
Ingoing spherical waves, 115, 267, 448, 455 Integral equation: 29, 134, 407, 449, 475, 492, 494, 495, 497; homogeneous, 503, 506; inhomogeneous, 503 Interaction energy, 417, 440
shift,
375, 381, 425, 427
Land6 splitting factor, 373 Larmor precession, 176, 372 Legendre, polynomials, 108, 430 Lepton: 186; charge, 186, 187, 281; charge content, 319; charge density, 281; conservation law, 186 Lie algebra, 36
Light cone, 49, 316 Linear operator, 6, 16 Linear Stark effect, 390 Liouville's equation, 135, 136
Lorentz Lorentz Lorentz Lorentz
condition, 300, 372 co variance, 208, 288
equations of motion, 27
group: 45, 46, 93, 248, 251, 257, 258, 287, 301, 325, 326, 330; homogeneous, 208; irreducible representations of the, 259, 326; inhomogeneous, 259; proper, 49,
52 Lorentz matrix: 47, 48, 216; proper, 50 Lorentz space, 227, 240 Lorentz transformation: 4, 45, 47, 48, 54;
homogeneous, 48; improper, 208, 216; homogeneous, 48
in-
Interaction picture: 396, 397, 402; and state
Magnetic moment, 147, 148, 305, 371, 431 Majorana exchange, 190 Majorana neutrino, 325 Majorana representation, 238
vector, 469 Intermediate states, 404
Many-particle systems, 433 Mass renormalization, 425
Intrinsic I-spin, 181 Invariant coordinate system, 208 Invariant gauge, 247
Maxwell's equations: 4, 57, 61, 62, 64, 232, 247; four-dimensional form of, 297 Mean occupation numbers, 177
Subject Index
516 Measure
Orthonormal
of spread, 143, 148, 206 Metric tensor, 46 Microscopic reversibility, 166 Mirror nuclei, 191 Mirror representations, 252
Momentum Momentum
Oscillator strength, 421
Outgoing scattered waves, 472 Outgoing spherical waves, 115, 267, 448, 455 Outgoing wave boundary condition, 453, 482
operator, 125, 127, 130, 140
representation, 132, 138, 203,
Pair interaction energy, 436 Parabolic coordinates, 391
453
Mott
set of basic vectors, 11, 19, 33,
291
460 Multipoles, 263, 282 Muon, 186, 279, 380 scattering,
Parity: 137, 166, 216; intrinsic, 165, 166; of a level, 383; nonconservation of, 279;
Natural width, 499 Negative energy, 65, 275, 281, 296, 304, 328 Neutral plane, 232, 278 Neutrino: see also Majorana neutrino; 165, 186, 275, 276, 280, 282, 299, 318, 321, 326; longitudinality of the, 279; left-handed, 277, 279; propagator of, 321, 331; trans-
formation group of, 328 wave function 286, 328 Neutrino equation, 300 Neutrino Hamiltonian, 284 Nonpure state, 174 Nonrelativistic limit, 293, 304, 359, 418 Norm, 23, 492 Number of final states, 400, 401 Number of initial states, 420 ;
of,
orbital, 166 Parity operator, 123, 165, 286, 358, 423
Parity selection rule, 381 Partial waves, 454 Partial width, 147, 410
Permutation, 31, 197, 200 Perturbation theory, 385, 481
Phase factor, 103, 342, 438 Phase shift, 190, 454, 457, 479, 480 Phase shift operator, 475, 487 Phase space, 135, 173 Photoelectric effect, 340 Photon: 58, 59, 63, 71, 143, 165, 173, 182, 261, 263, 266, 297, 326; covariance group of the, 328; Hamiltonian operator of the, 311; left circularly polarized, 65, 66, 268, 279, 313; parity of the, 266; propagator of,
315, 331; polarization
of,
60; right cir-
cularly polarized, 65, 66, 268, 279, 313; 2, 68, 96, 110, 137, 304 Occupation number operator, 199, 200, 201, 313, 314, 319, 370 Occupation numbers, 197, 200 Odd parity levels, 384 Odd permutation, 13, 210, 437 Odd states, 272
Observable,
wave function Photon
gas,
of,
66
59
Physical reality, 94 Pion, 165, 181, 187, 273, 275 Planck's formula, 59, 133, 207, 445
Planck's law, 49, 136, 315, 415 Plane electromagnetic wave, 61
four-
Plane wave: 113, 114, 115, 139; distorted, 486; partial wave expansion of a> 115, 454 Poisson bracket, 73, 74, 76, 91, 97 Poisson formula, 332
dimensional, 48, 248, 252, 366, 367; irreducible representations of, 256; proper
Polarization: 53, 144, 173, 261, 421; degree of, 269; of the electrons, 279; direction of,
Operators, 5 Optical theorem, 507
Ordered operator, 158 Orthogonal group, 123,
real,
124,
242;
10
Orthogonality conditions, 24, 25, 30, 48 Orthogonality relations, 7, 114
Orthogonal transformations: 9, 11, 97, 123, 366; improper, 10, 32, 43 Orthogonal transformation operator: 9, 10, 35, 104, 111, 112, 124; proper, 25
Orthonormal coordinate system, 5, 7 Orthonormality relations, 9 Orthonormal representation, 84, 91
321; linear, 173; longitudinal, 282; parallel, 379; partial, 268; perpendicular, 378 Polar vector, 11, 14 Position operator, 127, 130, 132, 137, 140,
438 Positive definite, 327 Positron, 186
Positronium: 275, 380; ground state Potential energy, 92, 205, 359 Potential scattering, 499
of,
381
Subject Index
517
P-representation, 131, 138 Principle of detailed balancing, 415, 416, 488, 489 Principle of general covariance, 93
Principal part, 86 Principal
quantum number,
362, 383, 393,
427
Rotation group: 10, 25, 36, 41, 102; doublevalued representation of, 228; proper, 35; three-dimensional, 42, 103, 363, 366, 370
Probability: 67, 68, 82, 129, 135, 136, 137, 196, 333, 338, 341, 346; Laplacian rule for the,
Resonance phenomenon, 406, 411 Resonance transition, 405 Resonance velocity, 411 Retarded potentials, 429
342
Rotation in time, 126 Rotation-invariant, 3, 121
Rotation operator: 29, 97, 112, 167; proper,
Probability amplitude, 136, 304, 338, 342, 344, 397, 407, 410, 481, 489 Probability density, 139 Projection operator,
7,
20,
189,
199, 268,
291, 299, 319
31 Rotation-reflection group, 43, 44, 122, 383
Scalar product, 5, 21, 22, 131 Scattered amplitude, 452, 454, 484
Propagator, 152, 154, 328 Proper time, 75
Scattered outgoing wave, 456
Pseudoscalar, 14, 123, 209, 236, 241, 261,
elastic, 405, 411 Scattering amplitude, 451, 493
Scattering: 447; Coulomb, 458, 463, 466;
273 Pseudovector, 236, 241, 261 Pseudovector interaction, 299 Pure state, 173, 269
Scattering amplitude
operator: 494; for-
ward, 507 Scattering cross-section, 448, 492 Scattering process, 28, 453, 496
Q-representation, 131, 138, 153 Quadratic Stark effect, 393
Quadrupole
transitions,
Scattering theory, 31
Schrodinger state vector, 141 Schrodinger's equation, 63, 133, 134, 135,
424
Quantization, 58 Quantized photon wave equation, 312 Quantum conditions, 71, 310
Quantum
136, 138, 159, 310, 355, 417, 438, 449 Schrodinger's picture, 335, 474, 485
Schrodinger's representation, 68, 131, 133,
distribution function, 438, 439,
440
310, 449
Schrodinger's representation of energy, 69
Quantum Harmltonian, 29 Quantum Hamilton-Jacobi equation, Quantum path, 345, 351, 444 Quantum Poisson bracket, 92
160
,
Quasi-stationary state, 411 of the lines, 394
Quenching
Schrodinger's representation of 69
momentum,
Schwarz's inequality, 23, 24, 145 Schwinger's action principle, 157, 158, 349 Screening factor, 364 Selection rules, 119, 377 Selection rules for electric dipole radiation,
Radiation: 58; electric dipole, 266, 430; electric quadrupole, 430; emission of, 415;
magnetic
dipole, 266, 423, 432; multipole,
424; quadrupole, 424
Radiation oscillator, 314, 420 Reaction operator, 477, 478 Reflection of coordinates, 14 Reflection transformations, 10, 44 Relativistic, 74 Relativistic Hamilton-Jacobi equation, 80 Relativistic mechanics, 94
Relativistic Poisson bracket relations, 75, 92 Relativistic wave equations, 248
Relativity, 71
Resonance, 406
377 Self-conjugate particles, 278 Self-consistent field method, 435 Self energy, 408, 426 Shutter, 338
Similarity transformation, 223
Simple scalar product, 21 Singlet antisymmetric state, 116 Singlet positronium S-state, 380 Singlet S-state, 275 Singlet state, 116 Small system, 94 S-matrix, 170, 468, 469, 481, 486, 487, 491,
506
Space inversion,
4, 261, 279,
490
518
Subject Index
Spacelike case, 49, 257, 308 Special relativity, 45, 93 Specific heat,
446
Translation invariance, 449
Transposed complex conjugate, 6 Transversality condition, 65, 270 273, 297 298, 311 7
Spherical Bessel functions, 430, 459 Spherical harmonics: 109, 114, 166, 265, 479, 495; addition theorem for, 109, 114, 166, 265, 479, 495 Spin: 53, 71, 97, 137, 148, 208; of pion, 490; and statistics, 325, 330 Spin coordinates, 106
Spin degeneracy, 490 Spin exchange operator, 190 Spin matrices: 40, 102, 175, 236, 329, 370; left-handed, 282 Spin momenta, 106 Spin nmltiplicity, 401 Spin-orbit coupling, 110, 305, 373 Spin-orbit forces, 190
Transverse spherical waves, 265 Triplet positronium S-state, 380 Triplet state, 116
Triplet symmetric states, 116
Tunnel
effect,
205
Two-component neutrino, 325 Two-component spinors, 220, 223, 275, 282 Two-component spinor wave equations, 290, 298
Two-photon system: 269; polarization states of,
273; Schrodinger equation for, 270;
wave equation for, 269 Two-photon wave function: states
of,
270;
odd parity
273
Spinor plane, 229 Spinors, 42, 137, 222, 238, 370
Spinor spherical harmonics, 282 Spin wave functions, 183
Spontaneous emission, 415
Ultraspherical polynomial, 249 Ultra-violet difficulty, 57
Uncertainty relations, 67, 95, 125, 146, 148. 151, 369, 370
State vector, 131, 133, 135
Unitarity condition, 126, 127 Unitary group: 38; two-dimensional, 107; unimodular, 38, 41
Stationary state, 64, 142, 196, 275, 313, 477 Stationary state perturbation, 386 Stereographic projection, 228, 229, 331, 366 Stern-Gerlach experiment, 147, 150 Strangeness, 185, 379
Unitary matrix, 35, 78, 104, 117 Unitary operator, 22, 104, 112, 127, 142, 157, 237, 351, 355, 395, 402, 481 Unitary space, 35, 104 Unitary transformation: 34, 36, 37, 40, 77,
S-state, 150
Stark
effect, 386,
391
Strong reflection, 219 Superposition of plane waves, 335
Symmetric states, 115, 171, 179, 197 Symmetric triplet states, 190 Symmetrical wave function, 462 Three-dimensional inhomogeneous rotation group, 251, 252, 261
Time-dependent perturbation theory, 396, 400, 406, 474
Time
inversion, 4, 262, 330
Timelike case, 49, 256
Time Time
reversal invariance, 167 reversal operation, 168, 169, 267, 282
Time-reversal operator, 218, 240, 296, 488 Total cross-section, 458, 479 Trace, 18, 19, 102, 175, 501
Transformation function, 127, 128, 129, 131, 152, 155, 157, 159, 347, 351, 356, 403 Transition probability, 400, 403, 405, 407, 408, 418, 420, 421, 470, 478, 482 Transition probability amplitude, 169, 473
41^
110, 116, 126, 156, 290; two-dimensional
41 Unitary transformation operator: 37, 39, 107; infinitesimal, 155
Unit operator,
7, 20, 21, 25, 37, 83, 84, 88, 100, 103, 104, 105, 110, 126, 127, 133, 151, 168, 264, 441, 469
Vacuum state, 198, 199, 281, 320 Vector spherical harmonics, 263, 264 Vector spherical waves, 264 Velocity operator, 144 Virtual photon, 425
"Wave function: 63, 67, 70, 115, 137, 139, 205, 340, 353, 447; momentum space, 139, 204; triplet charge state, 190 Wave packet, 67, 95, 125, 139, 142, 145, 146, 150, 335, 340, 449, 457, 483,
Weak
interactions, 185, 382
Zeernan
effect, 371,
374
Zero point energy, 428, 445 Zero point oscillations, 427
487
