Quantum mechanics, Second Edition

At some point P, the individual sinusoidal waves are all in phase and build up to a large resultant amplitude. Outside a region

ro
surrounding the point P, the individual waves interfere destructively. From the Fourier integral theorem one knows that increasing the size of the momentum interval Ll po of the group of plane waves enables one to decrease the extent Llro of the wave packet. � plane _wave represents the extreme case of a particle whose position is completely l!�own but whose momentum is defined with absolute accuracy. Superposing a large number of plane waves (i.e. large Ll po), so that the momentum is now badly defined, enables one to restrict Llro to a narrow interval, i.e. the position of the particle is well defined. We have here a first example of Heisenberg'S uneertain!J printiple, which states that it is not possible to have an accurate knowledge of the simultaneous position and momentum of a particle. For a monochromatic wave u = ei",

we define a phase veloci!J c" by

(ax) at

c,, = -

'P--'.

WA = - = VA; 2n

c" is the velocity wit� which a particular point of the wave, say one of • In practice, one urually deals with beams of many particles. For these, one can urually approximate the actual wave function by a plane wave, as explained above, 10 that one can avoid the somewhat complicated analysis in terms of wave packets.

29

6. TilE SCHRODINOER WAVE EQUATION : DISCUSSION the crests, moves along. the wavelength -

[n general the phase velocity ,.. depends on C.. -Cqo( A) ;

this is the wcll�known phenomenon of dispersion. As a result of this dispersion a wavc packet docs not retain irs shape accurately, but if it is a narrow wave packet it does retain its form approximately. 'Ve can then talk ofa gOO ue/oci{y c, of the wave packet, which is the velocity with which this pulse propagates itself. It is not difficult to show that ilL hc l limiting case of a sufficiently narrow wave packet the group velocity is given by_ (8) (if Exercise 11.3). Ifwe identify the group velocity " with the particle velocity II we have, using equations (8) and (5.1), ,

Mv (b, - ---d(l /')

md,

and integrating this equation, liw_ltv_imIl2=E.

(5.3)

We have thus obtained an independent verification that equation (5.3) givcs a consistent interpretalion of w in tcrms of the energy. So r.'lr we have considered one fixed instant of time 1 -/0, We must now turn to the time dependence of the wave function. Ifwe exclude processes in which particles arc annihilated or created, the normaliza� tion integral, equation (6), must remain constant throughout lime, (9) Now (10) From the SchrOdinger equalion . all

112

!liat - - 2mV2u + V(r)u and its complex conjugate

. eii

- !!i.el

-

112

- 2m'il2ii + V( r)ii

(we arc always assuming that the potential is real). we obtain, multiply� ing by Ii and II respectively and subtracting, a Ii a;11l12 - - 2mi (liV211 - ll'il2li) . 30

( 1 1)

7.

j

ENERGY EIGENFUNCTIONS : THEORY

If we define a vector by the equation

(12) j = 1i .(u grad u - u grad u) 2 and integrate ( 1 1 ) over a volume V bounded by a surface S, we obtain, using Green's theorem (Appendix A, equation (Al l »,

�

dT

m,

IL UI2dT - LdiV jdT - - f!"tL\'

( 13)

�

j

where is a volume element within the volume V, tiS a surface element on the surface S, and jn the component of the vector in the direction of the outward normal n of the surface element tiS. We interpretj as a c-!,rrent vector, !.e.jn(r, t)tiS is #le rC,\t� at wlllch pa�!icles �oss a surf�ce elemenLd.L{normal to�e vector � situated at r, at time t. Equa­ tion (13) expresses the fact that the rate of decrease of particles within the volume V equals the rate at which they escape across the boundary surface S. If = 1, is the probabili!)' current vector;intiS defines the p-robability_Qt�ui_electr�n crps�rng a.i�em_ent Qf ar�Cl. tiS i.n. the directio� I! normal tQ .dS Eet.uqi! time.. If our system is localized, we can choose the surface S so that u = 0 everywhere on and outside S, i.e. from (12) =0. Then the right-hand side of (13) vanishes and equation ( 9) follows. Even if u(r, t) extends throughout space, equation ( 9) remains true, provided u(r, t) has a finite normalization integral. For, in this case, u -+ 0 sufficiently rapidly at infinity for the right-hand side of (13) again to vanish *. Thus the normalization persists whenever the normalization integral is finite. But these are the only cases of interest; for if is infinite, it is meaningless to ask for its rate of change, nor can such wave functions represent an actual physical system.

N

j

j

N

N

,. Energy eigenfunctions : theory For a system in a state with definite energy E, we can put u(r, t) F(r) e iEI/... (1) For it follows from the discussion of equation (6.5) and from equa­ tion (6.3) defining the energy operator that a measurement of the energy of the system in the state (1) necessarily gives the result E. We have

-

=

i1t= =Eu

and the SchrOdinger equation (5.8b) becomes

HV'=( _�V2+ �}V':EV'._

(2)

• Theae conditions are mathematically not quite general enough, but they are sufficient for the functions one meets in quantum mecJuuics i .

31

7.

ENERGY mGENFUNcnoNS : THEORY

E9.uati�.!�jg). is.call4!_c! the_time�indep.c:mlent Schrodinger eguation. It represents a typical eigenvalue problem of quantum mechanics. The solutions represent stationary energy states, the energy being one of the possible eigenvalues of (2) ; the corresponding wave functions are called energy eigenfunctions. To define the eigenvalue problem completely, we must specify the fundamental domain V within which (2) is to be solved and the boundary conditions which 'I' must satisfy on the boundary S of V. In Chapter III we shall solve some energy eigen­ value problems and we shall then discuss these boundary conditions in more detail. For the present we want to discuss some general aspects of (2), assuming always that suitable boundary conditions are imposed.

(i) Discreteness

The eigenvalue problem defined above is very general, and it does not necessarily possess a discrete spectrum of eigenvalues. We want to restrict ourselves. for the time being, to a discrete spectrum of eig�!!:..... va1ues�

_

El <.E2 <

. . . <.Ea < . . .

(3)

and corresponding eigenfunctions

'l'h 'P2J which we take as normalized :

•

•

•

'l'a'

('I'll' 'I'll) = 1 .

That the

•

•

•

(4) (5)

V'a can be normed depends on a general theorem of eigenvalue

t11�:j'-an eig�._PIobltm has.a com/llelelJ discrete_spectrum_the__eigen­ ffi-nctions C4R be normed�_aml vice veJ',ta___ This theorem does not help us to decide whether a particular eigenvalue problem has a discrete spectrum not. However, in the following important classes of cases -the -spectrum is always discrete : or

(a) if ���t�� .�s_ localizecl, _�e� ther�_ is �. !"_eg!on l'J)oundec!.P� a surface S such that '1' =0 on and outside ,s'; . (�) for pI'Qblems_.YIitlt i!1finite ..fundam«mtal domaiDs,_ pr()�ded �e potential V(rL� CX) \!heneyer J.. L� �t,- We can think of �.�ase as'aq�uiSi:localized_p!obl�IIl; the potential increasing at large distances prevents any p articles from ���piDg�to l��· diStariEeS� --From -tile general theorem stated above, it follows that for problems with infinite fundamental domains and discrete spectra, 'I'{r) � 0 sufficiently rapidly at infinity for 'I'{r) to have a finite norm. If 'P{r) and x(r) are two functions with finite norms, V is a finite domain • One can show that there always exists a lowest energy eigenvalue, corresponding to the ground state of the system. t For criteria (/I) and (6) to apply, the potential VCr) must not have too strong negative singularities. We shall not state these coDditiOIlS explicitly, as they are usually satisfied in quantum mechauic:a.

32

dS,

7.

ENERGY EIGENFUNCTIONS : THEORY

bounded by a closed surface $, and surface then the surface integral

D

is

the normal

J ,,(r) :nX(r)dS

to

an element of

s

vanishes as we let the surface $ move off to infinity in (Cj footnote on page 3 1 .)

all directions,

(ii) Reality The_ eigenvalues of the above eigenvalue problem are all real. If E is an eigenvalue and 'P the corresponding eigenfunction, we have from (2) and (5) that

E-

-

:�J !'"'Y'dT + J .VI'!'I"'T.

(6)

In (6), the second integral is real, and we can transform the first by means of equation (A 10) of Appendix A, to give -

J

grad ;; grad .

v4T+ J.;;:dS.

The first of these integrals is real and the second vanishes, the last paragraph. Hence, the eigenvalues are real.

as

we saw in

(iii) Orthogonality If--.:§",_!l!!c:.I fill _�e _t�o difr�rent eigenvalues of (2), then any eigen­ r.unctiC;>ll�_1J'm _,!-_nd 'Pn belo}!ging to these eigenvalues respectively are_ orthogonal, j.e. ('P"" 'PII) = 0, m :;en. (7)

_

From the Schrodinger equation (2) we have

y",( ::V2 + V) 'Pn =E"Ym'P. ; -

interchanging m and n, taking complex conjugates and using the reality of the eigenvalues, we obtain

( ::V2 + V) Y.. =E..Y..'P".

'PII -

Subtracting these two equations, integrating over the fundamental domain V and using equation (A 1 1 ) of Appendix A. we obtain 33

EXERCISES II

h'J (

---

Since

En :f;:E..,

-0.

this equation implies

"an� ""all )dS

0..-" - 'P.-"

s

2m

('P"" 'P.) -0.

(8)

(iv) ComJ"tlellcss

Suppose we know the wave function u(r, to) at some time to. The time· dependent Schrodinger equation thcn completely determines the wave function rorm

tI(r, t)

at later times

1>/0,

If u(r,

10)

can be written in the

•

tI(r, to) - Lan'P.(r)

(9)

then we have at once the general solution •

It(r, I) - La.'P. (r)e-iF.,,(I-
( 1 0)

." '

as this runcliOH satisfies the wave equation (2) and the initial conditions Given that tI(r, to) is a linear superposition or energy eigen· runctions, equation (9) complelcly determines the coefficients Q.:

(9).

a.�(•• (r). ,,(r. 10) ) -

5

•.(r),,(r.

,.

lo)dT.

( 1 1)

The im ortanc, or Ihe above analysis is due to the fact that the energy elgen unctIOns orm a complete set: any arbitrary function J(r) with finite norm can be expanded in an infinite series or the rorm Hence equation (1O) gives a general solution or the time·dependent problem ror all physically admissible initial conditions.

(9).

EXERCISES 11

W

Derive the
J1.2. Write down the SchrOdinger equation ror the helium atom. I f one represents a one·dimensional wave packet by a Fourier

lJL

integral

11(.\', 1) -

["+''''

(I(fJ) exp

. ,._"�l

34

[�(p

.

.... -

]

EI) dl!

(IT.S)

EXERCISES 11

where a and E are continuous functions of p, derive equation (6.8) for the group velocity of an infinitely sharp wave packet (i.e. LJpo � 0) . �I.4. If e(r, t) and j(r, t) are the particle density and particle current vector at (1', t) respectively, prove the ' continuity equation '

11.5. i.e;

Of!

eli · "" 0.

ot + V J Suppose equation (L6a) (Exercise 1.6), describes a single particle,

J:..If(X) 1 2dx .

""

( I I. 4-)

1.

Verify Heisenberg'S uncertainty principle for this wave function; i.e. if LJ x and LJp are the uncertainties in the knowledge of the position and momentum of the particle, then (LJx) (LJp) is of the order of magnitude of Planck's constant h : (LJx) (LJp) �h.

(N.B. In Chapter IV we shall consider the uncertainty principle more critically.)

35

III ENERGY EIGENFUNCTIONS : EXAMPLES

ONLY a few of the energy eigenvalue problems which occur in quantum mechanics admit simple rigorous solutions. Yet these problems are of great importance, as they can in many cases be taken as the basis of an approximate method of solution. We shall, however, not deal exhaustively with these ' standard problems ' which can, for example, be found in the books by Schiff12 or Sommerfeld 1 3. We only want to deal with a few of them to make the reader more familiar with the general theory so far developed. The examples selected will be fundamental to our treatment of general topics, such as angular momentum, scattering theory, in later chapters.

etc,

8.

One-dimensional potential barrier

For a one-dimensional problem the time-indeJSendent Schrodinger equation takes the form

(7.2)

d2tp(x) dx2

Suppose

Vex)

as shown in

2m

+ 1£2 [E - V(x)]tp(x) =0.

(1 )

represents a potential barrier of finite height

Figuer

xO

Vex) =0, Vex) = Vo >O,

}

(2)

VrJ;l

Vo 1----

Figure 3.

Otl4-dimmsional

potential barrier

-------r�o-r from the left-hand side, xO. At x = O, Vex) is discontinuous. But since �L �� and ",-�L¥e everywhere� fi1!!� _(V'(-!)�_that �� llrobili!y_ illte!]!retation of In2

36

B.

ONE-DIMENSIONAL POTENTIAL BARRIER

s�ould have a meaniIlg), it follo� from ( 1) that 1p"(x) is everywhere __finite._ Hence !p' (!) andLtljoItiori, Y?� are every:y{het�_finite and con­ tinuous,._including � =:=:,J� These continuity conditions are generally valid. Glyen a gener�L(!hr!!.e-dimen!lional)�.!lIf.!lc� .� Q! Lwhich. me, potential experi�nces a fin!��jump d����tinuity, the .wave func- _ tion '" and its normaLderivati'ye _01J1/CJ!L(81p/an denotes differentiation atong the normal to S) must be cop.tim,lous it�Yery poin! of S. (The tangential derivatives are in any case continuous so that we can say 'I' and gradY' are everywhere continuous. ) Hence the density Itpl 2 and a the cuqent j, equa�o� (6. 1 2), are ev_erywlt�re �nite. and . continuous, . physically reasonablCLcQndition...... The solution of ( 1 ) and (2) differs considerably according as or If we write

VCr)

E>Vo

E Vo,

kl = +

(i� r

E , k2 =

and obtain the solution

+ (i�(E - Vo)y,

",(x) ==tP..1(�) = A _+)!e- ' (x

O). ",(x) = 1p2(X) ��C!�_11t eik�1t

1t

(3)

(4a) (4b) No term e-ik)'o occurs in '1'2 as no waves incident from = + CX) are physically allowed in our problem. From the continuity conditions (5) '1'1 (0) = '1'2(0), '1'1' (0) = '1'2' ( 0) we obtain i!

_

.�._ _

x

(6) Now eik'1t and e-il'1t correspond to an incident and a reflected wave respectively, the incident and reflected currents being, according to (6. 1 2), (7) equation (4b) defines a transmitted beam, the transmitted current being Tik iT = 21C1 2. (8)

m

We define the reflection and transmission coefficients R

=� 1�1 (!: �!:r 1I =y; =

2

=

iT k2 C 4klk2 = � A = (kl + k2P' and R + T = 1, as expected. T

37

(9)

( 1 0)

9.

ONE-DIME.'iSIONAL SQ.UARE

\VELL POTEr.'TJAL WITH RIGID

WALLS

If E< "0. the solution of the eigenvalue problem is still given by equations (4) to (6). But now kz-iK is purely imaginary. In the region x>O, the solution c-a:.o ... c.... is still excluded as it diverges as x _ co. Equation (7) still holds, but, from (6.12), (8) is replaced by iT= O, :md the reflection and transmission coefficients become

IBI' I'" I

II --. jR - }t 11

-

T_J:{_O.

i.

-- K '

1.-.

+11(

_I

(II)

(12)

j,

Thus all the particles arc renected, as ODC would also expect classically in the case £Vo differs from that of classical mechanics. Classically one requires T"", I , R -0, if E > Vo. If we consider the limit as Vo _ 00, keeping E finite, then I( -+ co, but kl remains finite, and in the limit -- - I ' A

B

I.t.

c

- �O' A

tp -�rl -a sin kl'-': (xO) �- 'h- O a being an arbitrary constant. Thus, the boundary conditions at a boundary where there is an infinite potential barrier arc that the wave function is zero (and is continuous), whereas the derivativc 'fi' is undetermined (and is, in general, discontinuous). Physically,_Vo '" a:> means that the barrier is complctcly impenetrable, the particles arc always confiIiCa to the region x
v

J

Onc--dimensional square well potential with rigid walls

We consider UIC one·dimensional motion of a particlc restricted to the region I.-.:IL and the corresponding eigenvalue problem is thc samc as the eigenvalue problem

( )!

2m k- + fji E

for the region - L <x
-

33

(I)

(2)

(3)

9.

ONE-DIMENSIONAL

SQ.UARE

WELL POTENTIAL WITH

RIGID

WALLS

From section 7 (i) we know that this eigenvalue problem has a completely discrete spectrum of eigenvalues. The general solution of ( 1 ) is 'I' = A cos kx + B sin kx ( 4) and the boundary conditions (3) give at once A cos kL = B sin kL = O. (5) Hence we have two possible types of solution, either or

(I.) A =0, (ii) B =O,

M M M k" = 2 ' 'I'" = A,, cos 2 x, k" = TIn 2 L'

. '1',, = B,. sln 2Lx,

n = 2, 4-, 6,

.

•

•

(6a)

n = 1, 3, 5, . . . (6b) L L the coefficients All and B" being determined from the normalization of the wave functions. The corresponding energy eigenvalues are given by the same expression for both types of solution, liZ 1 M 2 EII = 2ink,,2 = '2m 2L n2, (n = 1, 2, 3, . . . ) (7)

( )

. Thus the enet:gy_�igenvallt�.Jorm an increasing infiJlJJ_e_s,eguence of dis-

crete energy levels. In Figure 4 the first few eigenfunctions are shown (with arbitrary normalizations). Figure 4 illustrates the f�ct�asjl}' verified Jr..Q.m _u equations {61, that the nth eigenfunction (ordered so that El<E2� -'--�_�) �as .ex�cJ1-Y" _en - l >-�eros (�Ln.9c.l�st �nsige . tile fundamental domain. This_ theQ�m is _ generally _�!i� for _ .one-dimensional eigenvalue proble�.!ritlt iliscrete�Rectr_a--,_ But no analogous theorem about nodal lines or surfaces exists for partial differential equations. The above eigl.alsQlutiQ.ns_are_ n.on-degen�ra,te. This again_is _generally_.true_for �igenvalue problems comprising _an ordin� seI:9n�-orde�differenJi� eguation and_two_boundary _conditionsL Eor the two boundary con­ ditions uniquely determine the two arbitrary constants which occur in the general solution of the differential equation, i.e. the eigensolution. The two types of solution, equations (6), show an important differ­ ence. The sine and cosine solutions have different pari�. Generally, we define the parity of a function f(x"y" z" xz, , x", y", z,,) of 3n spatial coordinates x.,y" Zh X2, xlI,y", z,., as follows. We consider inversion of the coordinate-Mte�, i.e. _a_l"efkctiQ�o[�coordinate... axes i�t the...Qriirln. A point with position vector r has the position vector - r after inversion. Thus, under inversion, the function !(x"y" . . z,.) becomes transformed into f( - x" -y" . . . - z,.) We .��r_�at[J�" ".. z,.) is an even function and llas. paritJI t), !f (8a) f( - x" . . . - z,,) =f(Xh ' " z..) , and it is !d4J!lllction an9 has_PJlnb'-=.J,jf (8b) f( - x" . . - z,.) = -f(Xh z,.) . __

an

•

•

.

an

•

•

•

•

•

•

•

.

39

--

•

•

-- .

.

10.

FREE PARTICLE WAVE FUNCTIONS, ETC

We see that the sine and cosine solutions are odd and even, and have parities - I and + I respectively. In g�neral, a function will be neither odd nor even, but a linear combination of the. two types of function. But the interactions with which one deals ill quantu!ll_mechanics always have a definite parity (ill our_ abov� example, V( - x) == V(x), hence V(x) is an even function an.d has parity + 1), and the corresponding wave functions then also always have a . definite parity [if Exercise 111. 1]. Different stateS can then be classified according to their parity. The importance of this classification is due to the fact that, when considering a particular atomic or nuclear process, states with different parity behave quite differently. We shall consider these questions in much greater detail later on.

__

10.

Free

particle wave functions and three-dimensional square well potendal In section 5 we obtained as the wave function of a particle of mass in free space (I) k where the momentum and the wave vector are related by

m

tp(r}==eib', p p=iik. (2) The wave function ", (r) is an eigenfunction of momentulll .and energy�\\Ii�_eigenvalues p and (p2/2m) respectively, for all values of !h� Ill�Il!entum ve��Clr. p, ���e (3a) grad tp=ptp 1i2 V 1 P (3b) - 2m ltp=2m ltp. That 1p(r) is a �i!!!�ltaneous eigC!!lf!1Jlcti9n of momen�m�atld �ergy is due to the fact thaJfgr a free particle the energy His a function of the m�rn�!!��-P °I'llY, 1 H= 2j � _ . knowlec;lgCL_Q{ .the: mom�ntuIIL�Q�letely �_ir L cl8.!Si�L physig, -

-

__

- iii

�_

.

c:ielel'llliJ!� the energr under t1!�e�oll
(eik'r, eib') == J:(k-k')rdT (2n)3 =

IJIlICe

c5 (k - k')

•

(4)

It is, however, possible to avoid intrQcl\J.cing.th�s.«!�ontinuum_eigen­ functions and toJorc.e th�hlem to hav��c!iscrete sp�£trum. This 40

10.

FREE PARTICLE

WAVE

FUNCTIONS, ETC

is achieved by eIl�IQ!ling the_&ystem in_a sufficiently large, but fini� box and im}:losing suitll.ble boundary conditions OJ! the walls of th� box. Since an}'. actuatphysical system is always to !ome extent loc�ed, it is clear that introducing a container the dimensions of which are suffi­ ciently large compared to those of significance for the system con­ sidered, will affect the system to an ar]>itrati!y small degree. Thus,

Figure 4. Energy eigenfumtions 'I. oj the one-tiimensi01UJI square well potential with rigid u'OIIs, for n = l, 2, 3, 4

r-----+--+--�----,

n·J

r-------�--�

n·;

'-------�--�

n-'

we can have a ' physically plane wave ' inside a box, provided the box is very large compared with the wavelength of the wave, as we saw in example 1.6 and in section 6. A physicaUy natural boundary condition to assume is that the wave function vanishes on the walls of the container. A mathcmatically.­ IDor�_conv�nienueprest!ntation is o_htained if one takes the box as a cube of side of length L and imposes periodic boundary conditions OJ) the wave function. The eigenvalue problem of a free particle then becomes _

(5 ) 41

10. where

FREE I'ARTICL� WAY!; I'UNCTIONS, ETC

1. 2 _ 2mE h2 '

"

_

(6)

logc.thc witl} the boundary conditions that 'P(r) be periodic with period

L,

i.e. (7)

The normcd eigensolutions or this eigenvalue problem are easily obtained by separating the variables. They arc 'P1I.{r)

-

L�n: exp (ikr),

(8)

where k is the vector with components (k.., k), k�) given by (9) TIl> 1/2 and III being restricted to the natlll'al numbcrs 0, ± I, ±2, . . " so that the periodicity condition (7) should hold. Ir I: is the magnitude or the vector k, i.e. k "' lk l ""' y(k�2 +ki + k�2) _

Y(1I1 2 + llZ2+lIJ 2), (0,) L

( 1 0)

the energy eigenvalue of the state (8) is given by equation (6). Comparing equations (1) and (8) illustrates our general disClission that introducing a sufficiently huge box docs not affect a physical problem. Though equation (5) has a discrete spectrum, whereas ( 1 ) has a continuous spectrum, the different momentum cigenrullcliolls (8) lie arbitrarily closely together ir L is sufficiently large; that s i , any eigen­ function or ( 1 ) can be approximated arbitrarily closely. When dealing with a continuum or states (whether one describes these by a continuum or cigenrunctions or by a discrete set, as above) one is not in general interested in individual states, but only in groups of neighbouring states lying close together. To distinguish individual statcs would require measuremenst or :lbsolutc :lccuracy. :MC:lsure­ ments or only limited accuracy enable one to say that a system s i in one ora small group orneighbouring states. One then requires the number of states in such a group. From the above analysis one can find this density or states. � classical physics, it is often convenient to introduce 1I!o1l!t:1IIu1I! JPac� tl� rectangular coordinatcs of which arc the components of momcntum (Pa. P" p�). Each point of this space corresponds to a definite momentum vector. It follows rrom em that the quantum-mechanically allowed momenta form a latlice or points in momentum space with coordinates

42

11.

LINEAR HARMONIC OSCILLATOR

Hence the volume per point, i.e. the volume per state is ( 2mt/LP and the number of states in an elementary volume dp of momentum space is

(:r

(11)

� (P)dp = 2 li dp.

Introducing polar coordinates (P, 0, 11') , defined by A =p sin O cos p, P7 =p sin O sin p. p� =p cos O, (12) we obtain the number of states lying in the elementary volume dp =p2dpdQ =p2dp sin 0 dOdp. From ( 1 1 ) , it is

� (p)dp = � (p)p2dPdD =

(�) 3p2dpdD = (�) 3k2dkdD

__

( IS)

where, as is often convenient, we have introduced the wave vector k = p/Ii. ( I S) gives the number of states f�� �hic!!__t1!e _Dlom�E-tum vector p lies within the �tement of solid aIlgt��::=:siIl 0 d.O�tp in !1te_ direction (0, 11') and its magnitude lies in the interval (p, p + dp) . WCShalI sUDsequeritlfseehilililthe lengtll-L occurtirfgin-(lS) cancels out identically in all physically significant quantities. From the way we introduced it this was to be expected and indeed had to be so in order that the above treatment should make sense. For this reason one often talks of the density of states I>er:_ unit VO�uIl}�e(P)fP which _is indej>endent �f L an�, from (!1),_�s si_l!lply-1fh�. __

�

II. Linear harmonic oscillator We consider the one-dimensional motion of a point particle of mass m in the potential field

V(x) =tJlX2•

(1)

According t o classical mechanics, this particle experiences the force () V =

- ()x

- px

and performs simple harmonic motion with circular frequency Wo given by

(2 ) The corresponding quantum mechanical problem consists of the Schrodinger equation 1i,2 (S) = (x)

��m"'''(x) +t���) �tp

and the boundary conditions,

", (x) - 0,

as

x

-

(4)

± co .

As stated in section 7(i), this problem has a completely discrete spectrum -

�f eigenvalues. since V�L'7"

__

ex:> �..LX

__

-

43

± co.__

--

- -

11.

LINEAR HARMONIC OSCILLATOR

To solve (3), we first eliminate the constants ,,2/2m and p by intro­ ducing new units. We let

a4= mli2l'' = 2IiE(tE)i = liCll2EO E =ax, V'(X) =V'( E/a) = 91( E) ; .it

p

equation (3) then assumes the simpler form

dE2 + (.it - E2)91=0.

}

(5)

d291

(6)

91( E) -

We now require the eigensolutions of (6), for which 0 as - ± 00, and the corresponding eigenvalues .it. We shall only indicate the method of solution of this eigenvalue problem, the details of which can be found in the books by Courant­ Hilbert3 , Schiff12 or Sommerfeld 1 3 , for example. As this eigenvalue problem is of considerable importance in many applications, we shall however state the eigensolutions and their most important properties. To solve (6), we consider�
E

I EI -

00 :

E2

lEI.

P..

__

=

e ±f2/2.

(7)

Of these two solutions exp ( + E2/2) must be rejected as it diverges at large distances, whereas we have imposed the boundary condition that 91 - 0 at large distances. Substituting

(8) into (6), we find at once that it represents an exact eigenfunction corre­ sponding to the eigenValue (9) Since the eigenfunction (8) possesses no nodes, this is the lowest eigen­ value, the ground-state of the harmonic oscillator according to quantum mechanics. From (5) and (9) we have for the energy of the groundstate (1 0)

Eo

Eo = O: �JO=!llCllO'_

Classically, we would have expected in the state of lowest energy the particle is at rest. According to quantum mechanics, the This positive zero­ oscillator possesses a �ero-point poinLenergy is intimately connected with�Heisenberg's uncertainty E!iE�le Uh�J)��r�<;I�_ �
__

__

=

11.

LINEAR HARMONIC OSCILLATOR

To obtain further eigensolutions we try to find solutions of the form (1 1 ) 9' (E) = e-t2/2x ( E) . This leads to the following differential equation for X( E) : dX d2X ( 12) dE2 - 2EdE + ( ). - I ) x = 0. To solve equation...(l2). .. o .- ne_a.sswne�L a_solution in the form of a power series expansion. Either the series terminates or it does not. lfit does notJ �itnecess�riIy diverges. -Hence_tli�only_solutionLwhich converge _. as IE1 ---±.gLoccudCthe po'Wer series .breaks off after a finite number _ o[terms, i.e.-x (E) is a polynomial. The details of this analysis can be found in the books mentioned above. The eigcnfuncti9ns oL( 12) are_ knoWILaS. Hermite polyrwmials,

_

__ XfJ{�) = HfJ<'E),

(n = O, 1, 2,_

.

and the corresp.IDldin�genvalues are

An = (2n + I ) ,

(n =O, I , 2,

•

•

•

)

( 1 3)

)

(14)

.

•

i.e., from (5), the corresponding energy levels are EA = (n + t)luu",

(n = O, 1, 2,

•

•

•

)

( 15)

.

We state the most important properties of the Hermite polynomials and of the oscillator wave functions, given by equations ( 1 1 ) and (13) : 9'n (E) = e-t2/2Hn (E)

(n = O, 1, 2,

.

•

•

( 16)

).

(i) The l-Iermite polynomials H..( E) are defined explicitly by

;;(e-t2)

l!nCE) = ( _ I )Aef2

(n = O, 1 , 2,

•

•

( 1 7)

).

•

(ii) Fro�l1ations ( 1 2) a.nd ( 1�), _ HA(E) satisfies the differential equation Hn" (E) - 2EHA'(E) + 2nHA(E) = 0 (18) allcl the_I:e.f.urrence . reJation ( 19) Hn+l (E) - 2 EHn (E) + 2.nHn-l (f) = 0. (iii) From (I 7}. one sees that_Hn ( E) is a real polynomial in E (i.e. all th� coeffi�i�nts are r�&l!t.1lllbersl,
--

�----

Ho = l Hl = 2E (20) H3 =8E3 - I2E H2 = 4E2 - 2 H4 = 16E4 - 48E2 + 12 Hs = 32Es - 1 60E3 + 1 20E. (iv) The oscillator wave functions ( 1 6), which are the solutions of our original eigenvalue problem (6), are mutually orthogonal, as we expect from the general theory of section 7(iii),

}

45

II.

(�••

LINEAR IIARMONIC OSCILLATOR

P.) � r �.p.d'-(:-'·fl.(')fl'(')d'�O. .

The oscillator wave functions (16) are not yet normalized. functions normed wilh respect to x, for which

(,•• (x), ,.,(x)) _ f:"

,

(X)�'(X)dX -�[':_(')P.(')d'

, -

-, t ·x

The wave

- 6_.

(22)

.-J

,

,

(2 1 )

m""'"

-,

-J

,

-,

,

.

-, ,

-

,

,-

,

-,

-x

-3

t

J

-,

,

_

n"

'-xC-_� . <:- :--

-�: �

J

3

-f

-.]

-2

=

C.-:1{1 H.(�)

-t---'--::'-�J-�'-�X-I

� '

.] .� -,

x

Figure 5. Harmonie oouiila/ol WmOtfUlUliol1S rp.(�)

, -

� Oo

. I

f

-

0

2

,]

,

, x I

(23) first Figure 5.

The

few unnorrnnlizcd

oscillator 46

wave functions are shown in

12.

12.

SPHERICALLY SYMMETRIC POTENTIALS IN THREE DIMENSIONS

Spherically symmetric potentials in three dUnensions

Finally we consider the motion of a point mass m in a field of force derived from a spherically symmetric or central potential VCr), r being the distance of the particle from a fixed origin of coordinates. Atomic systems and, to a lesser degree, nuclear systems can be adequately described in terms of such central potentials which are consequently of great importance. As in classical physics, a central potential implies that the angular momentum of the system has some special properties. Since the angular momentum properties of a system are very important in quantum mechanics and as we shall investigate them in great detail in later chapters, sections 1 2 to 1 5, dealing with spherically symmetric potentials and related topics, should be studied very carefully. z

Figur4

6.

Spherical polar coordinaft> CT, 8, 9')

�----�----y

To study the Schrodinger equation

{V2+ ��[E-

}

V(r)] tp = o,

(1 )

it is clearly most convenient to introduce spherical polar coordinates (r, 8, 91), shown in Figure 6, defined by the equations :

x = r sin

Y = r sin

(J cos 91

9 sin 91 z = r cos 9

}

where the ranges of r, 8 and 91 are r>O, 0 <.9 <:11:, 0 <91 <.2n. 47

( 2)

(3)

12. SPHERICALLY Sy!t.I!t.IETRIC POTENTIALS IN THREE DIMENSIONS In spherical polar coordinates, the Schrooinger equation ( I ) becomes

)

I 1 {J. � (r2!.. ) + __ !'(sin O ! + r1 ar ar r2 sin 0 ao ao r2 sin20

+ 2111 Ii' (E

-

iF},,(,

alP! ,

r

,

I (')].(',

°,

m) T

o, �) -0,

(4)

where the expression in the curly bracket is the differential operator \,2 expressed in polar cordinatcs (if, for example, Abraham-Becker6, p. 43, or Hague', p. 1 12). We can separate Ihe radial and angular variables in (4), by writing (5) �(" 0, �) - R(,)S(O, �); substituting this in (4) and dividing by V' one obtains

( )

1 d dR r il (ir d,

+

2m r2[E - V(r)) 1i2

- -HSil:0 :o(sin O�

+ si� 2

0

:��.

(6)

Since the left-hand side of this equation depends only on r and is in­ dependent of 0 and tp, whereas the right-hand side depends only on 0 and rp and is independent ofr, it follows that the two sides must be equal 10 the same constant, which we shall call ).. Equation (6) then gives the pair of differential equations

(,, dR -) + [2m(E

1 d rZ dr

--

�

dr

1.1

-

V(,))

--rZA] R-D

(7) (8)

The radial equation (7) depends on the central potential V(r) which must be specified in order that (7) may be solved. Only in a few cases c.'1.fI the solutions be expressed in a simple closed form. These include: (i) a free particle, V(r) -0, (ii) a particle in a three-dimensional spherical square well, V(r) Vo for rro, (iii) a charged particle in a Coulomb ficld, V(r) - const./r. In general, equation (7) must be solvcd by some rather claborate method, such as a series solution, numerical integration, a method of approximation, tic. For these reasons, we shall not consider the radial wave equation in detail. (Sec, for example, the books by Sehiff12 or SommerfeldlJ, 14.) We shall only consider the asymptotic solutions for large values or r, which we shall require later. Jr V(r) tends to zero sufficiently rapidly, as r -+ 00 (to be precise, if V(r) -+ 0 more strongly than I/T as T -+ co), Ihen the asymptotic solutions of (7) do not depend on V(r) : we can oblain them by considering the asymptotic solutions of the case V(r) -0. We obtain these by a method given by Sommerfeld (reference 13, _

18

1 3.

SPHERICAL

HAllMONICS

p. 22, or 14, p. 13) which is not mathematically rigorous but is simple and often useful. For large values of r equation (7) becomes RIJ + k2R = 0,

(k2 = 2� ,

(9)

with two linearly independent solutions R =AeUr,

R =Be-w•

( 10)

One can improve the solutions ( 1 0) by taking A and B not as con­ stants, but as slowly varying functions of r, i.e. we allow for terms in­ volving the first derivative A', but neglect terms in An, A'lr, A/r2. Substituting (10) into (7) one then easily finds A (r) =const.lr

(11)

and similarly for B(r}, so that we obtain as the two possible asymptotic solutions ( 12) R(r} = e±w/r. Any solution of (7) has an asymptotic form which is a linear combina­ tion of the two asymptotic forms (12). We see their physical signifi­ cance by introducing the time-dependent factor exp ( - imt) , equa­ tion (7. 1 ), where m =EIA. From ( 12) it follows that the surfaces of constant phase are ± kr mt const.,

-

=

i.e. eW/r represents an outgoing, diverging spherical wave and e-w/r represents an incoming, converging spherical wave. (CfExercise III.2.) 13.

Spherical harmonics We begin by considering equation ( 12.8) which holds for any central potential VCr). The unknown function in ( 1 2.8), S=S(O, tp), depends on the polar angles 0, tp only, and can therefore be thought of as a func­ tion on the surface of the unit sphere. Equation ( 1 2.8) first arose in harmonic analysis, i.e. the study of solutions of the potential equation of electrostatics, V2U = 0,

and its solutions are called surface harmonics or spherical harmonics. Equation ( 1 2.8) represents an eigenvalue problem. From it we shall obtain a complete set of normed orthogonal spherical harmonics and corresponding eigenvalues ; i.e. any sufficiently well-behaved function f(O, '1'), defined on the surface of the unit sphere, can be expanded in terms of these spherical harmonics. To solve ( 12.8) we again separate variables ; putting S(O, '1') = 8(0)4>

49

(1)

13.

SPHERICAL HARMONICS

it leads, by the same method as above, to the two equations dUb 0 drp2 +p.q').,,

Si� 0 110(Sin O��) + (.�

:

(2)

)

(3) 0 si 2 O e = where J.l is a second constant, introduced in separating the angular variables 0 and rp. We must restrict the solutions q'J(rp) of equation (2) to be sillgle-valued functions ofrp as the coordinates rp and 1P +2nn ( n - O, ± l , ±2, . . .) refer to the same point on the unit sphere and we rcquire single-valued wave functions to lead to unambiguous physical interpretations. The required solutions are >.(m) 't T

_

with

I _c'� ( ,/2l't

,"

_

-

0 ±1 ±2 '

"

. . .

)

(t',)

(5) We shall hereafter always assume that m has one of the values given in ('�) ; m -O, ± I , ±2, . . . The sCl ofsolutions (4) are the complex ex­ ponential functions which we have already mct in section 3, equa­ tion (3.16). As we stated there, they form a complete orthonormal set for the interval 0 <9' <2:r. To solve equation (3), with tt -m2, we introducc new variables (6) x-cos 0 61(0) -P(x). Equation (3) then becomes dP d m' (7) (I -x')+ 1. - -d.t d.'.; 1 - :r, P - O and the domain of x is - I <x< + I corresponding 10 0 <0 <no Equa­ tion (7) is a second-order differclltial equ:ltion containing the unknown parameter J., whereas "', as stated, has one of the values 111 -0, ± 1 , ±2, . . . In general, the solUlions of (7) become infinite at X = ± l and are IIms physically not acceptable. Only in the special case when (8) 1. -1(/+ I ) 1 -0, 1, 2, . . . (9) 1"'1
[

] {

}

pr(x) �(I -",')r�"dx'�p,(x), (1.01 <1; 1-0, 1 , 2, . . . ) where P,(x) are the Legelldre po!;'nomials of order I: P,(x) - 2�1!

:�[(X' - I)') 50

(1-0, I , . . ). .

( 1 0)

( I I)

13.

SPHERICAL

HARMONICS

pr(x)

The associated Legendre functions are derived from the Legendre polynomials by a process of repeated differentiation, as shown in equation one has, from For

P,(x) (10). m =O,

(10), PP(x) =P,(x) (7) , � [ (1 _X2) :/] +/(l+ I ) PI=O.

i.e. the Legendre polynomials satisfy equation

with

(12)

m =0, i.e. (13)

We shall not give the analysis which proves the above results. It is a rather tedious standard analysis which can be found in many books, e.g. those by Courant-Hilbertl, Schiff l 2 and Sommerfeld4• 13• We shall confine ourselves to a summary of those properties of associated Legendre functions and of spherical harmonics which are most im­ portant for quantum-mechanical applications. (i) The Legendre polynomials are real polynomials in of )'. The first few polynomials are degree I and parity

(ii)

P,(x) (-1 PI (x) =X 3 Po(x) =1 P2(x) =-i(3x2 - 1) P3(x) =-i(5x -3x). } Pl"(x) (I - 1m !)], (1 _ x2)lm/l2 ( - 1 )Hm!. P1:1: I X) = (1 _X2)i, P2:1: 11 ((X) =( I - x2)t3x, P3:1: (X) = (1 _X2)ii (5x2 - 1), P :1:2(X) = (1 -x2)3, P23:1:2(X) =(I -x2)15x, P3 :1: 3 (X) = (1 - x2)31215. - 1 <x <1, f::pl" (x) PI'''' (x) dx =0, 1' :f.:1 (10) . (l + lm!) ! {P, (x)} 2dx = [21 2 1 (I1m!) !] . f+l (17)

The associated Legendre functions

and have parity tions are

x

x,

(14)

are of the form

[a real polynomial of degree

The first few associated Legendre func­

(15)

(iii) The associated Legendre functions are mutually orthogonal over the fundamental interval

but they are not normalized as defined in equation shown that

-

I

m

+

(16)

It can be

(17)

Equation enables us to define normalized associated Legendre functions, apart from an arbitrary phase factor. The choice of such a factor of modulus unity does not affect the physical interpretation of the wave function (cJ p. 28 and section In quantum mechanics it is most convenient to define the normalized

16).

51

13.

SPHERICAL HARMONICS

associated Legendre functions by ( 18) (2n)I.N}"p; (x) , (1 ) l m l ) , where 2 1 if m >o' _ 1) m ( 1 9) N';' = 21 + 1 (I - 1m!) 1'f m <0, (I + 1m!) ' I.e. • Nr _ 1Cm+1m1) 21 + 1 (I - 1m!) = ( 1) ( 1 9a) I (I + 1m!) ! (iv) For a fixed value ofm, the associated Legendre functions, defined by equations (18) and ( 19), form a complete normed orthogonal sequence for the fundamental interval - 1 <x<:l . (v) Combining the above results, i t follows that the set of spherical harmonics, defined by = .N'j'P'I' (cos O) e�, ( l m l
[ � ��� :::� if, [ 1] 1 4n 1 [ 4n

j(

•

1] 1

•

11'(8, 9') = 0,

j

.

.

•

•

•

.

N,-m = ( 11'(8, 9') = (

9') .

( 11.01', 11m) = JunJ,dD1I.m·(O, 9') 1'r(6, 9') IPhere

= N,.m·.NrJ+dxPr"" (x)P,.,,(x)Jo(lnd9'd(m-m"J"

•

= dll'dlllf1J'

-

I

(2 1)

Th is form i s particularly convenient i n calculations as it i s valid for positive and negative

values ofm.

52

13.

SPHERICAL

HARMONICS

where d!J denotes an element of solid angle. That the spherical harmonics r,"'(0, '1') are complete implies that any function f(O, '1') for which

I.

is finite,

can

d!J/f(O, '1') / 2

WI aphcre

be expanded in a series CD

,

f(O, '1') = 2: 2:c,"'1i"'(O, '1') ,- 0 ",--'

(22)

and the coefficients c,m are determined from (2 1 ) Cr = ( 1i"',j) =

J.

d!J r,"'(O, tp)f(O, tp}.

(23)

wI aphero

The parity of the spherical harmonics is easily found from the fact that an inversion which transforms r into - r is expressed in polar coordinates by the transformation o -+ n - 6, 'I' -+ 'I' + n. (24) Now cos (n - 6) = - cos 6, so that pr(cos [n - O] ) = ( _ I ) l-Imlpr(cos 6),

and so that

, r,m (n - 0, 'I' + n) ""' ( _ 1) r,m ( O, '1') i.e. 1',.. has parity ( - 1 ) '.

(25)

We state the first few spherical harmonics explicitly :

1 ro° .... v4n

1'1° = (!r cos 6,

(�r

r1 ± 1 "," T

1'2° = l (3 COs2 0 - 1), r2± 1 = T n

(inr sin 0 e±i" (�!r cos 0 sin 0 e±i".

(26)

To obtain the complete solution of the eigenvalue problem, equa­ tion ( 1 2. 1 ), we must combine the spherical harmonic solutions (20) with the radial solutions of (12.7). For a given value of A =/(1 + 1), equation ( 12.7) possesses a series of radial eigenfunctions R,., ,(r) which we shall take as normalized, corresponding to energy eigenvalues E", , of the system. These energy levels may form a discrete set or a con­ tinuum depending on the nature of the potential VCr). The solutions of the eigenvalue problem ( 1 2.1 ) are then V'E". ,. '. '" (r, 0, '1') =RA, ,(r) 1i"' ( O, '1'). (27) 53

14.

ANOULAR MOMEI\"TUM

From the general discussion in Chapter II we know that these form a complete set of orthonormal eigenfunctions. 14.

Angular momentum Classically, a point panicle of mass Tn moving in a spherically symmetric potential VCr) possesses a constant angular momentum. Thus, if •

p -mi, p

then

- - grad V-

- W I r, Or ,

1 _ rAp,

OV I l-rAp +rAp-m (rAr) - :;- - (rAr) - 0, vU '

.

•

•

•

i.t. the angular momentum vector J is constant. As there c.xislS a close correspondence between classical observablcs and their quantum mechankal operators, one might expect that, quantum mechanically too, systems with centml potentials have some special angular momen­ tum properties. Using the results of section 6, one finds for the angular momentum operator (if Exercise 11.1), (Ia) l - rA ( - ;A grad), I.t.

1. -

-

i (� -i), etc, /i

(Ib)

or, changing to polar coordinates,

( :0 + cot 0 cos '1':'1') [, -ill ( - cos rp: O + cot 0 sin : ) -ilia�'

[. -iii sin '1'

'P

1. -

(2)

'1'

and for the operator III denoting the magnitude of the angular momentum (3) Consider a wave function �r(r, 0, 0) -R(r) rr(O, 0),

(lml <1, 1 - 0, I,

. . . ),

(4)

where the angular dependence is to be given by the spherical harmonic 1"r(O, '1'), but the radial wave function, R(T), in which we are not interested, is not specified any funher. From equations (2) and 54

14.

ANGULAR MOMENTUM

( 1 3.20) it follows that It.tpi'" = (lim)tp,"', ( l m l
•

•

.

•

•

II I

I

tpl = "2/mtpl", m--l but we are not at liberty to choose the linear combination arbitrarily ; • The case 1 = 0 represents an exception. Equation (6) now states that the total angular momentum III is exactly zero, and one would expect any component of I also to be exactly zero. In fact, since V'oo =R(r)/v'(4n), i t foUows at once from (2) that tpoO is a simultaneous eigenfunction of aU three components of angular momentum, with the common eigenvalue zero.

3 + Q·M .

55

EXERCISES III

it is determined by the physical conditions. By a suitable choice of the coefficients em we can make tpl represent a state with a definite com­ ponent of angular momentum, lim, in any prescribed direction. For a spherically symmetric system we must average over all possible values of em, i.e. average over all possible orientations. The spherical harmonics, i.e. the tpr, are themselves already the correct linear com­ binations of wave functions if there exists a preferred direction in the the system which is taken as �-axis ; for example, if a magnetic field is applied in the �-direction. The properties of angular momentum which we have so far con­ sidered have all referred to one instant of time. In Chapter IV we shall see that theorems analogous to the classical conservation theorems hold in quantum theory also. EXERCISES III Suppose that the Hamiltonian function H of a system is a function of n -position and momentum vectors r" , " rAJ Ph , PII'

�I I. I .

'

•

•

H = H(rh " , rll, Ph ' . . PII)

and that it is invariant under inversion of coodinates i.e,

H( - r"

.

.

•

- ra, - p" ' , . -PII) =H(r"

'

,

•

rm P., , . . PII)' (III. I )

If tp(r" ' . . rn) is an eigenfunction of H belonging to the non-degenerate eigenvalue E, show that tp has a definite parity, i,I,

�I,2. -

tp( - rio ' "

- rn) = :!:tp(r., . . . rJ.

Show that the wave functions

�±� r

represent spherical outgoing and incoming waves, respectively, by consider­ ing the corresponding probability currents. \ III.�, The SchrOdinger equation of two interacting particles of mass mJ and m2 respectively is

[

]

li2 1i2 8 ma,tp(rh r2, t) ... - V,2 - V22 + V(rl - r2) tp(r., r2J t), (III.S) 2m2 2ml where rl and rz are the position coordinates of the particles and 82 82 8z V,2 = _ + - + _, (i ... I, 2) . 8xj2 8y,2 OZ;2

Show that if one introduces the centre of gravity coordinate,

mlr, + m2r2 , m l + mZ and the relative coordinate r = r, - rz of the two particles, then the R ...

SchrMinger equation (III.S) is separable. Interpret the two resulting equations. 'tII,4_�-' In section 8 we considered a plane wave incident on a potential jump; equation (8.2),

56

EXERCISES III Consider the same problem for a potential barrier of finite thickness:

Vex) = 0, V(x) ,.. Yo. Vex) = 0,

x< O, ( Vo> 0), O < x< a, x> a,

and obtain the reflection and transmission coefficients for both the cases E> Vo and 0< E< Yo. 111.5. Obtain the eigenfunctions and eigenvalues of the eigenvalue pro61cm

where

Vex) = 0, I xl 0) , I xl >a,

which lie in the interval O < E< Yo. (In this energy range, this problem has a discrete spectrum of eigenvalues.) (Hint : The solution of this eigenvalue problem can be simplified in two respects: (i) by using the fact that the solutions must be even or odd (if Example 111. 1), one need only solve the problem for x> 0 ; (ii) instead of matching the wave function and its derivative at x a, we need only match its logarithmic dmuative ; for =

implies

i.e.

'P1(a) = 'P2 (a) ,

'PI '{a) ='P2'{a)

'P1'{a) 1f'2'(a) 1f'1 (a) ,.. 1f'2{a)

[t,

ln

"'I (x)

]"_,, [t. =

ln "'2 (x)

L-,..

This procedure of matching logarithmic derivatives is generally applicable and often advantageous.)

111.6. Obtain the Schrodinger wave equation of the hydrogen atom. Obtain the eigenfunction corresponding to the lowest energy, which is spherically symmetric. In this ground-state, what is the most probable distance between the electron and the proton ? Obtain the ground-state wave function of the helium atom, neglecting the Coulomb repulsion between the two electrons.

57

IV GENERAL PRINCIPLES OF QUANTUM MECHANICS IN Chapter II we introduced Schrodinger's wave equation and dealt with its more important mathematical consequences and, to a lesser extent, with its physical interpretation. Our main object in this chapter is to investigate systematically the physical interpretation, the relations between the mathematical symbols and their laws of operation on the one hand, and the physical observables and their measurement on the other. At the same time we shall extend the mathematical formalism to admit much more general questions than

i.e.

we have considered so far. It will turn out that all questions con­ cerning the interpretation of quantum mechanics can be summarized in five basic rules or interpretative postulates, which we shall state and discuss in section 1 6. We shall try to make these axioms appear physically reasonable, and show that they are often natural generali­ zations of concepts we have already met. These axioms will merely serve as simple and concise rules for the interpretation of quantum theory. We shall not discuss the difficult question of whether these axioms are the minimum number of necessary axioms, or examine their reasonableness in terms of highly idealized measurements. Such analysis is of course of great importance, but falls entirely outside the scope of this book. On the basis of these postulates we shall, in the remainder of the chapter, consider general questions of interpretation, such as the measurement of several observables, the uncertainty principle,

etc.

15.

Resume of classical mechanics *

Before considering the general principles of quantum mechanics, we shall briefly restate the laws of classical mechanics and their consequences. The configuration of a system of N point masses mit • • • mN is com­ pletely given by the 3N Cartesian coordinates (XhYh �1 )' (X2,Y2, �2) (XN,_VN, �N) of the masses. Frequently it is not convenient to restrict oneself to these Cartesian coordinates, but to specify the configuration of the system in terms of other parameters, called we sometimes prefer to work in polar coordinates (r, 6, 9') ' This is •

•

•

generalized coordinates,

e.g.

• Section 15 will not be required subsequently, but has been included to bring out the essential contrast between classical and quantum mechanics. For a more detailed treat­ ment see, for example, M. Planck, General Alechanics, London : Macmillan, 1933.

58

1 5 . RESUME

OF

CLASSICAL MECHANICS

particularly the case if the system is subject to constraints, so that the 3N Cartesian coordinates are not independent variables. If n is the leas1l!l!m�c:r ofJ)aramet�rs 1'le��sary. J�spe�it)r . the. c.Qnfigl!l"ation _oL. the system, then the system is liaid to haye_n .tlegrees offreedom . (Thus a point mass has 3 degrees of freedom, a rigid body has 6 degrees of freedom.) The configuration of a system with n degrees of freedom is fully specified by n generalized position coordinates gh g2, gn, and we express all quantities, in particular the Cartesian coordinates (xo, Yo, �), in terms of these generalized coordinates. The kinetic energy T and the potential energy V of the system become N dxo dZo 2 (1) + = T(qj, qi), + T dt dt dt •

{( ) 2 (4)'0)2 ( ) } a =!�:n

•

•

.

V = V(q;, qi) (2) where we have abbreviated the set of coordinates (qh q2, qn) to (gj) and the dot denotes differentiation with respect to the time : qj = dqi/dt. From ( 1 ) , we define the generalized momenta by � La ?, : r t� (3) Pi = 8q ( T - V), ( i = I , 2 , . . . n), j analogous to their definition in Cartesian coordinates for systems for which, as is generally the case, the potential energy V is velocity­ independent (i.e. 8 V/8qi = O, i = 1, 2, n), •

a :r,

P"o = maX

=

.

•

•

•

.

( a = 1, . . . N) ,

and similarly for the other components. W e now define the Hamilto[lian function H of the system as its .energy functio,! cxEressed hU�!.ms of th� �n gener'!lized position. aIld m()mentum �Qordjnates, (4) H = H(qb Pi) ' In terms of this Hamiltonian function, the equations of motion, Hamilton's equations, become . 8H , 8H n) . (5) Pi = - 8g ' qi = 8 ' (i = I , i Pi Hamilton's equations are the generalization of Newton's equations of dynamics, and for Cartesian coordinates they reduce to these (if Exercises IV, 1 and 2 ) . Hamil�Q.,!'�. equiLtions form a systeIll . o( 2B. . partial..4ifferential eql1atio�s of" the .firsJ ()r.der fQ.l'Jhe. �n Jlnkrtown fu!,c­ tions qi(t), A(t) , (i = 1 , n) ; given the values of �_he 2n functions at some instant ()f time to, l!amilt9n's eq\1ati().!l!"po�ess JL\1nique.s()lutioll, i.e. they cc!!1lple�c:.ly specify tlle�Hion and l!lomelltum c()()rdinate��.� �ll other times. In classical mechanics all dynamical quantities, such as energy, angular momentum, etc, are well-defined functions of the generalized coordinates. Hence, given the initial values of these at •

.

•

•

59

.

.

16.

INTERPRETATIVE POSTULATES OF QUANTUM MECHANICS

= to, the subsequent behaviour and properties of the system are com­ pletely determined by Hamilton's equations. The fundamental laws of classical mechanics are completely deterministic. Classical statistical mechanics is merely the result of averaging, when one does not know the initial values of all the 2n coordinates of a macroscopic system. 16.

Interpretative postulates of quantmn mechanics

We tum to the quantum mechanical description of a system with n degrees of freedom which is classically specified by n generalized position coordinates gil gz, gil. In quantum mechanics such a system is specified at a fixed instant of gil) whose norm is unity, i.e. time by a wave function 1jI(qll •

•

•

•

('P, 'P) = [ I'P(qh

•

•

"'"

•

.

•

qll) 1:�qL

•

•

•

'qll .:. 1 ,

(1)

thejntegrationJ)eing�oY'e(nthe Jundam�nt�tqQmain, i.e. the accessible values of the coordinates Qh-"--..!...!.L.9n'!- The set of all such wave func­ tions with unit norm span a function space or Hilbert spac_e_�,jIUhe �hich we discussed in detail in Chapter I. Every possible state of the system is represented by a function in this Hilbert space, and, conversely, every vector in this space represents a possible state of the qll, the system. In addition to depending on the coordinates qll wave function depends also on the time I, but the dependence on the gs and on t are essentially different. The Hilbert space � is formed with respect to the spatial coordinates qh gil only, e.g. the inner products are formed with respect to the g's only, and one wave function g,,) states its complete spatial dependence. On the other 'P(ql l hand the states of the system at different instants of time 11 1 12, are given by different wave functions 'PII(qh . . . q,,) , 'P/l (qll of gn), the Hilbert space. This mode of description which treats the time t as a parameter on an essentially different footing from the position coordinates results from the non-relativistic way in which we are formulating quantum mechanics. The normalization of the state vectors '1'1 is closely connected with the probability interpretation of the wave function. This normalization, however, still leaves undetermin�.Jlfuc.tor ofmodulus unity in the waY.e functiQU.. If a is real, then lIeia'P/li = 11'1',11. We shall see that such a factor of modulus ynity in the wave function does not affect the p�_ dictions of observable_physi�l _pr()perties--, i.l._oL�pecimenYi .-:L� •

.

•

•

.

•

.

•

.

•

•

•

•

•

•

•

•

•

the choicc_a(a-pbase fact01"-�th�wave1imctionJs_arbitrary�utjD­

a quantum mechanical calculation involving different wave functions one must of course 11."( these phase factors in a consistent manner, as they determine the inj�nerence betw_e_en _thC!Se_,",-avcs,,_ It is one of the most basic assumptions of quantum theory that a complete knowledge of the wave function represents the maximum in­ formation of the system that is possible. Any information that one

M _______ _

60

16.

INTERPRETATIVE POSTULATES OF Q.UANTUM MECHANICS

can give about the system can be derived from its wave function, and no experiments can give more information than can be represented by a wave function. In this_statement is ah-eady implied, as_we shalLsee _ latcr._the inherent inability of quantum mechanics to specify a system ilS fully as is the case in�Jassic(t1 theory. We shall see presently how one derives information about the system from its wave function. Here we only stress the severe limitation that there exists no way of obtaining additional information about the system beyond that which can be derived from the wave function. We summarize these results in :

Po-rtulate I. A system with n delQ:ees of freedom is completely specified by a normed state-V�'PI(qh qg) which contains an arbltrarylactOr�o.f m���lu_s--uni� ffi�p�sil?�e. info!I!l�tion . abO\!t the syst�!!!. canJ:l�deriv�d frQ.m thi!; \V.aye function. •

.

•

The state of a system, specified by a vector in Hilbert space, repre­ sents a theoretical abstraction. One cannot measure the state directly in any way. What one does do is to measure certain physical quanti­ ties such as energies, momenta, etc. Following Dirac, we shall call these observables. From these observations one then has to infer the state of the system. Qbservables_�r_e ,r_ep-resented in guantum mechanics by operators. MMhematically, an operator_A_definesAttansfur:mah tio.nin e functWlL sp-acc des�ribing the sys!em, which transf.9J'msJLIl,LWilVe_flm_�tiQJJ..JPjnJo_ �nQther_ wave function x :

_

We can think of A as defining a correspondence between the two wave functi<:ms-;iii a rule, for �ocia!1ng a ' vector X 'WIth -the-vector �' In Chapter II we had some examples of operators. We defined the operator X'I> as ' multiplication by x' and X'I> operating on 'P(x) gives X(x) = x'P(x) , Or we can define the operator A as ' differentiation with respect to x ' ; in that case a x(x) =A'P(x) = 'P(x). ax The rule defining the correspondence between the object function 'P and the image function X need not of course be as simple as in these examples, but it is always useful to interpret it geometrically : 'P and X are two vectors in Hilbert space, and A transforms, i.e. rotates and magnifies, 'P to become identical with X. The operators of quantum mechanics are linear, i.e. for any two complex numbers Cl and C2 A(Cl'P l + Cl'Pl) = c1AV'1 + C2AV'Z'

This property of being linear is a most important one which is con­ stantly used. 61

16.

INTERPRETATive POSTULATES OF

Q.UA!\lUIII MECHANICS

In Chapter II we considered the operators r�� and P�p corresponding to the classical obscrvablcs position vector r and momentum veClOr p. We saw that

P


r.�=r. so that

x(r) -r��(r) �r�(r) rind

x(r) _p" p(r) - - ;n gcad p(r),

and that the classical observable

F(xl> . . .

';;;,v;

.

p""

. . /\vJ

repre­

sented by the operator F(xl> . . . ';;;,v; - ilia/a.'!;!> . . . - ilia/a,;;;",). One can show that if one uses generalized coordinates qi. (i ... 1, . . . 1/) , the operators representing the generalized momenta /J;, equation (15.3), are - ilia/agi• and the classical observable F(qI J q�; PI> . . . Pn) is n:presentcd by the operator F(qJt . . . qn ; - ilW/aqh " - if�a/aqn)' .

•

.

This is a natural generalization of our results for Cartesian coordinates. We shall say that p is an eigenvector of with a as eigenvalue, if

-:4p _ap.

A

We shaH assume that operators corr�onding to ohservables possess a " and corresponding complete set of eigenvectors Ph 'h• . . . CfJn discrete eigenvalues ---.!!J , a1, 0", orderc�so that Re at
.

•

•

•

.

•

•

(2)

A�, - a,�,

_

and a!lY2t:t.le tp with finite norm can be expanded in a linear super­ position of these eigenstates of A •

( 3)

p - Lc,p,. .-,

rn addition to these operators with completely discrete spectra, one also requires operators with continuous spectra to represent observables. For example, the position vector of a particle has a continuous spectrum. In section we shall give the necessary generalizations to deal with continuous spectra. But the physical interpretation is not much affected by this change. Consequently we shall restrict oursclves to operators with discrete spectra to keep the mathematics as simple as possible · . To complete the definition of operators whidl,.. Lenresellt observables. we jmposej.he restriction that they areJ-fermilia[L or selfadjoillt, A is a is Hermitia ! l operator if for any vector , the scalar reduct (If', rea f

17

Aif')

T,'l.l.l

• ,Vhcrc the nalnre of the spectnun is not important, continuous .pectra :IS illuJlr:uive examples.

62

(4)

we shall also

use opcraton with

_

1 6.

INTERPRETATIVE POSTULATES OF Q.UANTUM MECHANICS

We shall see below that (11', A tp) represents the expectation value of the observable A for a system in a state '1', and this is the reason we restrict ourselves to Hermitian operators. We can see this differently by taking one of the eigenfunctions of A, say CPR' for 'I' in equation (4) . It then becomes a. = a,., (5) i.e. the eigenvalues of a Hermitian operator are real, and we know from Chapter II that the eigenvalue an is the value of the observable A in the state CPA' Since equation (4) is to hold for any vector '1', take lp = 'l Xl + '2X2, where '1 and '2 are arbitrary complex numbers and X l and X2 are arbitrary vectors. Then (' , '¥-. " ,' , ' 2 2 (11', Atp) = 1 '1 1 (xh A XI) + IC21 (x2' AX2) + {CIC2(X2, AXl) + Cl '2{Xh AX2)} and this is real for all values of '1 and '2 if, and only if,

(6) ( X2, AXI) = (AX2' X l )' Equation (6) represents a very important property of a Hermitian operator and may be taken for its definition. Examples of self-adjoint operators are the position and momentum coordinates. In the one-dimensional case one has

(X, Xtp) =

J: XX'I)(Jx = J:=(xX)tpdx = (XX, '1'), ..

x is real, and (X. P.�) � [x( - ili:x'P) dx� [ - iA%{ �. L(iIi�X)VHiX � (P.X' �). since [ mxtp] : =vanishes, as and X tend to zero when x ± since

-

'I'

_

00.

J)ne_ c_a�.,easi1Y_l:lrove that eigenfunctions belonging to different e!g�nvalues oCaJlerm�tia.n_�erat�r a!� OI-:thogonal; for

0 = (CPm, Acp,,) - (Acpm, CPR) = (a" - am) ('Pm, cp,,), and if a" :;:bam, then, (CPm, cp,,) = O. If a" = a"., i.e. for degenerate eigenfunctions, we can always choose mutually orthogonal linear combinations of eigenfunctions, using the Schmidt orthogonalization process (section 3). For the particular case of the energy operator we already proved the reality of the eigen­ values and the orthogonality of the eigenfunctions in section 7. We state our general results about observables and operators in : Pll1l1l1alLII. To every observable corresp-onds a Hermitian operator with a complete set of orthonormal eigenfu_n_c_tio_lls . .

3*

63

/.

' .

.1 -

16.

INTERPRETATIVE POSTULATES OF Q.UANTUM

MECHANICS

SO far we have only considered one instant of time. variation of a system is given by

The temporal

Postulate III. For every system there exists a Hermitian operator H, the Hamiltonian 0Rer�tor, which determines the_tim�.yl!ti�tion_of ��tem through the time-dependcnt SchrOding� e_quation

o"a'l'

til

at = n'P, u

(7)

provided the syste� iJ; !1o�fiish!!.becl;)lle��r�.m�nt . represents such a dis_t.YJJ>�nce. _

___

We have already discussed the time-dependent Schrodinger equation in Chapter II. It is the equation of motion of quantum mechanics. Like the equations of motion of classical mechanics it is completely deterministic : given the wave function at some time t = to, equation (7) uniquely determines the wave function V', at any other time t. Equa­ tion (7) only applies as long as the Hamiltonian H describes the system as a whole, i.e. the Hamiltonian must include any C external ' inter­ actions with the system. If, for example, we consider an atom in an external magnetic field, the Hamiltonian must contain a term repre­ senting the interaction of the atom with the magnetic field. In other words, like the classical equations, equation (7�or closed systems, i.e. J9.r...mtems_ QoJ_dis!ll.-.b..edJmIlLQutside. According to classical physics, we can carry out measurements on the system which interfere to an arbitrarily small extent with it. According to quantum theory this is not the case. The interactions between the system and the measuring apparatus cannot be made arbitrarily weak but there exists a definite finite lower limit. As a result, a measurement in general represents a disturbance of the system and its temporal develop­ ment is no longer given by equation (7). If '1'0 is the wave function at the initial time to =0, we can integrate equation (7) to give the solution ¥. - exp

{ -iLH�'!¥�.

(8)

We have not yet defined what we mean by the rather complicated operator

(9 ) operating on 'Po and we shall do this in section 18 dealing with functions of operators. But it follows from equation (7) that, geometrically speaking, for a given initial vector 11'0, 11', traces out a completely defined trajectory in function space. The operator S, relates the vectors

64

-------

16.

INTERPRETATIVE POSTULATES OF Q.UANTUM MECHANICS

associated with the initial Foint t �O _and _an_arbitrary poinLt on the _ �ajectory. In section 1 8 we shall also show that S, has the property that ( 1 0) (S,'I'O, S,'I'O) = ('1'0' '1'0) I .e.

(V'" '1'1) = ('1'0, '1'0) ; (11) in other words, the normalization of the wave function persists throu� out time, a result which we hacl aJ���clY establisjled i_� section 6. The general solution, equation (8) , of the Schrodinger equation simplifies if the Hamiltonian function H does not depend on the time explicitly ( oH/ol = O) ; it then reduces to ( 1 2) with T, e -iHl/lI• ( 1 3) =

We have introduced wave functions as representing states and operators as representing the physical observables of systems. We must now see how these quantities are related to actual measurements. Given a system in a definite state, described by a given wave function, how can we predict its physical properties, which are to be compared with experiments ? Conversely, how do we incorporate information about a system, gained as a result of measurements, into the wave function ? How, for example, do we determine the initial state '1'0 of a system which we must know to be able to integrate the wave equa­ tion (7) ? For this analysis, and in the rest of this chapter, we shall use Gothic letters to denote observables when we want to stress the distinction between the observables and the corresponding operators. In the first place, we consider the measurement of a single observable �. Only in sections 1 9 and 20 shall we consider the measurement of several observables. We shall again assume that the operator A, corre­ sponding to this observable �, has a complete normed orthogonal sequence of eigenfunctions rph rp2, rpn, and corresponding eigen­ values al
•

•

•

•

•

'"

(14) V' =n2/nrpn �1 where Cn = (rpn, '1'), i.e. the state 'I' is a linear superposition of the different eigenstates rp" rp2, , in each of which � would have a perfectly definite value ah a2, respectively. Measuring a on a system in a state V' no longer leads to a definite result ; rather there exists a proba­ bility distribution to find any one of the valuf's a" 02, This probability distribution is given by •

•

•

•

•

•

•

65

•

•

16.

INTEIU'RETATIVI!. l'OSTUJ.ATES OF QUANTUM MP.CIIA:-IICS

Postulate IV. If g is measurec:Lon a. s _l�m in a state :'Pol e ua­ tion (If)�prQbability. of findin for g a value a 1 IIlg 111 the interval a'
Postulate ]V contains the basic rule ror predicting the result or measurements. Its most important implication is that, in contradis­ linction to classical mechanics, a system willi the m:lxlmal specification of ils SL"l.te, namely with n given W:lve function, still shows dispersion: measuring g docs not neccssnrily lend to one definite result. Rnlher, there exists only a probability distribution, which can be calculated, of finding a particular result. This means that if we measured g on a large number of systems, all in the state 'P, then the different possible results Gh a2• • • • would occur with a frequency given by Postulate IV. Thus the diffraction pattern of an electron scattering experiment corre­ sponds to the probabilities of the electron being scattered into different final states with different momentum vectors. One can obtain such a p:Htern 'instantaneously by scauering a shower of electrons, or by scattering successive individu:lI electrons and recording these events on a photographic plate. We obtain a definite result ir the system is in an_eigenstate p.. of II before the measurement. In that case there is a probability I of observing the value an and a of getting any other result, as follows at once rrom (15). Conversely, if we know that the system is in a state 1p such that on measuring � one certainly finds a, then a must be an eigcnvalue of 11 and 1p a corresponding eigenfunction: I1VJ=mp, i.c. 1p is of the form 1p = 1:C.T.. thc summation being only over those values of 11 �for which a.=a. From (15) it follows that one can carry out measure.p ments of absolute .leeuracy in quantum mechanics, provided II has a discrete spectrum. For if the interval (a', a·) is sufficiently small so that it contains one cigenv:llue a. only, a'
y

\'t v' 0 Pl..p l'

-=

P( - <Xl.

<Xl )

-

•

.z:k.I' - I .

.-,

( 16)

i.t. the probabilities in equation (15) arc correctly normalized. Since g docs not necessarily have n dcfmite value, we arc interested in its statistics. We define its expectation val� in the state ...Y'....11L. •

(A) - .zJ.I 'a., .-,

66

( 1 7)

16.

INTERPRETATIVE POSTULATES OF Q.UANTUM MECHANICS

{4)_ re�resentL!he average value which o_�� finds on repeatedly m�asurin�, always. sJ�rting wit!!J!.!Ystem in the state tp. An alter­ native way of writing ( 1 7) is
( 1 8)

and we see from equations ( 1 7) or ( 1 8) that (A) is real, as required. If 'P = 9'RJ equation ( 1 8) gives (A) = an' In addition to the expectation value we are interested in the mean square deviation LlA which is a measure of the dispersion around the mean value (A). We defi n e (LlA)2 as the expectation value of [A - (A)] 2 in the state 'P, (LlA) 2 = ([A - (A)F) = ('P, [A - (A)F'P)

so that

= ('P, [A2 _ 2(A)A + (A)2]'P) = ( 'P, A2'P) _ (A)2 ( 19)

where, on account of equation (4) , (A2) = ('P, A �) = ('P, A(A 'P) ) = (A'P, A 'P) = IIA'P1I2.

The condition that the mean square deviation vanishes, LfA = 0 , implies [A - (A)] 'P = 0, i.e. A", = (A)"" i.e.
'Pr (r) = R(r) rr ( O, 9' ) ,

",(r) = R(r) ' } L:rr ( O, 9' ) Y 2 l + I m--I respectively, where we assume _

f:IR(r) 12r2dr = I

( 14.4) (20)

so that ",r and 'P are normed to unity. Both ",r and 'P are eigenstates of the operator 12 giving the square of the magnitude of the total orbital angular momentum of the electron [equation ( 1 4.6)] , hence in both states 'Pr and '" the angular momentum of the electron has the value "y {l(l + I )} . 67

16. INTERI'RETATIVE POSTULATES OF QUANTUM MECHANICS

V'{" is an eigenfunction OUt with eigenvalue lUll [equation ( l kS)]: '

ItIPt" -nm'Pr. ( 1 4.5) In Ihis case we have not only
(if Exercise

IVA)

(/d � O

(21 )

(�/.l' - (I;) _ih21(1 + I). (22) = For the probability of finding lim (m - I, . . . + I) for the valuc of It in the state 'P. onc has, from equation (l5), 1 P(hm) - 1 + 1 (m � -I, . . . I). (23) 2

\'\ie C;1I1 interpret 'Pr, somewhat classically, as a slate in which the angular momcntum vector 1, of magnitude 1iv'{I(I + I)}, makes a dcfinite angle wiLh the z-axis, where

a

(24)

{/(l

Wc might have c.xpected the ma:ximum value of I, to be liv + I)}, corresponding to i.e. the momentum vector 1 pointing i n the ximum value of is Iii, corre­ direction of the z-axis. That the ma. sponding to cos I)} , is closely related to the uncertainty principle. According to this, lhe exact valucs of I. and '7 cannot be specified simuilaneously with 1<. however, means that I" and 17 have the exact values zero. \,Ve shall return to this point in section 20. On the other hand, we c,'m interpret " as a state in which the angular momentum vector 1 has random orient'ation. It describes a spherically symmetric system in which no direction is preferred. This follows from cquations (21) to (23) ; equations (21) and (22), in particular, follow from the r.'lct that for a spherically symmctric system (I;> = 0, and

a c::: O,

It

a - v {l/(l+

a=O,

(I.') -(I,') -(I,') �t;, '/(1 + I ) , 2 since 1 has the value 1121(1+ I). Equation (23) c.xpresses the fact that all values arm, i.t. all orientalions of me vector 1 are equally probable; they arc independcnt of III .

\Ve come to the converse question of how information obtained about a system by measurements is incorporated in thc wave function. Suppose a ain that the observable g is measlIt.c
68

16.

INTERPRETATIVE POSTULATES OF Q.UANTml MECHANICS

for a sufficiently short interval of time between the two measurements. For if the interval is too long the system will have changed appreciably merely as a consequence of the time variation of the system according to Schrodinger's equation (7), and the value of � will also have changed. Only if � is a constant of the motion (ifsection 19) does it retain the value a found in the first measurement. This situation is entirely analogous to that in classical physics where an observable, say a position co­ ordinate q, is a function of the time, q = q(t), its variation being deter­ mined by the equations of motion. It follows from Postulate IY that if the second measurement is to have the certain result a, then the wave function tp', re resenting the s stem Imme late ter e first measurement, must e an el en­ function of A belon n to e elgenva ue a. a 15 a non-degenerate eigenvalue, this determines t e wave tion tp' completely. If a is v-fold degenerate, we only know that tp' lies in a v-dimeIlSional sub­ space, spanned by v linearly independent eigenfunctions belonging to the eigenvalue a (we shall take these as the normed orthogonal sequence . . tp, 15 9',+ It 9',+2, 9',+., correspond·109 to a,+I = a,+ 2 = . . . a,+_ = a) , I.e. of the form •

•

•

(25)

but the coefficients Cn are in general unknown. (The sum in the denominator ensures that tp' is correctly normed : IItp'lI = I .) However, if the initial wave function tp is known, we shall postulate that the wave function tp' after measuring � is also known, the coefficients Cn in (25) being given by (26) cn = (9'n, tp) (n = r + I, . . . r +v) , [and all other cn = O, as implied in (25)]. In other words, the expansion coefficients of 9',+ h 9',+ - are the same in tp and 'P" The reasons for this assumption will become apparent in section 19 dealing with simultaneous measurements of several observables. It follows from (25) and (26) that if 'P is an eigenfunction belonging to the eigenvalue a, then tp' = 'P, i.e. the wave function is unchanged by the measurement. We generalize these results somewhat in •

•

•

Postulate V. If the result or measutjgg_!l �o�_�_�}'st�m is a value a lyingjIl a ' �(L<�J .theIl, immediately after: measllfemeI1.t,JheEstem is represented by a wave function tp-'--of the_form _

,

'P =

2: cn9'n ' J{2:lcRI 2} 69

(2 7)

17.

OPERATORS WITII CONTINUOUS SPECTRA

th.U1llJlmruIDJl.sJldng_Jlll{y oxc.uhc...Yalucs of n for wbjcJL tL�a�a':. of the but the coefficients c. arc not known._ lLthUJlitial state s�lem, bcf Qtc _ tlu,t mcasurcment, is known, then V" is also completely known, the coefficients c. in

(27)

being the expansion coefficients of

(28) ])ostulatc V states how one incorporatcs information about a system, obtained as a result of experiments, into lhe wavc function describing that system. In general, such a measurement may lead to a variety of results and the wave function of the system must allow for all these. But once the experimcnt has been carried out and one particular rcsult obtaincd, wc can 'discard' most of the wave function, corresponding to aU the results which were not obscrved, and retain only that part which on immediate remeasuremcnt would lead to

thc same

result.

Thus Postulate V partly answers the question how we determine the state ofa systcm,

i.e.

its

state vector.

This dClcnnination is only com·

plete if the eigenvalue measured is non-degenerate; ill gencral i t will be degenerate and further measurements arc required to determine the state completcly.

Wc shall return to this problem in section

19

dealing with the simultaneous mcasuremCnl of several observablcs.

17.

Operators with continuous spectra

J n section :J. we dealt with the mathematical aspccts of operators with

\Ve now wanl to sec i n whal way

continuous spectra of eigenvalucs.

the physical interpretation given for operators with discrete spectra i n

1 6 has t o b e modified t o denl with the continuous case. A has a completely continuous spcctrum or eigenvalues fl. and corresponding eigenfunctions 9'(", x), the parameter

scction

Suppose the operator

I'

-

labeUing these eigenfunctions assuming all values in thc interval

00<1'< 00.

,"Ve have written .' as argument rather than as suffix of

the eigenfunctions to stress its cominuous nature as distinct from thc

suffx n ( .". I, 2, ) which occurred in the discrcte case. A gencral i stale 'l(.\') of the system will no longer be given by an infinitc serics,

3, . . .

equation

(16.1:�),

but by an intcgral

� (.,)

-

f.'(")�(v, .,)d.,.

(1)

For a system described by ( I ) we would expect as the rcs ult of measuring

m an)' one ofthc valucs fl.) which now form a continuum.

in the probability Postulate IV, equation P(o., a,) Equation

(2)

-

Accordingly,

(16.15) becomes replaced

f>(")12d.,.

by

(2)

gives the probability or observing a value a, lying i n the

illlcrval all
70

For

18.

FUNCTIONS OF OPERATORS

the expectation value of � we obtain, analogously to equation ( 1 6 . 1 7)

(A) =

J�",aplc(v) 1 2dv.

(3)

We see that there exists a close similarity between the discrete and continuous cases. To go from the discrete to the continuous case we replace summations over the integers n by integration over the con­ tinuous variable v. There exists however one principal difference. If A has a discrete spectrum we can measure it with absolute accuracy, as discussed in section 1 6. If A has a continuous spectrum this is not possible. We see this at once from equation (2) . An infinitely sharp measurement implies a = p, i.e. zero probability. So all we can do in the case of a continuous sp�ctrum is to measure � with arbitrarily high precision by makingJ[J - a) sufficiently small, but we cannot measure ifWIlli absolute--­ I!!ecisiQIl_,-_ Apart from -tms dffference;-tlle-interpret
_ ___

P(q. , ql + dql ;

qa, q" + dq,,) = 1 1J' (qh

gil) 1 2dql

dq" (4) is the probability of finding a system specified by the wave function tp(q., qa) in a state where the coordinate qi lies in the range i, n. Hence the expectation value (qi) is i + (q q dqi) , i = 1 , •

•

•

•

•

•

.

•

;

•

•

•

•

•

•

•

•

•

I

(gj) = qd1J' (qh

•

•

•

g,,) 1 2dg1

•

dq",

(5)

the integration being over the fundamental domains of the coordinate, qh . g.. Comparing equations (2) and (3) with (4) and (5) re­ spectively, we see that the probability interpretation of 11J' 1 2dg1 dg, is a special case of equation (2). .

.

•

•

•

18.

Functions of operators We now want to consider some of the properties of linear operators which are particularly important in quantum mechanics.

(a) Functions of operators For two operators A and B we define the operator (aA + bB) by

(I) (aA + hB) 1J' = aAtp + hB1J' for aU complex numbers a and h. The product (AB) is defined as the operation of successively applying first B and then A, i.e. (AB) tp = A (B1J') .

In general

71

18.

FUNCTIONS OF OPERATORS

In this case we say that A and B do not commute : the commutator [AJ B], defined by [A, B] =AB - BA, (2) does not vanish. This non-commutability of operators is closely related to the uncertainty principle. (Cf section 20.) If B =A, we can successively define powers of A, namely AZ,A 3 , , and using ( I ) we can then define polynomial operators (3) PN(A ) = co + c1A + . . . + cNAN where Co, Ch • • • CN are complex numbers. If we let N -+ co, we can define a general function of an operator f(A) . We had examples of such functions of operators in equations ( 16.9) or ( 16. 1 3) . If rp. is an eigenfunction of A , corresponding to the eigenvalue an, i.e. Arp" = a,.rpn, then (4) and, generally, two functionsf(A) and g (A) always commute [f(A), g(A)] = 0 .

•

.

and have a common set of eigenfunctions rp h rpz, 0 0 • rpn, • • • and the eigenvalues f(al ) ,f(az) , . . . and g(al ), g(az) , • • • respectively. All these results follow by considering an individual term in the formal power series expansions off(A) and g(A ) : f(A) =co + c1 A + • • • + cNAN + " 0, etc. Thus (4) follows from ANp,, = AN-l (Arpn) = AN- l (anrpn) = a.(AN- lrpn) = . . .

=

(an) N9'n.

(b) Acijoint operators In section 16 we defined a Hermitian or self-adjoint operator A by one of the equations ( 16.4), ( 1 6.5) or ( 1 6.6) . Let us take ( 16.6) as defini­ tion : for all "I' and X (5) ( x, A tp) = (A X, "1'). In general an operator will not be Hermitian or self-adjoint. Ifwe can define an operator A* such that (6) ( X, Atp) = (A*X, "1'), we call A* the acijoint operator to A. IfA is self-adjoint, then A* = A. (7) It follows at once that (A*) * =A, i.e. the adjoint of A* is A, and furthermore that {A + B) * =A* + B*, (8) (aA) * = t1A*, and that (9) (AB) * = B*A*.

}

72

18.

FUNCTIONS OF OPERATORS

These relations all follow at once from (6), e.g. one gets equation (9) : ( X, ABtp) = (A* X, Btp) = (B*A * X, tp) . If A and B are self-adjoint, then (9) gives (AB) * = BA. (1 0) If in addition A and B commute, [A, B] = 0, then (AB) is self-adjoint as well ; from (10) (AB) * =AB. (11) IfJ(x) is a complex function :ll , then the adjoint ofJ(A) is and ifJ is a real function

[J(A)] * =j(A*)

( 1 2)

[J(A)]* =J(A*) . ( 1 3) If in addition A is Hermitian, then J(A) is Hermitian too, as one sees at once from equation ( 1 3) . Equation ( 1 2) follows by considering a typical term in the expansion ofJ(A) and repeatedly using equations (8) and (9), with B = A. (c) Unitary and inverse operators So far we have considered Hermitian operators which correspond to the observables. We now want to consider a different type of operator which plays a part analogous to the coordinate transformations of classical physics and is consequently very important. We define the unitary operator U by the conditions UU* = U*U = I (14) where U* is the adjoint of U and I is the identity or unit operator, which leaves any vector tp unchanged, Itp =tp. From ( 14) it follows that for any vectors tp and X ( X, U*Utp) = ( X, tp) , and hence ( UX, Utp) = ( X, tp), (15) i.e. a unitary transformation leaves the inner product of any two vectors invariant. This is the most important property of unitary operators and one can use it to define unitary operators. It follows in particular from ( 1 5) that (1 6) i.e. unitary transformations preserve the length of a vector, and if 'I' and X are orthogonal, then ( UX, Utp) = ( X , tp) = 0, ( 1 7) • Iff(x) is a complex function, it can be written f(x) =u(x) + iv(x) , where u(x) and II(X) are real functions. The complex conjugate function is defined asl(x) =u(A') - ill(x). Iff(x) is :1 real function, then l(A'} =f(x}.

73

i.e.

19.

SIMULTANEOUS

MI!,\SURAIIII.ITY

Of.

OllSERVAllLRS

the transformed vectors are still orthogonal. \Ve see that unitary transformations have properties simil:u to rotations in ordinary space. They are in f:'lct the corresponding generalizations to function space. H A is Hermitian and J a real funcuon, then U =exp [if(A)] is a unitary operator. For from (12) it follows that U* = exp [ - if(A)], and hence eqnations (14) and (15) are satisfied identically. [Equa­ tion (15) becomes

lexp [!fIA) Jx. exp[if(A»)v,) - lexp[ - !fIA)] exp [!fIIl)]X. ¥) � Ix. '1').] We see now that the operators S, and T, introduced in (16.9) and (16.13) are unitary operators and equation (16.10) is only a special case of (15). l�rom equation (H) i t follows that if 11' = Urp, then rp = U*'/I, i.e. for a unitary operator U the adjoint U* has the property of being in a sense an inverse operator to U. This suggests that for any operator II we should try to define an inverse operator which we shall denote by A-I and which has the property that AA-I =A-IA =1 (18) so that 'P=Arp implies


(19)

SitnuItaneous measurability of observablcs So far we have dealt with the measurement of only one observable A . This does not in general specify a system completely and w e must now consider the measurement of several observablcs. According to classical mechanics, we can determine the state of a system with 1/ degrees of freedom at time t by measuring the 211 position and momentum coordinates q,(t), p,(t), i= I , . . . 11. We can think of these measurements as either simultaneous or as carried out in very rapid succession so that the coordinates have not varied appreciably, on account of their temporal development according to the equations ofmotion, during the interval ofume required for all the measurements. But it is inherent in the concepts of classical physics tbat the measure­ ments do not affect the values of the variables describing the system. Thus, lhe order of measuring the 211 coordinates is immaterial and remeasuring any one of them sufficiently quickly necessarily leads to the same result. Furthermore, aU conccivable observablcs arc func­ tions of lhe 211 generalized coordinates ; from these, any other quantity can be calculated, and specifying the system by 2n diITerent generalized coordinates, to begin with, docs not lead to any new or essentially din-erent results. In quantum mechanics the situation is entirely diITerent. It is no longer possible to specify the simultancous values of all obscrvables ofa

74

19.

SIMULTANEOUS MEASURABILITY OF OBSERVABLES

system. In general. observables are mutually incompatible, measu�ng op� 9Lth.enLp�r.tlY�QLc9rnpl«;1dy_od�Jr ys any: knowledge about the others. If two observables A and B possess a mutual eigenfunction rp, i.e. Arp arp and Brp = brp, then, in the state rp, A has the value a and B the value b, and it follows from Postulates IV and V (section 1 6) that successive measurements (performed sufficiently quickly for the time variation of the system to be unimportant) would give these results. However, if A and B have no common eigenfunctions, successive alternate measurements of A and B, which necessarily leave the system in eigenstates of A and B respectively, destroy the information obtained earlier. We shall deal with such incompatible observables in section 20. They are closely related to the uncertainty principle. In this section we consider under what conditions one can ascribe simultaneous values to several observables. To begin with, we again consider two operators A and B with com­ pletely discrete spectra of eigenvalues. Let al
•

.

.

•

•

•

Difinition 1. IfA and B are measured on a system, in quick suc­ c�ssion, with .the res�!�.!lr_and�sLand an immediate remeasureme_nt of A or B necessarily. reproduces these re��lts, whatever the initial state of the vstem, then we say tl:!at .A and II are c.omjJtljiJde. or simultaneouslY measurable.

From the postulates of section 1 6 the following theorem can be deduced which is of fundamental importance. Theorem 1. A necessary a.l1d_mfficient cpndition for A and B to be compatible is fot:.1�.!!! !!> POSs�!s_.a _c9I!1pl�tUl�rll!ec! 9rth9g��.�t�et or. . simultaneQ.us eigenfunctions. TafThe condition is necessary, i.e. if A and B are compatible, there exists such a set. Suppose the system is initially in a state tp. Mter a measurement of A, with result 0" it is in an eigenstate e, of A belonging to the eigenvalue a,. Expanding e, in terms of the eigenfunctions of B, we have __

(1)

where 'Y}s is a normed eigenfunction of B belonging to the eigenvalue b" (In general, b, will be degenerate so that there exist several linearly independent eigenfunctions belonging to this eigenvalue b" But the particular linear combination CS'Y/I occurring in ( 1 ) is of course uniquely determined by �,.) A measurement of B on the system in state ( 1 ) (i.e. we measure B immediately after we have measured A ) with the 75

1 9.

SIMULTANEOUS

MEASURABILITY

OF OBSERVABLES

result b. leaves the system in the state Y/.. But since A and B are com­ patible, a second measurement of A must necessarily give a, again, i.e. Y/, is a simultaneous eigenstate of A and B. Hence any eigenstate e, of A can be expanded in a series of mutual eigenfunctions of A and B. Since any state tp of the system can be written as a superposition of eigenstates of A, it can also be expanded in terms of simultaneous eigen­ functions of A and B, i.e. the set of mutual eigenfunctions is complete. (b) The condition is sufficient, i.e. if there exists a complete set of mutual orthonormal eigenfunctions Tb T2, • • •, T", • • • then A and B are compatible. Since the set Th T2, • • •, T", • • • is complete, any state tp of the system can be expanded in terms of this set,

After measuring

A,

with result 0" the state of the system is

the summations being only over those values of n for which eigenfunction of A belonging to the eigenvalue 0, :

Tn is

an

ATn =a,Tn'

A subsequent immediate measurement of B, result b" leads to the state

2: C"Tn ' j{2: l c" l z} the summations now being onlY over those values ojnfor which T" is a mutual eigenfunction ojA and B belonging to the eigenvalues Dr and b. : ",' =

ATn = a,TII' BTn = b.Tn·

Hence remeasuring A gives 0" and A and B are compatible. Theorem 1 gives the generalization of Postulate IV (section several observables. values

ar

and

If'P =

for

2�.!.1II then the probabi�i!y of observing the �. and. �i�2Jn I 2, the summation

b. simultaneousltior

again beJDg_only_�,,-er_those _vcl1u eL2f

BTn = b,T/!.

1 6)

n

fOL whi�Il._ATn = gy!n and

_

Theorem 1 has a simple geometrical interpretation. The complete normed orthogonal set Th T2, • • . Til, . • • spans the Hilbert space �. We can subdivide the whole Hilbert space into mutually orthogonal subspaces (0" b,) in each of which A and B have definite eigenvalues a, and h, for all vectors belonging to them. Each of these subspaces is spanned by the mutual eigenvectors T" belonging to the eigenvalues a, and b" and their number specifies the dimensionality of the subspace.

IDl

76

19.

SIMULTANEOUS MEASURABILITY OF OBSERVABLES

We can write this decomposition as CD

.t> = L

CD

LIDl(a" b,),

(2)

,- 1 ,-1

If we know the results a, and b, of a measurement of A and B, and the corresponding manifold IDl(a" b,) is many-dimensional, the state 'P' after the measurement depends on the coefficients c" of the relevant Tit in the original wave function ",. On the other hand, if IDl(a" b,) is one-dimensional, 'P' = Tn' i.e. the state after measurement is independent of the coefficients Cn of the original wave function '1'. ( It is of course implied that c,. :;60, so that we can obtain the results a, and b, postulated.) In section 18 we saw that the two operators A and B do not, in general, commute. Correspondingly, we have :

Definition 2. flln�gons 1p

Two operators A and B commul�� for all wave

_

__

A _B�=BAI/J. .

(3)

_ _

If this equation is to be identically satisfied for all '1', it follows that the commutator vanishes identically, [A, B] = 0,

(4)

i.e. it transforms any wave function into zero. Two operators A and B commute if (3) holds for every member of a complete set of functions Til T2, TIIJ For any other function 'I' can be written as a linear superposition of these functions Til T2 , Tn' We can then prove the very important, purely mathematical •

•

•

•

•

•

•

•

•

•

•

•

•

Theorem 2. A necessary and sufficient condition for the operators A and B to commute,

(4)

_is_ the_ exist(!llc(!. OfJL C_()JllPkte norm�lized _orthogQnaLset .of mutuJll eigenfunctions, Th T2, T.., {aJ 'Ilie condition is necessary, i.e. (4) implies the existence of the set Ti t T2, T,., Take any eigenfunction Er of A belonging to the eigenvalue ar• As in equation ( 1 ), it can be written _

•

•

•

•

•

•

•

•

•

•

•

•

•

•

(I)

where t}, is a normed eigenfunction of B belonging to the eigenvalue b,. From (4) and section 1 8, it follows that for any function f, Af(B) =f(B) A, i.e. Af(B) " =f(B) A E, =f(B) a" " hence (A - a,)f(B) E, = O. 77

19.

SIMULTANEOUS MEASURABlLrn.' OF OnSERVAIlLES

From equations (1) and ( l 8. �) it follows that •

'

2: (A - .,)/(b') " 'I, -0, and sinccJis an arbitrary function, it follows thai

(A -.,) " , �O,

i.e. all cigenfunClions 1], occurring in (1) arc also eigenfllllclions of A belonging to the eigenvalue n,. The proof then follows exactly as in part (a) of Theorem I . (0) The condition is sufficient, i.e. the existence of a complete set TI, T2• • • • T•• • • • of mutual eigenfunctions implies ('�). If T. belongs to the eigenvalues a, and h, of A and B, then AI1T. -OJ/h. -=a,b,TM BAT. =a,BT. =a,b,T., and the commutability of A and B follows at once. We can now combine theorems I and 2 into the general

Theorem 3.

Of the three conditions:

(a) A and B arc compatible, (ft) A and B possess a complete normed orthogonal sequence of simul­ �ane.ouu:igenfunclions, (.,.) -,J and...B_ commutc, any one implies the otller two. Suppose Tis a Hermitian operator with the complete nOl'med ortho­ gonal set of eigenfunctions T., T1• • • • T., . and corresponding eigen­ values 11 <11 < ' " <1. < . . . From section 18 it follows that if we define two operators II and B which arc functions of T .

.

.

A -/(T). B �g(T). then A and B have the same complete set of eigenfunctions Th T1, . . . T., • . . and the eigenvalues J(tI).1(12), . . . , "nd g(ll ), g(l:!) , . . ., re­ spectively. 1t can conversely be proved th"t if two operators A "nd B commute, they arc functions of a third operator T, II-J(T) and B -g { T) . We shall not prove this converse. But we state it, "s it represents a particularly simple and lucid formul"lion of the above results. The above ideas can easily be generalized, firstly to any number of observablcs with discrete spectra, secondly to operators with con­ tinuous spectra. In the latter case the only change that occurs is that we no longer have measurements of absolute accuracy, but only of arbitrarily high accuracy, the situation being essenti"Uy the same as in section 17.

78

19.

SIMULTANEOUS MEASURABILITY OF OBSERVABLES

Consider, for example, the position and momentum coordinates (Xh X2, X3) and (Ph h, P3) of a particle. Since p, = - ilialax" we have at once that (if Exercise IV.B) but

[x" x,] = 0,

[prJ P,] = 0,

[X" p,] = iMm

(r, s = I, 2, 3 ),

(r, s = l , 2, 3) .

(Sa) (5b)

These equations are of course always to be understood as operator equations : the two sides have the same effect when applied to a wave function. In contrast to equations (5) the position and momentum coordinates of two different particles always commute. If (Xh X2, X3), (Ph P2) P3 ) are the position and momentum coordinates of a second particle, then [x" X,] = [PrJ P,] = [x" P,] = [p" X,] 0. =-

Consider the complete set of eigensolutions of a particle moving in a central field of force, equation ( 1 3.27) . V'E",I. I. m er,

0, 9') = Rn• 1 (r) rt( o , tp)

(6) 2 is a simultaneous eigenfunction of l�, 1 and the Hamiltonian operator H, the eigenvalues being lim, 1i21(t + I ) (equations ( 1 4.5) and ( 14.6» and En, /) determined from ( 1 2.7). It follows that these three operators commute, [H, t�] = 0 [H, 12] = 0 (7) and [12, lJ = 0 (8) which can easily be proved directly (if Exercise IV.9) . We can now say how the state of a system at a given instant of time is determined. As we have seen we cannot increase our knowledge of the state of a system by successive measurements of any arbitrary observables. Rather, having measured A, we may only measure a second observable B if it commutes with A, and so on. After measure­ ment of A, with the N(ar)-fold degenerate eigenvalue ar as result, the wave function of the system necessarily lies in the N(a,)-dimensional subspace 9n (a,} of Hilbert space, comprising all eigenfunctions of A belonging to ar• Mter a subsequent immediate measurement of an observable B which commutes with A, with result b" the wave function of the system lies in a subspace 9n (a" b,) which is a subspace contained in 9n(a,) , whose dimensionality N(a" b,) is necessarily not greater than N(a,) : N(a" b,)
I t comprises all simultaneous eigenfunctions of A and B belonging to a, and b" In this way we continue measuring more and more com­ muting observables A, B, C, At each step we reduce the 79

19.

SlMULTANEOUS MEASURABIUTY OP OBSERVABLES

dimensionality of the manifold within which the wave function may lie; the degeneracy of the state is reduced ; the number of possible ways of resolving the state by a further measurement is reduced. Finally we have a set of commuting operators A, B, Q, such that to any set of eigenvalues a" bIt q�, there exists exactlY one eigenfunction, N(a" b" q�) = 1 . This state cannot be resolved further by more measurements . We call such an observation maximal or complete, and the corresponding set of observables (or operators) a complete set of commuting observables (or operators) . To determine the state of a system we must carry out a maximal observation. Thus, in the above example of a particle in a central field, H, and I� constitute a com­ plete set of commuting observables, but not H and by themselves, for example. For after a measurement of H and P, with the results En, and A 2/(1 + 1 ) , there is still a (21 + I)-fold degeneracy of the wave func­ tion corresponding to the (21 + I) possible values of I�. From our derivation of a complete set of observables A, B, Q, it is clear that after a maximal observation we do not know the exact values of all possible observables. We cannot ascribe a definite value to any observable W which does not commute with all observables Q of the set. Only if W commutes with A, B, Qcan we A, B, ascribe a definite value to it. Any function W =f(A, B, Q) is an example of such an observable. But we do of course know the exact wave function of the system after a maximal observation, hence we can calculate the statistics (mean values, probability distributions etc) of a'!)' observable. It is furthermore clear that there is not one unique complete set of commuting observables. Thus, if we had taken an operator W for the first measurement, which does not commute with Q, we would have obtained a every operator of the set A, B, different complete set of commuting observables. For the particle in a central field, H, and I� form a complete set of observables. Since Ix and I, do not commute with 1(, we cannot determine their values simultaneously with 1(. But H, 12, Ix or H, P, I, would be equally acceptable complete sets of commuting operators. We consider now the operators which commute with the Hamiltonian function of a system. We shall restrict ourselves to the case, sufficiently general for most problems, that the Hamiltonian operator H does not contain the time p.xplicitly (aH/at =O) . In this case the explicit solu­ tion of the SchrOdinger equation, giving the temporal development of the wave function, is [equations (16. 1 2) and ( 1 6. 1 3)] .

•

•

•

•

•

•

•

•

12

12

•

•

.

•

.

.

•

•

•

•

•

•

•

•

•

12

'1'1 =

(9) T,'I'o, where T, = exp ( - iHt/A) is a unitary operator. Suppose A is an operator which does not contain the time explicitly, and which com­ mutes with the Hamiltonian, [A, If] = 0. It follows from section 1 8 that A then also commutes with T, which is a function of H, [A, 7,] = 0, 80

1 9.

SIMULTANEOUS MEASURABILITY OF OBSERVABLES

and since T, is unitary so that T,- l = T,*, we have that A = T, *A T,.

(10)

If 'Po represents the state of the system at time 1 = 0, the expectation value of A at t = O is (1 1)

(1 2 )

( 1 3)
and

11', =

TlPo, then

Atpo = atpo

(14)

( 1 5) Atp, = atp" i.e. if the system is in an eigenstate of A belonging to the eigenvalue a at time 1 = 0, then, under the undisturbed temporal development, it remains in an eigenstate of A belonging to the same eigenvalue a. Thus A has a constant expectation value for this system, and if the system is in an eigenstate of A, then it remains in that eigenstate. Hence we call an operator A which does not contain the time ex­ plicitly and which commutes with the Hamiltonian a constant of the motion, and the above results are the quantum mechanical analogues of the classical conservation theorems. An important consequence of the above results is that, if we specify the system by a complete set of commuting operators containing the Hamiltonian H as one of its members, a complete observation gives us the values of these observables not only immediately after the measurement, but for all subsequent times, as long as the system is not disturbed. In the case of a particle in a central field of force, we see now that the magnitude of the angular momentum II I and its z-component l� are constants of the motion [equations (7)] as already mentioned in section 14, analogously to the classical conservation of angular momentum in this case. But unlike the classical case, we cannot say anything about I" and I;,. For though relations analogous to (7) hold, [H, I..] = [H, I;,] = 0,

as follows simply from symmetry arguments (the z-direction is in no way preferred), we cannot specify I.. and I, simultaneously with I" since these operators do not commute with 1(, and hence there is no sense in talking of their constancy. 81

20.

20.

UNCERTAINTY PRINCIPI.E

Uncertainty principle

We now come to consider twO operators A and B which do not com· mute. We know already that the corresponding observables arc not compatible. However, it may happen that A and B havc somc eigcn. functions in common and for these states one can ascribc definitc values to A and n, but they may not possess a complete sct of simultaneous eigcnfunctions. H '/' is a mutual eigenstatc, then il!p_mp. and B'P """ h'P. i.e. (AB - BA)'I''''''' O, i.e. 'P is nn eigenfunction of the operator [A, Bl, belonging to the eigenvalue o. This situation is illustrated by the angular momentum operators lA, IJ> I�. Tbese do nOI commute with each other, but the state 'Poo, cquation (14.4), is a simultaneous eigen· function or aU three operators with the common eigenvalue 0, and [l�, /,]'1'00 =0, etc. Ir A and B arc two sclf·adjoint operators which do not commute, and 'P is a normecl wave runetion of the system for which (I) [II, Blv,-ia�, then the mean square deviations [cquation (16.19)] AA and AB satisfy L1A L1B >t la l. (2) The operator [A, 8] is not Hermitian, since [A, B]*-(AB -BA) * _ BA -IIB- - [II, B]. Taking the adjoint operators of (I). we get - [A, B] - - i" i.e. (i_a; a is real. [This was ofcollrse our object in writing jn rather than a in (1).] To prove (2), we write a _A - (II), P - B - (B) where (A) = (V', A'l'), elc. One then gclS from ( I ) ap - p a = ia (3) since (A) and (8) arc ordinary numbers which commute with all operators. Consider the imaginary part of the inncr product (P'P. 0'1')' 2bn(Pv', a,) - (P�, a�) - ( a,p, P'p) - (�, [pa - aPl,') _ -la, (4) using equation (3) . Tnking moduli or (4), wc havc at once

lal - 2 1Im(P" a.)I <2 I (P'" a�) 1

<2I1p.ll lla�1I -2�A LIB, where the last step but one follows from Schwartz's inequality (if Exercise 1.7). 82

20.

UNCERTAINTY PRINCIPLE

Equation (2) is a statement of the uncertainV' principle first derived by Heisenberg. It expresses the fundamental limitation to the accuracy with which one may hope to know the simultaneous values of two in­ compatible observables. If one calculates the product of the ' un­ certainties ' Ll A LIB, for a given state tp, one finds that in general the inequality holds. Only under particularly favourable conditions does the equality hold. From equations ( 1 9.5b) we obtain an uncertainty relation between the position and momentum coordinates of a particle, Ii Llx, LlP' >2'

(r = I , 2, 3) .

(5)

Introducing the velocity vector v" we can write (5) Llx, LllI, >

Ii 2m'

(6)

where m is the mass of the particle considered. Equation (6) shows why the uncertainty principle is not of importance for macroscopic bodies. Since Ii � 10-27 erg sec, Ii/2m is exceedingly small in this case. That we cannot know the position and momentum of a particle accurately at the same time, implies that any accurate position measure­ ment partly alters the momentum of the particle i n a manner which cannot be completely specified. One can theoretically analyse ex­ periments to confirm this and to verify (5), qualitatively at least. Plane HlC/tlenl lll1w. lt1OI11enia'l1 (P.ao)

y

- --T --ti----

Figure 7. Positioll measuremetlt of a platle wave of particles

_.t__ _

Heisenberg 1 6 has given a rigorous detailed quantum-mechanical analysis of such experiments. Here we shall limit ourselves to a simple example. We consider the position mcasurement ofa particle whose momentum (P, 0, 0) is known accurately. Such a particle is described by a plane wavc, i.e. its position is completely unknown. To determine the posi­ tion of the particle in the plane x = 0, we place a partition with a slit of width d in this plane, as indicated in Figure 7. Mter passing through 83

20.

UNCERTAINTY PRINCIPLE

the slit they-coordinate of the particle is known with an uncertainty equal to the width of the slit, LJy � d. (7) But in passing the slit the beam suffers diffraction, and hence its momentum, after going through the slit, is no longer (P, O, 0), but has a component in they-direction. The first minimum of the diffraction pattern occurs at an angle a such that d sin a = A.

(8)

Corresponding to this deflection, the particle has a y-component of momentum and hence there is an uncertainty in the y-component of momentum (corresponding to all angles of deflection between a and + a) 2" . A (9) DP 7 � 1 sm a. -

Combining equations (7), (8) and (9), we obtain LJyLJp, � 2h,

as verification of the uncertainty relation. It is clear that this sort of analysis is rather qualitative. For example, we could equally well have taken an angle a defined by d sin a = ),/2, corresponding to interference between the extreme rays at the two edges, to define the uncertainty in the momentum. We would then have obtained iJyiJp, � h. We obtain another example of the uncertainty relations by consider­ ing the orbital angular momentum of a particle. Using the commu­ tation relations ( 19.5), one easily shows that the angular momentum operators I", I, and I� satisfy the commutation relations (10) [I", I,] = ilil�

and two similar relations obtained by permuting x, y and .t cyclically (if Exercise IV.13). If we consider the state tpl", equation ( 14.4) , (10) gives [I", I,]tpl" = iIi2mtpl", and hence, from (2), the uncertainty relation

�

iJ I,.iJ 17 >1i 2 m l .

follows.

(11)

20 .

UNCERTAINTY PRINCIPLE

Another way of looking at this uncertainty relation is to say that in the state "Pr we do not know I" or I, precisely, but only that 1,,2 + 1,2 =12 - 1�2 has the exact value A 2(/2 + 1 - m2}. Thus, we know the angle a, equation ( 1 6.24), between the vector 1 and the z-axis but not the direction of the component v' (/,,2 + 1,2) in the (x,�'JI}-plane. We illus­ trate this by the vector diagram in Figure 8 showing the angular momentum vector in the case l = 5. This figure is axially symmetric about the z-axis and must be imagined rotated about the z-axis as indicated by the dotted circles. The azimuthal angle rp of the (I, z)­ plane with the (x, z}-plane is not known. I

I

"

:

... ... .. .... _- .. _ - - -

J

:I

Figur(8. Vector diagram of orbital angular momentum of a singlt particle for the case 1 = 5

I I I I I I I I I I

.. t.. ..

- - - - - ---_ ..

_-

"

... ... ... _ - -.. -

Since from ( 1 6. 1 9),

<1,, 2) = (/,,) 2 + (.£1/.) 2, and from symmetry considerations

.£1l" = .£1 l,, it follows from ( I I ) that the value of (1,,2 + 1,2) in the state "Pi'" is 1i2 (/2 + 1 - m2} > (.£1 I,,} 2 + (.£1 /J1) 2 >1i2 I m l.

Hence if m = l, i.e. in the state, "Pi, (/..2 + 1J'2) has the minimum value compatible with the uncertainty principle. I� has correspondingly its maximum value Iil, and on account of the uncertainty principle it cannot attain the value Av'/(l + 1 ) as discussed in connection with equation (16.24) . 85

20.

UNCERTAINTY PRINCIPLE

An uncertainty relation, similar to

(5),

cxists between energy and

time

LIt LIE �li.

( 12)

This relation cannot b e derived rigorously within the framework of nOIH'elativistic quantum mechanics which we have developed,

\Ve

have trcated the time on an entirely different footing from the other coordinates.

\'\Thereas

these

latter

were

described

by

operators

operating on wavc functions, thc time was always a parameLCr and as such an ordinary number. But equation

I-Iencc it commutes with all operators_

(2), of which (12)

is to be a special case, was derived for

non-commuting operators. That an uncertainty relation

(12)

is to be expected can be seen

formally by going over to a relativistic notation, In the special theory of rc1ativity one supplements the position vector (.\'10 .\'2, .'\:) ofa panicle by the ' time coordinate' .t'4 =ict, to form the ' four-vector' ('\'10 and similarly one forms the four-vector momentum coordinates Equation

( 1 2) is

relativistic case.

(p" Pz, h)

(PI , /h, h, P4)

.'\:2,

the energy component /'4

then the generalization of

(5)

'\'3, X4),

by adding to the

to include

r='�

=iEjc.

in the

Furthermore with this notation we can write equa­

tion (6.3) in the f01'1ll

" a P4 = - lrl-, ax,

similar to equation (6.1 a),

P

,

...

" a ax,

-lri-,

(r= I , 2,

3),

showing that X4 and /'4 satisfy the same commutation relations as Xr and I),

(r= 1 , 2,

3), i.c. we replace equations [X., x,] = 0

[p" p,) �O [" , p,) � i"6" ,,,,e ,I must now interpret equation

}

(12).

( 1 9.5)

by

(r, s � l"

"

4).

It implies a limitation on

the acccllracy with which the energy or a system can be measured in a finite time.

(LlE=O)

For an absolutely accurate determination of the energy

we must take infinitely long over the measurement.

If the

state we measure is stable, we can indeed determine its energy level very accurately, as we call take as long as we like over the measure­ ment. But suppose now the state is ullstable with a life time . (i.t. the transition probability pCI' unit Lime is y "", l/.). In this case the average time available for an energy measurement of the state is .; consequently there given by

( 1 2) ,

s i

an uncertainty

LIE in

the energy determination,

of (13) 86

2 0.

UNCERTAINTY PRINCIPLE

If we introduce the circular frequency 00, corresponding to the energy E= 1ioo, then ( 13) implies an uncertainty in the frequency Lfw � ". ( 1 4) In practice, energy levels ofsystems are often observed spectroscopically. Equation (14) then tells us that for a finite life time T, the spectral line has a finite width on account of the uncertainty principle. We have already met spectral lines of finite width in connection with wave trains of finite length (Exercise 1.6 and Chapter II), and we use the same method of analysis in the present case. This approach is

Figure 9. Energy distrihution of a non.statioMry statB

particularly useful in certain problems of nuclear physics where one considers unstable excited nuclear states. (This analysis leads to the Breit-Wigner formula.) The wave function "PI = "Poe-iEotIA represents a stationary state with energy Eo, its amplitude is constant in time, 1"P, 1 2 = "Po 2. If we consider a time-dependent wave function "PI = "Poe-YlI2e-iEotIIl, (15) this represents a ' decaying ' state, its intensity decreases exponentially, I "PI 1 2 = l "Po I 2e-),I, and T = 1 11' is a time characteristic for the rate of decay. Equation ( 1 5) does not represent one stationary state, but we can write it as a Fourier integral, i.e. as a superposition of stationary states; i.e. we write

I I

( 16) f:.,A (E)e-IEtIAdE, where A (E) is the amplitude of the stationary state with energy E in "PI =

(15). A (E) is given by Fourier's integral theorem, and the corre­ sponding intensity distribution is "Po 2 1 ( 1 7) 1 2 = l 2l (E - Eo) 2 + F2/4 4n

IA (E)

87

EXERCISES I'\'

where r_'q.

This energy spcctrum is shown in Figllre

half.width of the level,

i.e.

9.

r is

the

the maximum intensity occurs at Eo, nnd

this intensity has fallen off to half the maximum value at energies Eo ±r/2. We sec that it is reasonable to take the energy uncertainty of the state (15) as LIE ::::. r-!iy, in agreement with equation ( 1 3 ) .

EXERCISES IV.I.

The Hamiltonian function

IV

ofa system is

i-

H - .m(PIZ +pz' +PJ2) + 1'(910 91' 91) j

(IV. I )

obtain Hamilton's equations and interpret them.

IV.2. The motion of a point mass III in a central field JI(r) is described in spherical polar coordinates, equations ( 1 2.2). Ir these are L.1.ken as generalized coordinates, define the corresponding generalized momenta, the I · Ia ian function, and derive Hamilton's equations.

�

�

In section 16 we sho�ved that the eigenvalues of a Hermitian operator arc all real. Prove the converse, i.e. that an operator with a com­ f real eigenvalues is Hermitian. ple :J' Prove equation (16.22).

�

I�

( �?; Prove that. ifU and I'are unitary operators. then (i) ana li ii)

(VV- I) arc unitary OPCI";1tOrs.

V*, (ii) (VJ!)

IV.G. If the operators A and B have the inverse operators A-I and B-1, show that A- I is unique, and find the inverse of (AB).

CUl.�f!p h

!Pl, . . . rp�, . . . is a complete normalized ortllogonal sequence .. off unctions, show that the sct

V'i - Urpi, (i - 1 , 2, . . . n, . . .) generated by the unitary operator V, is also a complete normalized ortho· gonal set.

� .

vV I)rove equations (19.5).

-r

rove equation (19.8) direclly.

If tp(x) represents a non·degenerate energy eigenfunction of a (1 par ide, moving in a potential field V(x) of definite parity, prove that the expectation value of the position coordinate x is zero. •

IV.I I .

Ifa beam of particles is represented by the normed wave function

'I'(r), what is the momentum distribution ofthi! ueam, ;.e. what is the proba. bilit)' of a particle having IV.12.

a.

momentum in the range

p to p + dp?

If A, B and C are any operators, prove the following identities [A, B +C] - [A, B]

+ [A, C]

[A , DC] _ [A, B]C + D[A, C]

[DC,

A] -I1[C, A] + [11, A]C.

BB

(IV.12a) (IV.12b) (IV.12c)

EXERCISES IV

�

Prove the following commutation relations for the orbital angular m� operators I., I" l� : [I., x] = 0, [/.e, Y] = iIiz, (IV. 1 3a) [I.e, z] = - if!>', (IV. 1 3b) [/.e, p..] ",, 0, [/.., p,] = ilip�, [1., pJ = - ilip" (IV. 1 3c) [I., I,] = i1il�, (IV. 1 3d) [I.., 12] [I., r2] [I.., p 2] 0. All the other commutation relations between the components of r, p and I can be derived from (IV. 13a-d) by permuting the indices x, y and z. =

=

89

"'"

v MATRIX MECHANICS TN Chapter I we discussed two methods of describing vector spaces or function spaces. The simpler and morc direct method is to usc the physic.t1ly significant quantities, stich as the \'ectol's, inner products, tic directly. Alternatively we can introduce a coordinate system to which all quantities arc referred. This latter method is orten can· vcnicnt i n practice though it involves a largely :lrbitrary coordinalt: system. In this chapter, we shall deduce this forlll of quantum mechanics using coordinale systems and components frolll the co· ordinate-free formulation which we have so f.'\r developed. In this way, matrix mechanics, as this formulation of quantum mechanics is orten called, will appear as a natural and simple consequence of the pre\'ious theory. T-listoricall)', it was actually the fin;t form of quantum mechanics, having been developed in 1925 by Heisenberg, Born and Jordan, and independently by Dirac. 21. Mntrh. representation of wave functions and operators 12 obtain matrix mechanics, we choose a complete orthonormal sct of functions " '2, . . . €M . . . to form a coordinate system or basis, spanning our Hilbert space, and expand €2, . . . €u . . . Later we shall also admit other bases, and we shall have to consider the transformations between these equally acceptable coordinate systems. We shall always limit ourselves to orthonormal sets as these arc most convenient in Illost applications. We can expand any vector 'P of the Hilbert space in the form •

9 - L'.'.. .-,

(\)

i.e. the \'ect0LY' has the components 'It '2, . . . c., . " , these fully specify the vector 11' (since we arc throughout referring to the basis

90

21. �" �2'

lolATRIX REPRESENTATION OF WAVE FUNCTIONS AND OPERATORS •

•

•

the vector

�n' V'

column vector

) and vice versa. by its components c"

•

•

In matrix algebra, we specify Cn, and write it as a

.

C2,

•

•

•

•

•

•

(2)

'1' =

which is often abbreviated to [c,,]. We call (2) the matrix representa­ Ell, tion of fjJ referred to the basis E" E2, 0 •

Given two vectors co

CD

fjJ = 2.c"�,,

•

•

0

•

CD

and X = 2.d"E", their inner product is .. - I

,, - 1

(X, '1') = 2.d"c"o Anticipating matrix multiplication, with which we ,, - 1

shall deal in section 22, we write this inner product in the form (3)

i.e. we write the first factor in the inner product (X, '1') as a row vector, taking complex conjugates. We_sh�ll
91

21.

a.�TRlX REPRESENTATION OF WAVE FUNCTIONS AND OPERATORS

i.e. c,, = (En, tp} and d,, = ( En, X}. co

Substitution in (4) then gives '"

'"

J.d",Em = AJ.cIII Em = J.(AEm} cm Dial I m-I m- J and taking the inner product with.respect to E", since (En, Em) = 8_, co

d,, = J.(E", AEm) clII III- I co

= ZAnmcm (n = I, 2, m-I

•

•

(5)

)

•

where we have introduced the abbreviation

AIIIII = (Ell, AEm) .

(6)

(Note the order of the suffixes n and m, as in general Anm ::;6A.....) Equation (5) is the matrix equivalent of (4) . It specifies how the nth component of X is derived from the components of tp. We see that a knowledge of the square array of numbers

Au A I 3 A I4 Au A23 A3Z •

•

•

•

•

•

•

•

•

•

•

•

•

•

•

•

•

•

•

•

•

•

•

•

•

•

•

•

•

(7)

is equivalent to a complete specification of the operator A ; it is the matrix representation of A with respect to the basis Ell Ez, Em Given (7), we can, on account of (5), transform any vector tp by means of A. We call the array (7) a matrix operator, or simply a matrix, and the Anm occurring in (7), and defined by (6), matrix elements. We can write (5) in a different manner, using (7) : •

]

[dl] [A ll Au A13 . . . . . . [Cl] dz AZI Azz Cz . d3 = A31 • C3 ' . ............ .... . •

•

•

•

•

•

•

•

•

•

•

•

•

•

•

•

•

•

•

•

•

•

•

•

•

•

(5*)

this notation brings out particularly clearly that the matrix operator [A_] transforms the vector [CD] into the vector [d,,]. The law of multiplication on the right-hand side of (5*) is as follows : the rth component of X is obtained by multiplying the terms in the rth row of A.... by the corresponding terms of V' and adding these terms. Above we have considered an infinitely dimensional function space whose vectors and matrices have infinitely many components and elements. But we can equally well apply the above formalism to a

92

22.

PROPERTIES OF MATRIX OPERATORS

finite n-dimensional vector space whose vectors have n components and whose matrices have n2 elements. We can now express wave functions, inner products and operators in matrix form, enabling us to express many of our previous results in this language. Thus, the time-dependent SchrOdinger equation ( 1 6.7), ." mp, = H'P" z 81 is replaced by the system of simultaneous differential equations CD

ift,,, (t) = 2.H"""", (t)

(n = 1 , 2 ,

.

•

•

)

(8)

for the time-dependent expansion coefficients c"(t) of the wave function CD

t = 0, c,,(O), the system of equations (8) completely determines the c,,(t), i.e. the way i n which the

'P, = 2.c"(t) E,,.

n�1

Given the coefficients at

individual components of the wave function vary with time. To be able to use matrix mechanics generally we must consider the form which the most important properties of operators assume in matrix language. This we shall do in the next section. 22.

Properties of matrix operators We now derive the matrix form of some of the more important properties of the operators of quantum mechanics.

(a) Matrix Multiplication

To begin with, let us find the matrix corresponding to the product AB of two operators. For any vector tp, let 'P = Btp and X = A'P' C =AB is the operator which transforms tp into X : X =Ctp = ABtp. Suppose CD

CD

CD

x = 2.d"�,,.

tp = 2.b"�,,, 'P = 2.c"�,,,

"-I

n �1

,, - I

We want the matrix representation of the operator C = AB, in terms of the matrices [A_J and [B_J . Equation ( 2 1 . 5 ) gives at once CD

CD

I-I

m-I

Cm = 2.Bmlb" d" = 2.A"mcm and hence

...

CD

dn = 2.2.AJ""b,• m - I '�I

( l a)

But if C_ = (En, C�m) is the matrix representation of the transformation C which transforms tp into X =Ctp, we also have ( l b) 93

22.

PROPERTIES OF MATRIX OPERATORS

CoI-2A (11, /=1, 2, . . . (2) [Cn C" C"Cn [An A" A" [ J J J

Now equations (la) and ( I b) arc 10 hold for all vectors since Cis simply defined as lhe product AB. Hence it follows that •

01',

using

tJ1C

..,

B..J.

)

....

complete matrix notation, we can write this

. . . . . . . . . ... A21 A22 liB • • • e21 Cn All An . . . . . Cll Cn . . . . . . . . . . . . . . . . . .. .... ....

Bn IJ" B" . . . B21 B22 Bll . . . . (3) Bll Bn · . . . . . . .. . ... ....

X

Equation (2) Slates the law or matrix multiplication, which we have here deduced from the general transformation law of linear operators. We obtain the (I, I ) clement of AB by multiplying the first row of II by the first column or n, term by term, ridding these terms. Simibrly ,�obtain the (m, n) clement (i.e. the element in the mth row and nth column) of AB from the mth row of A and the 11th column of B. But one must :llways multiply a row in the first factor by a column in the second f.'l.clor and not in the rcverse order. As we would expect from the f:lct th:lt matrices represem operators, matrix multiplication s i in general not commutative, [A_1[IJ_1 *[B_1[A_1· We give two examples of this:

[0 1] [1 0] _ [0 - 1] [�-�] [� �] - [ -� �] I 0 0- I I 0 ' [� :] [: �] - [� :]. [: �] [� :J - [: �]. For certain speci:ll matrices, multiplication is commutative, 'We define a diagonal matrix as being one in which all clements off the principal diagonal arc zero:

[ 00 0 J [ 00 J

.... .. d, 0 . . . . . dJ " " . ·

[D_1 - [d.8_1 - d, o

o

In

,... ,.--, unit articular, the

. . .... matrix, denoted b' I is ,

[1_1 - [8_1 -

'

,

1

o o

,

.

,

I

'

.

. . . . . . .

,

,

.

,

,

All matrices commute with the unit matrix, for

. ..

[A 1[8 1 - [8_1[A_1 -[A J, ••

••

••

94

"

. . . . . . . 0 .. . . . . .

I

0

...

,

(4)

.

(5)

22.

PROPERTIES OF MATRIX OPERATORS

and any_t�o_clli!gmt�LJ!l�tti���t comlll\lJ�_ancl their_ product is also. a diagonaL matrix (if E�ercisc V. l ) . (b) Adjoint Matrices If A * is the adjoint of an operator A, then for any two vectors tp and X ( X, Atp) = (A * X, '1') ;

in particular for two basic vectors tp = Em' X = En,

(6)

i,-�. the adjoint matrixjS3.lbtained _by interchanging. row£ and columns li,e. tl1lJlsposing thc matrix) and taking complex conjugates. If A is _ self-adjoint, (7)

j��. tl!� dia�n.al elem.ents (",-== n)
where Xl and X2 are real and

[it x�]

z

(8)

may be complex.

(c) Change of Basis Sincc the choice of basis is largely arbitrary, different representations being physically equally acceptable, we must now consider coordinate transformations from one complete orthonormal set of basic vectors to another such set Eh f2' • • • En, • • • • En, E., E2, In Exercise IV.7 wc saw how to generate such new bases from the original basis E., E2, En • . • : allY unitary transformation U trans­ forms Eh E2, • • • En, into a new complete orthonormal set of v"e t��-�t� .E�, . '. : J", .. .. . 'Y_�erc - - - -- .�.- .-.- - •

•

•

.

•

•

•

�

.

•

•

•

____

•

(n = I , 2,

E,, = UEnJ

•

.

.

)

which we can take as new basis. Conversely, given any two complete orthonormal sets of vectors and eh E2, Eh E2, • • • En, En, . • we can define an operator U completely by thc correspondence •

•

•

•

•

E" = Ue,,,

•

•

(n = I , 2,

• • •

)

(9)

i.e. we define U as the operator which transforms En into Ell' Since the set of vectors e., e2, • • • e., . . . is complete, this also tells us the 95

22.

PROPERTIES OF MATRIX OPERATORS

vector 2; ;p into which any vector '" GO

tp = cnEn, then

is transformed by U. For

E 2; '" GO ip = Utp = cn UEn = 2cn n. 13 - 1 n-I

.-1

if

(10)

One easily sees that U possesses an inverse operator U- I which is com­ E pletely defined by the equations (1 1) E.. = U-' ". '"

The operator U which we have defined is unitary; for, if '" = 2cnE. 13-1 GO

and X = 2dnEnI then

d

d 2; '"

d 2; '" ( X, "') = 2 ,.cn, ( UX, U"') = nCm( ", Em) = ncn• A-I 11-1 m, .. -I I i and equation ( 1 1) becomes Hence equation ( 18.1 9) holds, U- = U*, ( 12) E,, = u* n. 13-1

i

From the way in which we have introduced coordinate systems, it is clear that if we consider two vectors '" and X, whose column vectors, referred to the two bases, are [cn2 ] ,d..[dn] and 2;2 '[en], [till] respectively, then

( X, "') =

'"

a-I

cn =

"' -

n- I

" n;

the scalar product of two vectors � and X is indepelldent of th� ch�ice oLbasitJ() whi�h �c: refe_r '" and x . This must of course be the case since scalar products give the physically observable quantities (expecta­ tion values, probabilities, etc) . To find the effect of a change of basis on the matrix representation of an optrator A we consider the transformation of the vector ", mto X =A 'f' referred to two different bases. We arc, of course, in both cases dealing with the same operator A and the same vectors '" and X , and only have different matrix representations of these. Referred to 2 the basis El l E2, •

2 '"

• •

En,

• • •

'"

, '" and X are given by '" = CnE.. and

x = d..En, and the equation x =A", becomes 13 - 1

where

d.. = 2A"""", (n = 1 , 2, '"

",-1

Referred to the basis El l E2,

• . •

..- 1

)

(2 1 .5) (21 .6)

• • •

EA'

. • •

96

the same vectors '" and X are

22. CD

PROPERTIES OF MATRIX OPERATORS

CD

dnEII' and equations �En and X =2. tp =#2. -1 a-I dll

'"

=

2.Anm;m,

m-I

(21 .5) become replaced by

(n = I , 2, . . . ) ,

(2 1 .5a)

where A nm is the matrix representation of A referred to the new basis : (21 .6a) From equations (2 1 .6), (2 1 .6a) and (9) we obtain Anm = <E", A i...) = (UE,u A UE...) = ( Ell, U*A UE...) = (U*AU) ....., or using the rule for matrix multiplication Allm

'"

=

'"

( U*A U)"", = 2. 2, U",*A"Umo•

( 1 3)

. - 1 ,-1

From (13) one can find the matrix representation of an operator with respect to a-new basis. The operator U we have so far considered can be thought of as a transformation (a kind of rotation) of the coordinate system in the function space, keeping the state vectors fixed. W� obt�in the_ sa�rn� relative motion of coordinate system and state vectors if w(! �onsider the coordinate system as fixed and subject all state vectors to the inverse transformation U*' . Since U* is also a unitary operator (if Exercise IV.5), we can generally think of a unitary operator U as a transformation which replaces all wave functions tp by ip = Utp, leaving all inner products invariant, and replacing the matrix representation [A....J. of an operator A by [A....]. = [( UA U*)_]. For, if X = Atp, then X = A � implies UX =AUtp, i.e. A = UA U*. (14) (T.h��order of'the factors U and U* here �ppears i!l the . reverse order to that in ( 1 3), as we are rotating the vectors, instead of the .co.gr!!inate system.)

__

(d) Eigenvalue Problems We must now consider the form the eigenvalue problem takes in matrix mechanics. Let us, for the moment, assume we have already solved the eigenvalue problem of a Hermitian operator A, i.e. we know the eigenvalues aJ
97

•

•

22. PROPERTn:s OF MATRIX OPERATORS o

0

a, 0 o a,

( 1 6)

and the eigenfunction qJl has the matrix representation [ 01..], etc Though by a suitable choice of eigenfunctions we can always diagonalize [Anm], degenerate eigenfunctions belonging to the same eigenvalue will in general not be orthogonal. Hence the matrix A wiU not be diagonal, but consist ofsquares, containing non-zero clements, arranged along the principal diagonal. The eigenfunctions labelling the rows and columns of any square belong to the same eigcnvalue. They span the corresponding manifold in I-Tilbert space. We illustrate this for the casc al =a2 #-aJ ""a� #-as =F . . ., [A••] � rAil Al2 A21 A2!

0

(17 ) .. . ... ... .

o

...

,

....

, ....

...

To express the eigenvalue problem AqJ ""'alp ( 18) in matrix. form, we introduce a matrix representation in terms of some complete orthononnal basic set of vectors �l' �2• . . . . �nJ If we •

•

•

o

write the eigenfunction qJ as Ip = 'Ic.�•• we must solve the infinite set of linear homogeneous equations o

-at. (n ... I .

..

'IA_c

.-,

.-,

• . . .

2

)

(19)

for the eigenvectors [tn]. In general, this set of equations only has the trivial solution [en] =[0]. and only for certain values of a, the eigenvalucs, arc there non-tdvial solutions. We shall not give a general treatment of the method of solution of a system of linear homogeneolls equations. but shall only discuss the general features of the corresponding problem il l an ;V-dimensional 98

22.

PROPERTIES OF MATRIX OPERATORS

vector space mp• This is the problem most frequently encountered in practice, the X-dimensional vector space being a given subspace of the infinitely dimensional function space. This more restricted eigen­ value problem leads to a finite number of equations for a finite number of unknowns. Its solution can always be put in the form of N orthonormal eigenvectors and corresponding eigenvalues. We can formulate this eigenvalue problem as follows. Given the operator A, we can find its matrix representation An., = (�,,, A �.,) with respect to an arbitrary basis �I > �2' • • • �'" • • • �p. We want to find a unitary transformation U which transforms this basis into the set of eigenfunctions fPl> fP2' • • • fP.., fPN, which are initially unknown, to act as new basis : fP,, = UEm (n = I , 2, .N) . (20) •

•

•

.

.

Referred to this new basis, the matrix of A will have to be a diagonal matrix a, i.e.

U*A U= a, A U= Ua,

or, in terms of matrix elements, N

p

LA,.,U.,,, = 2 U,., (a., 8"",) m--l m- I = U",a" N

= 28r.. Umna" hence

N

.. - I

2(A'm - 8,m a,,) U..,, = 0,

., � I

(I, n = 1 ,

(I, n = 1 ,

• . .

. . •

N) .

N),

(2 1 )

For a fixed value of n, equations (2 1 ) represent a system of N homo­ geneous linear equations. They contain the unknown parameter a" and the N unknown quantities U.", Uln, • • • Up". But since the equations (2 1 ) are homogeneous they cannot determine the absolute values of the U""" but only the (N - 1 ) ratios U... : Uln : • • • • : UN", Hence equations (2 1 ) are a system of N equations for (N - 1 ) un­ knowns. Thus we have an over-determined problem and, in general, equations (2TL w�ll be inconsis!�Jli� '!'o l!l!ye_�a n�n�vi�I�9jution, they_ must satisfy a consistency �oD:dition (\VlUch ess�Il!japy �tat� tl1�t one of the equations (2 1 ) is simply a line�r.com�!!��9E-. of t!te re_�ai�­ ing (N - 1 ) �quations)� This consistency . conditio� is �ll�t the d�!�!": minant of the coefficients of U"", in (2 1 ) should vanish : __

IArm - 8/m a..1 = 0 ;

(22)

this is an equation of the Nth degree in a" which has N generally distinct solutions which we shall denote by ab a2, aN' These are •

99

•

•

23.

THE

EQ.UATIONS OF MOTION OF OPERATORS

the eigenvalues of, A. For each value all' one can solve the set of equations (2 1 ) to obtain the corresponding eigenfunction (20), N N (23) N). tpll = UEn = 2.Em (Em, UEn) = 2.EmUmIl' (n = 1,

m-l

mClOll I

•

•

•

If the N roots of the C secular equation ', as equation (22) is called, are not all distinct, then we have a degenerate case. To solve the problem for a degenerate eigenvalue requires further analysis which we shall however omit. One finds that corresponding to an r-fold degenerate eigenvalue, the eigensolutions span an r-dimensional subspace of tRN, and one can choose r orthonormal eigenfunctions to span this subspace. With the aid of the above analysis all our previous results can be expressed in matrix form. One can easily show directly that a Her­ mitian operator has real eigenvalues and orthogonal eigenfunctions (if Exercises V.2 and 3). Our resJ,llts about comlllu_ting operators �cti()JL1�, theorel!l 2) take a paJ"!iC!llarly siElple form. For if � set _oLobserv.ables possesses a ..cQmplete_s�t of simultaneo� eigenfu!1.ctions, then, taking these eigenfunctions as basis, the_ observabj(!s of theJlet are all represented by_ diagonal matrices and th�e necessCirily commute. 23.

The equatioDs of motioD of operators

In section 16 we dealt with the time variation of a system. It is determined by the Schrodinger equation (16.7) or, in a particular representation, by the set of equations (2 1 .8) . In this description, known as the Schrodinger picture, the temporal variation of a system is given by a time-dependent wave function, the observables being described by time-independent operators. In formulating matrix mechanics, Heisenberg adopted an alternative approach which we shall now develop for operators generally, rather than for a matrix representation. This alternative mode of description, known as the Heisenberg picture, arises because the physically significant quantities are the inner pro­ ducts, of the form ( X, Atp), as we have repeatedly stressed. We can then ascribe the time variation of such inner products to a time­ deR-endence of the wave functions or of the operators. - In the Schrodinger picture, an observable a, which does not contain the time explicitly, is represented by a constant operator Ao, the timedependence entering the wave functions tp, and X, through the Schro­ dinger equation (16.7) or (16.12) and ( 1 6.13)*. In the Heisenberg picture, the states of the system are represented by time-independent wave functions 1J'0 and Xo, and the time variation

rI I

\

• We are throughout restricting ourselves to systems whose Hamiltonian functioDJ do not contain the time explicitly.

100

(

24. THE DIRAC

NOTATION FOR WAVE FUNCTIONS AND OPERATORS

of the inner products arises from time-dependent operators A ,. The two modes of description agree at all times if ( Xo, A ,tpo) = (X" A oV',) = ( T,Xo, Ao T,V'o) (I) = (Xo, T,*Ao T,V'o) , i.e. we define the operator A , at time t i n the Heisenberg picture by (2) A, = T,*Ao T,. From (2) and ( 1 6. 1 3) one easily obtains the differential equation which A, satisfies : dA dT, dT,* A 0 To, + ·" T.*A ." , ." 'n dt = " 6 T ''' , oTt = - HT,*Ao T, + T,*Ao T,H = - HA, + A,H (3) = [A" H). (In deriving (3) we have used the fact that H commutes with T, and T,* but not, in general, with A o.) Equation (3t_is th�!leisenb�e1"g equation of motion of an operator which does not explicitly depend on the time (aA ,/at = O)._ One can easily generalize these r�sults to the c
l

24.

The Dirac notation for wave functions and operators We now want to consider some developments in notation. In writing eigenfunctions we have so far written the corresponding eigenvalues as suffixes and have often explicitly written out the arguments of the wave functions. Since the emphasis is usually on the eigenvalues, it is often more convenient to write these as C arguments ' and omit the position coordinates. Thus we would write for the orbital angular momentum eigenfunctions ( 1 3.27) and ( 1 4.4), 'P(E". " I, m ) and '1'(1, m) instead of 'P£,.. I, I, ,. and V'rJ as we have done so far. 101

EXERCISES V

A similar change of notation is desirable for matrix clements. Wc shall, however, not write the matrix element 11 ... = (E.., II�.) as lI(m, 11), bUl ralher as

<m IAI_)

and the matrix clement ('Pr'" , Hlp/') say, as
II, m). If p=I), wc denote lhe complex conjugate vcctor p not by I) in general, but by
correspond to the column and row vectors of matrix algebra. scalar product ofv'=lh) and IP= Ja) is written (p, Y') _
The


If A is an operator, it is natural to write the scalar product of I,)- A la) and Ib) a. and generally for the matrix element between two states IP' and fP" labelled by the sets of eigenvalucs �I/ f2', . . . �: and �L"' ;2", ' .

. . . ,:IAI,,', ,,',

. ,:) . If11* is the operator adjoint to.d, thenioUll)' two veclors_la)_and_lh),

(p', Ap') -(f"

" ,

. .


being the statcment analogolls to cquation (IS.G).

EXERCISES V

(}V. I)

Prove that two .N x .N diagonal m:!.trices commute :!.nd their product :!.!so :!. diagonal matrix. v� Prove from equation (22.19) that the eigell\'alues or a Hermitian matrix arc real. Prove rrom equation (22.19) that Ihe eigenrunctions or a Hermitian matrix belonging to different eigenvalues :!.re orthogonal.

is

tv�

102

EXERCISES V

'\[.1) The angular momentum operators for a particle with angular momentum 1i,/{1-(t + I)}, spin t as i t is ealled, can be represented by the following matrices,

'. _ -�[ OI

' _�2 [O -i] ' ,� _![I 2 7

j

0

0

Show that s� and S2 =S,,2 +S,2 +S�2 are simultaneously observable and give their eigenvalues. Show that S,,' S,' St do lIot commute with each other and obtain expressions for the commutators [s". s,], ltC. V.5. Solve the eigenvalue problem i n the two-dimensional space !Hz of the operator 11 whose ma trix, referred to the orthononnal vectors �l and '2 as basis, is

[A_l - � ';]

ri.,. ). - (I" A,,) - (I" A,,), p - (i" A,,) - (i" A,,)]. V.G. Derive the Heisenberg equation of motion (23.4) for an operator which depends c;"plicitly on the time, the Hamiltonian of the system being time-independent. Derive the matri.x representation of the position coordinate x r to thc oscillator wa,'e functions V,�(.t). equation ( 1 1 .23), as basis.

(\7:" "'"7.) �

103

VI SYSTEMS OF MANY PARTICLES

SYSTEMS of many particles are of importance in many problems of atomic and nuclear physics. In these rather complicated problems a direct complete solution of the Schrodinger wave equation is not usually practicable. It is therefore important to gain as much in­ formation as possible by studying general aspects such as the angular momentum and symmetry properties of the system. 25.

Angular momentum In section 1 4 we considered the orbital angular momentum of a single particle, and we defined the operators I", l�, I� which, as is shown in Exercise IV. I 3, satisfy the commutation relations

_[I", I,l = iJi/� etc [12, I,,] = 0 etc

where

(1)

__

(2)

(3) _ 1�= 1,,2_+ 1�� + li. _ We found in section 1 4 that, for a single particle in a spherically !ymmetric field, the operators I.e, 12 and tIie HamiltomanHTorm a complete �et_<>.f cotllI1lu!ing _o1>�eI"\'a.blc:!.___T�� simultaneouS eigeo:' functions (13.27), in the notation of �ec!ion 24, are - (4)

and have the eigenvalues lim, li 2/(1 + 1 ) and n respectively, m and I - - - - - " " -_ . E . takfn� �n the values _ m . = �, �/ + !'.-"-' ,i -:l�/.L I��I ,_�._�" Using the wave functions (4) as basis of a matrix representation we find that (En'.r, I', m' I HIEII." /, m> =�I1. , 8�, S..'" 8mm, " "

.

(Elf.,"

-.

--

�----.

._" " " " ._"

.

I', m' 112 IEII." !... !1!!_= 1i21(l + 1 ) 8�, 811, 8mm,

(En'." , I', m' 1/�IEn." I, m>= lim8"", 8u' 8""",

}

(5)

and using equations (14.2) defining the angular momentum operators in polar coordinates one easily finds that I" and Ir are not diagonal matrices in this representation.

1 04

25.

ANGULAR MOMENTUM

We have derived equations (5) using the eigenfunctions (4) explicitly. However, equations analogous to ( 1 ) , (2) and (3) hold for the angular momentum of any system, not just a single particle, and we shall see that a result analogous to (5) can be derived from them directly. The o�er--1!Q.inL to note is that equations (5) give a representatio.!lJ�y (2l + I ) x (21 t I) matrices of the angular momentu� for a.giyen_eQ!!rgy eigenvalue En. I. Since l can take on the value 0, 1 , 2, . . this restikts us to ' x I , 3 x 3, 5 x 5, . • . matrices. But we can clearly write down similar representations with matrices having an even number of rows and columns. We shall see presently that this generalization to l values of half an odd integer is connected, with th� intrinsic angular momentum, the spin, of elementary particles. To distinguish it from the orbital angular momentum 1 of a single particle we shall denote the general angular momentum vector by M. Assuming that M satisfies relations analogous to ( I ) and (2), .

[M,., M,,] = iliM, etc

(6)

[M2, M,.] = 0 etc

(7)

_

where M is the magnitude of the angular momentum vector,

M� =M: = A!,,2 + M,,2 + M,2, .

(8)

we shall prove the following result which is sufficiently general to deal with particles with spin, systems of particles, etc : Let Q., M2, M, form a complete set of commuting observables * for a system, which satisfy equations (6), (7), and assume that Q, commutes not only with M, but also with M,. and M",

[Q., M,.] = [Q., M,,] = [Q., M,] = Ot.

(9)

We shall see that the eigenvalues of M2 are of the form

1i2n(n + l )

n = 0, !, I, i, 2, . . .

\:Y..hich of these eig(!nv�!ues actuallY occur i!L�. _PJlrtic!!lal'�yst�Dl d�I>ends on t�e2yst�!D� (Thus we saw that for the orbital angular momentum of a single particle we can only have n = 0, 1 , 2, . • • . ) Furthermore, for a given eigenvalue q of Q" an .eigenvalue 1i2n(n + 1 )

as abbreviation for a set of commuting observables Q. . Q2, • • . , if neces· In general the Hamiltonian H will be one of these observables and it may happen that it is the only one, as in the orbital angular momentum case, above. t This corresponds to the assumption in deriving (4) that the system is spherically symmetric. The Hamiltonian H is then invariant under a rotation of axes, i.e. It is inde· pendent of the angles 8 and rp and hence commutes with I", I, and It which, according to (14.2), only involve differentiations with respect to 8 and rp. Conversely, if H commutes with /'" I, and I, the Hami1tonian is spherically symmetric, i.e. independent of 6 and rp. Thus, on account of (14.2), ItH'I' = 111:'1' implies aHlarp �O, i.I. H is independent ofrp.

• Q here stands

sary.

105

2S. ANGULAR MO:'IENTUM of M2 is (211 + I )-fold deg�lcratc eorrcsponding ....!Q (211 + 1) eigenstatcs ...JJ(q, II, 111) of A1� with cigenvaluc.� !j1ll,

I.e.

111 = - fl, - n + I , . . " fI - I , II,

QIJJ(q, lI, 111) =qlj/{q, 11, m) _<\f2lJ1(q, fI, m) =/12n(1I + I ) IjJ(q, II, m), 1\l�!J1«(J, n, m} _limlJl(q, II, III),

(m _

{II = 0, t, I , . . . },

-II,

. . . II).

( l Oa)

( l Ob) ( l Oe)

We sec that, as in the orbital angular momcntum easc, corresponding to a given pair of eigenvalues q , 11 there exists a (211 + I)-dimensional subspace in the Hilbert space of om system which is spanned by the (211 + 1 ) orthogonal eigenfunctions J l 1(q, 11. 111) of k(. --IQ provc these results from equaLions (6) to (9) we define the _operators 11+ and iL by Ai-lH� + iAlp A_

so that (A+)*-iL.

=

AI!" - iA'I"

( I I)

One easily verifies the following commutntion

7elation sfor these operators (if Exercise VI.I)

[A+ . MJ - - liA+ [A_, MJ - IlL

( 1 2a) ( 1 2b)

[/4, M'J - [/L, M'J -O and

A+ A_ _ Ml _ M�2 +'iM� iLA+ _M2 _ A/�2 -liM�.

( 1 3) ( 1 4n) (14b)

Let ljJ(q" t2, .IIJ be an eigenfunction of Q, M2 and 1\1. with eigen­ "nlues q, It2 nnd 11� respectively, i.t. Q}J1(q, l,l, Il.) _qlJ1(q, pl, II�) .>\.f21J1{q, Ill, ll,) _lt 21f1(q, p2, /1,) M/J1(q, p2, /1,) "",p,IjJ(q, p2, II.) .

( ISa) (ISb) ( I Sc)

Opemting on ( l Sc) with M, nnd subtracting from ( lSb) we get (M.2 + Mi) jl 1(q, 1,2, /1,) = (1,2 -It,l) P l (q, pl, It,) .

(16)

Since (lH�2 + .M,l) is a positive definite operator it cannot have negative eigenvalues, i.e. ( 1 7) where p >0 is the magnitude of M in the stale lP(q, ,,2, J1.�). II follows from (17) that the eigenvalues of A'r. corresponding to a given eigen­ value of J\11 arc bounded above und below. 106

25.

ANGULAR MOMENTUM

Considering the wave functions * A± lJI(q, p,2, p,�) we easily prove, using equations ( 1 1 ) to (1 5), that ( ISa) QA± lJI(q, p,2, p,�) = qA± lJI(q, p,2, p,�), M'2A± lJI(q, p,2, p,�) = p,2A± lJI(q, p,2, p,�), (ISb) ,M�± lJI(q, p, 2, p,�) = A ± (M� ±Ii) lJI(q, p, 2, p,�) ( I Sc) = (p� ± Ii)A± lJI(q, p,2, p,�), i.e. A± 'P(q, p,2, p�) are eigenfunctions of Q, lv/2 and M� with the eigen­ values q, p2 and (p� ± 1i) respectively. Thus operating on lJI(q, p2, I'�) with A± gives new simultaneous eigenfunctions, belonging to the same eigenvalues of Q and 1\12 as lJI(q, 1'2, I'�) itself, but to different eigen­ values of 1\1�. Thus by repeatedly operating with A+ on lJI(q, p, 2, p,�) we generate a whole set of eigenfunctions of M� belonging to the eigenvalues I'�, p,� + Ii, I'� + 21i, • • • and all belonging to the eigenvalues q and p.2 of Q and M2. Similarly, the operator A_ generates a set of eigen­ functions of lv/� belonging to the eigenvalues p�, I'� Ii, p,� 21i, and to the eigenvalues q and 1'2 of Qand M2. But on account of ( 1 7), 1p,� 1 cannot assume arbitrarily large values. Hence for a given total angular momentum p, 2, there must exist some maximum possible eigenvalue of M� which we shall call lin, so that A+ operating on the corresponding eigenfunction ':P(q, 1'2, lin) does not give an eigenfunction of M� corresponding to Ii(n + 1 ) , but vanishes identically : ( 1 9) A+ ':P(q, 1'2, lin) =0. If (19) holds identically then (20) A_A+ ':P(q, 1'2, lin) =0, i.e. from ( 1 4b) -

-

•

•

•

i.e 1'2 =li2n(n + 1 ) . (2 1 ) Equation ( 2 1 ) defines the value of 1'2 for a given n and vice versa. Similarly there must exist a least eigenvalue lin' of M� for which A _ ':P(q, 1'2, lin') = 0 ( 1 9a) so that A+A_ ':P(q, 1'2, lin') = 0. Using (14a) and (2 1 ) one finds n (n + l ) = n'(n' - l ) • In the followin we shall use the symbols ± and 1= as abbreviations for two alternatives . � Thus A ±IJI(q , p2, Pr) stands for the two wave functions A + IJI(q, I,Z, llt) and A _ IJI(q, l'z, Pr) and an equation such as ( l ac) stands for the pair of equations obtained by taking the upper or lower signs throughout :

MrA+ 'P(q, p2, Pr) "'" CPr +/i) A + 'P('l, I'Z , Pr) Mr, L lJI(q, P Z, l'r) - CPr - 1i)A _ IJI(q, pZ, Pr) .

1 07

25. ANGULAR MOMENTUM

n' = n + 1 n' = -n.

n' =n + 1 M�, n' = -no M2 Ti2n(n + 1) 1) !P(g, Ti2n(n + 1), Tim), (m = -n, -n + 1, . . . n - 1, n), cOrr��oriding to th.e eigenvalues lim, (m= -n, -n+I, n - I, n), respectively. Sitlce tlie dimension-aliti-of thiS manifold is (2n + I ), (2n + I) must be an integer, i.e. n can only have one of the values n=O, ;, I, I, 2, t, . . . completes the proof of equations (10), where we have for con­ venience written !P(g, n, m) for the wave function !P(g, Ti2n(n + I), lim) Thuorresponding matrix re�resentation of Q, M2. an(LM�l!_ (22a)
.

r'

t\,-

•

.

This

0'1"-"

If)

.

/'

•

•

this

t' • •

'

'

All

•

• We have omitted some of the rather tediom allJebra leading to equations (24) and (25). The reader should easily be able to carry out the mtermediate steps if he so desires. The more lazy reader may prefer to find it in Condon and Shortleyll, p. 48.

l OS

25.

ANGULAR MOMENTUM

In section 14 we have shown that the eigenfunctions of the orbital angular momentum of a single particle are essentially the spherical harmonics Tr(O, equation ( 13.20), 0 and being the polar angles of the particle. The normalization and phase factor of the spherical harmonics were fixed by equations ( 1 3. 19) . In our present notation we can write the complete eigenfunctions as

91),

91

'P(g, n, m) =R" n T,t(O,

91),

(n = O, 1 , 2 . . . ),

(26)

91,

where R" " is independent of m, and of 0 and and depends only on other coordinates and commuting observables not of interest to us. The equations (22) and (25) completely specify the eigenfunctions 'P(g, n, m), apart from a common phase factor. If for integral values of n, we choose this phase factor so that 'P(g, n, n) = 'P(g, n, n),

(n = 0, 1 , . . .) ,

(2 7)

then it can be shown that the eigenfunctions (26) are identical with the 'P(g, n, m) for all values of m : 'P(g, n, m) = 'P(g, n, m),

(m = - n, , n) ( n = 0, I, . . . ) . .

.

•

(28)

We now see the reason for the choice of phase factors in our definition of spherical harmonics in section 1 3 : the operators A ± generate the spherical harmonics with these phase factors. We have considered this question of the choice of phase factor in some detail as it is important to choose the phase factors in a consistent manner. This will become apparent when we consider the addition .. of angular momentum vectors in section 27. As an example of the above theory we consider a system with angular momentum t. (It is usual to refer to a system as having angular momentum n or lin if the magnitude M of the angular momentum is Iiv' {n(n + I )} .) In section 26, where we shall deal with the interpreta­ tion of this case in detail, we shall see that it corresponds to an elemen­ tary particle of spin i, such as an electron or proton or neutron. The matrix representation (25) is in this case (2n + I ) = 2-dimensional. If we write the eigenstates of M� which have eigenvalues ± lij2 as (29) the matrix representation of M, ( 30) 1 09

2 6. SPIN

becomes

(3 1 ) £1. = [� �] , £1, = [� -�] , £1� = [� �] , where the Pauli spin matrices are the components of the _

£1"L5L £1.3;

_v.ector _G._ _

- -------- -

Using the Pauli matrices one can verify the general properties of angular momentum derived above, such as the commutation relations (6) which now are (32) One easily finds that (33)

so that

M2 = 1i2! (! + I ),

(34)

A± defined in ( 1 1 ) now become A± =�{[� �] ± i [� -�1 } I.e. (35) A+ = � [� �J ' A_= � [� �] , (as required (A±)* =A=F)' We can check the process used above for generating eigenfunctions by means of the operators A±. Thus A+ [�] = � [� �] [�] =g [�] , A_ [�] =� [� �] [�] =,� [�] , as required.

The operators

agreeing with equations ( 1 9) and ( 1 9a) .

Similarly

[�] = � [� �] [�] =1i [�] A_ [�] =� [� �] [�] =1i [�]

�

in agreement with (23) and (24) . Another useful property of the Pauli matrices is that they anticommute, i.e. 26.

(36)

Spin

In discussing spin we shall assume that the reader is familiar with the experimental facts in spectroscopy, such as the anomalous Zeeman effect, which led Goudsmit and Uhlenbeck to the hypothesis of the electron spin . We shall only state briefly the main facts which led to this hypothesis and shall then consider its interpretation and mathematical formulation.

1 10

26.

SPIN

Being electrically charged, an electron interacts with a magnetic field. For a homogeneous field H, in the z-direction, the interaction energy, which enters the Schrodinger equation as an additional term, is

2;lP,

(1)

where I, is the z-component of the orbital angular momentum of the electron ( - e = electronic charge, p = electronic mass, c = velocity of light) . We can interpret ( I ) by ascribing a magnetic moment

- el

(2)

2pc

to the electron. One obtains this magnetic moment classically by thinking of the electron in its orbit as defining a magnetic shell. The effect 2f.�C! !!!�_ractiol1_�rm (!L�S_!<:l cause a shifting in the atomic ene�Jeve!s which results ln�__c_1.!ang� in ihe line spectrum when an extem�l magnetic_ field is applied_to an atoffi.. This is the normal Zeeman effect which we shall consicier in Chapter YII. The observed spectrum of an atom in a magnetic field, known as the anomalous Zeeman effect, is much more complicated and cannot be explained in the above way. Many other spectroscopic phenomena such as the multiplicity of many spectral lines, which on the above theory one would have expected to be single lines, also defy any simple explanation without the introduction of a new hypothesis. The hypothesis which resolved many of these difficulties, and which is essential to an understanding of many atomic and nuclear properties, is that of the electron spin. We assume that associated witlLtbe electron is an intrinsic angtllar momentum or_s-piti:J.e._1It_� magnitJJd� . ,I s of the spin vector 5 is given by s2 = 1i2!(t + I ) .

)

:

-�

The spin thus representS an example of an angular momentum ! treated at the end of section 25, and we can write the spin vector 5 in terms of the Pauli vector a, (25.30),

s =!1ia. Associated with the spi� ��� int���!C magnetic moment

-e 5. pc

(3) (4)

The theory of the spin of the electron based on the above postulates of spin and magnetic moment is due to Pauli. That quantum mechanics was successful in explaining many atomic properties without the introduction of spin is due to the fact that, in atomic systems, the spin interaction energy is a small correction term in the Hamiltonian of the system : to a first approximation the effect of spin can be neglected.

III

2 6. SPIN

Pauli's theory is a non-relativistic theory. It is one of its pecu­ liarities that one must postulate the intrinsic magnetic moment (4) to obtain agreement with experiment. For, by comparison with (2), we should have expected an intrinsic magnetic moment of half the magni­ tude, - es/2flc. In Chapter X we shall consider the relativistic theory of the electron due to Dirac. This theory predicts the correct spin and magnetic moment of the electron. In this chapter we shall only con­ sider the non-relativistic theory due to Pauli, which is of course a limiting form of Dirac's theory. For many applications one only requires the non-relativistic theory since relativistic effects are negligible. Apart from electrons, there are other elementary particles which possess spin i, for example protons and neutrons. Though we shall for definiteness often speak of electrons, many of our considerations will apply to these other cases as well. The limiting criterion will in all cases be that a non-relativistic theory should be adequate, i.e. that the energies of importance in the phenomena considered should be small compared to the rest energy of the elementary particles involved. For electrons this rest energy is 0·5 MeV, for protons and neutrons it is 940 MeV. As the binding energies of atomic electrons are of the order of a few electron volts and the binding energies of protons and neutrons in nuclei are of the order of a few MeV, the non-relativistic theories are applicable in a very wide field of atomic, molecular and nuclear physics. On account of the spin. the electron now has four degrees offreedom. To specify its state we must supplement the position coordinates x,y, z by one other observable. We can take for t4is th� ��fQ�p�o1!el!Lr� Qf the spin angular mo.mentum. The_ electron wave function is now d>(r,l'� Whereas r can assume -- ----all ---values, s� only has the two eigenvalues ±1i/2, -- - -. ..

.-

s�d>(r, ±1i/2) = ± !1i d>(r, ±1i/2) ;

(5)

d>(r, ±1i/2) represent states of the electron in which the z-component of its spin has the exact values ±1i/2,· we say, the spin points in the direction of the ± z-axis. In general we have a linear superposition of these states, and there exist definite probabilities of obtaining the values ±1i/2 as a result of measuring s�. We can think of d>(r, s�) as a wave function with two components (/>1 and d>2, and write

[I] o d>2 = d>(r, - li/2) = 9'2 (r) P = 9'2 (r) [�] .

d>1 = d>(r, + 1i/2) = 9'I (r) a = 9'I (r)

}

'?\l"";'� 'J-. 'Q

'-NJ:-- ' �(6) "

�L,\,)"'''

• One often measures spin and angular momentum in units of Ii and refers to these eigen­ values as ±t. Some authors we at instead of Sf as fourth observable with corresponding changes in notation.

1 12

26. SPIN _�n (�) we �xplicitly introduced th(!_ spi� ei�(!��unc�� J2 5.2�) p= a=

[�]

[�]

_�

(7)

of section 25, and the wave -functions q,l-(r) -and 9'2 (rfin (6Jare now orcHilary sp�tial · wavi-functions-whicli a�e -iriaej>ena_C!jrofspin. We can combine equations (6) and put (8) + 9'2 = tP = tPl + t/>2 = 9'l a + 9'2P = 9'1

[�]

[�] [:�.

We have here introduced several alternative notations which all express the same facts. It is important to be familiar with all of them as they are all frequently used. From (6) we see that s�t/> (r, +1i/2) =s�9'I (r) a = + !li9'l (r) a s�t/> (r, - li/2) =s�9'2 (r)fJ = !1i9'2 (r)fJ, i.e. 9'1(r) a and 9'2 (r)fJ are the parts of the wave function representing spin states with s� = ± 1i/2. The probability of finding an electron in the volume with �-component of spin ±1i/2 is -

V

fl' l 9'i(r) 1 2d-r

with i = I , 2 respectively. The probability of finding an electron in a volume regardless of its spin state is

V

I" { 19'1 (r) 1 2 + 19'2(r) 1 2} d-r,

and for the normalization of the wave function we obtain

fall { 19'l (r) 1 2 + 19'2 (r) IZ} d-r = 1 .

(9)

The combination of spin and spatial wave functions allows a par­ ticularly lucid formulation in terms of the ideas of vector spaces con­ sidered in Chapter I. We described particles without spin by ordinary wave functions 9' (r), VJ(r), x(r), . . in a function space � spanned by a complete orthonormal set of functions El (r), Ez (r) , . . . En(r), • . • We can describe the spin state of a particle by a vector in a two- r» , ,_� dimensional spin space 6. If we span 6 by the basic vectors a and fJ such a spin vector has the form aa + bfJ. A general electron wave function, equation (8) , can then be thought ofas a vector in the product space � x e formed from � and 6. The product space � x e consists of the products of all wave functions of � by all wave functions of6 ; it is spanned by the complete orthonormal set E1 (r) a, �1 (r)fJ, . • • ,,,(r) a, E,,(r)fJ, . . If 9'1 (r) = cnE" and 9'2(r) = dnEm we can write .

.

2

2 n

"

t/> = 9'1 (r) a + 9'z (r)fJ = 2En(r) {c"a + d.,{J} •

1 13

26.

SPIN

and we can interpret f/J either as a vector in spin space 6, the ampli­ tudes of the two spin states (i.e. the amounts of spin and - t) being functions ofposition, or as a vector in ,v, with each orbital wave function which occurs in f/J being a superposition of the two spin states. We shall usually find it most convenient to introduce the representa­ tion with a and (J as basic vectors in the spin space <S. In ,v we shall usually prefer the coordinate-free formulation so as to be able to treat each problem in the manner best suited to it. This leads to a difference in the appearance of quantities referring to spin and spatial coordinates. The former occur naturally in matrix form, the latter in coordinate-free form. We define the inner product of two vectors

+!

! Mr)

qJ = [:�

f/J =

[::]

in the product space §) x <S by_

(f/J , !l') =

f{[9ih 9i2] [:j}dT= f{911'Pl +Q;2'P2}dT (9'h 'PI) + (9'2' 'P2) e' =

(10) i.t. the scalar produft!�. ,v_x <s is f()JlTl�d_by adding th scalar produc_ts of the correspondin�!!!P9J}��!! of. 4> _�l!..d 1Jl in . .\>J in.. �greement �with_ (9). Operators in the product space e x <S can now be subdivided into operators which operate in <S only, in e only and combinations of these two which operate in both spaces. The operators in spin space are 2 x 2 matrices with constant co-. efficients

[: :] [:� = [::: ::::J

i.e. such operators transform each component 'Pi into a linear com­ �a_�on �f .�I Ha_n(L9'2' .�Lthe operator represents an observable it

must be Hermitian,j-,�,--a _�nd_d_anueaLand_b =" . The Paul; matrices are Hermitian as reqmred . . Apart from these operators in <S we have operators in e which are functions of the coordinates and momenta just as in the spin-free theory. These operators are now understood to be applied to both components of the wave function tfJ, e.g. �1;. grad) 9'll ,. i1;. ad) 9'1 1 = 9'� - lli grad) 9'� , gr more accurately we should write such operators as multiples of the unit matrix in spin space, i.e. if we simply write for i1;. grad), (11) H

F(r

_

[

[FF((r,r,

1 14

-

F F(r,

-

26.

SPIN

From (1 1 ) i t follows that a n operator in � only, commutes with an Qpentoun_e� onlYdince any opera!or oU� a �!l!!!P!� 2LtJje unit � _ Il!atrix ip. the spin spac�, Jmd a!!y oEerntor in��n�ain� _only__c�mst�m... matm . .�kments_._and_ hen�e alwaYLcommuteL.Wit1L aD.}'- operator.- iii grad). Thus the operators referring to the orbjfaLangular momentum of an electron, lit, I" In 12, always commute� with operators � - r��fe_rring � the �e l�ct1".�� spi� sK� !;' s;'S2; _ - - - ( 1 2) [Sit Ij] = 0 ( i,j =x, y, z) etc. We can also define operators which operate in both .v and is, e.g. F(r, - iii grad) , O (r, - in grad) , H(r, - iii. grad) , J(r, - iii. grad)

___F(�,

�

. -

-

-��

-

- -- -- - � - --

[

- - - - . � -- , - - - - - -

]

but we shall not require these. With the above definitio�s of operato_rs, inn(!! E!.2ql:lcts,�_�e obtailL-. expectation values, probabilities, etc in exactlY Jhe same way as for particles without spin. Consider the wave function . l/J = CIPI (r) a + cZP2 (r){J with (PII PI) = (pz, P2) = 1 . The normaliz
I c, 1 2 + ICz I 2 = 1 .

The probability of the electron descr!l��d�y_ l/J l1�vi}!g�z-compOIle1J.t of spin ±t is Ic, 1 2, I C2 1 2 respectively. Tlt..
�(cIPla + C2P2P) ) = ICI IZ(PII ApI) + I C2!� (p2'PKtt'2) ' (

� !


i.e. it is the mean of the expectation values oC p.. injh�states�-9'l� ancL P2, .suitably weighted.

As second example we consider an atomic electron whose Hamil­ tonian function can be written H = Ho + H, ( 1 3) where Ho is a term independent of spin which is spherically symmetric and H, represents a spin-dependent term which, in general, is a small correction compared with Ho. Since Ho is spherically symmetric, it follows from section 25 that Ho commutes with the orbital angular momentum operators ( 14) [Ho, lit] = 0 etc. In the spin-free theory we neglect spin completely. Ho then repre­ sents the complete Hamiltonian of the system and, on account of ( 1 4), the total orbital angular momentum and its z-component are constants of the motion. The energy levels of the system, i.e. the eigenvalues Ell" of Ho are (21 + 1 )-fold degenerate, as follows from equations (25. 10) .

1 15

27.

ADDmON OF ANGULAR MOMENTA

As the second approximation we allow for electron spin, but still ignore the spin term H, in the Hamiltonian. Each eigenfunction tp(E... " I, m) of Ho gives us two eigenfunctions tp(E", ,, I, m) a and tp(E... " I, m)p belonging to the same energy level Ell, ,, whose degeneracy thus becomes doubled to 2(21 + 1 ) . In addition to (14) we now have (15) [Ho, s..] = [Ho, s,,] = [Ho, sJ = [Ho, 8 2] =0 since Ho is a multiple of the unit matrix in spin space, i.e. we can take the operators Ho, P, I� and s�, which are constants of the motion, as a complete set specifying the system * . Finally, we consider the accurate Hamiltonian function (13). The effect of the spin term H, is to shift and in general split the energy levels E", ,, as different spin states have different interaction energies. In general, the orbital and spin angular momenta are no longer con­ stants of motion. Instead, the tQt�1 angular momentum of the elec!!"Qn_J=It� �olllJ!lutes_�th the HamiltQn H. _ The statc_oLthe system is now specifiedk_the_eigenvalues _ ofj 2 amLjc,_each-such-state­ being_a....i\lp_eJ"p9si_tiQIl Qf_J!jgens!aJ�Lthe orbital and spin an�__ momentum operators. _To investigate this problem more fully we must be able to add angular momenta in quantum mechanics. We consider this problem in section 27. 27.

Addition of angular momenta We must next consider how the angular momentum of a composite system is related to the angular momenta of its constituents, i.e. the addition of angular momenta. This problem is of great practical importance in nuclear and atomic physics where we often deal with systems of many particles (protons, neutrons, electrons), and it is the resultant angular momentum of the nucleus or atom which is of direct physical interest. In section 26 we considered in detail the combination of spatial and spin wave functions of a particle and found that the combined system is described in the product space � x e, which is a natural generaliza­ tion of the spaces $j and e appropriate for the description of spatial or spin properties by themselves. Before considering addition of angular momenta, we must deal with this problem of combining two systems EI and E2 generally. The situation is analogous to that of section 26. By itself, each system E. (k = 1 , 2) is described in a function space iJ�, spanned by a complete The combined EtlC�), set of orthonormal vectors EIC!), E2(!), -.-System E is described in the product space ff =iJL�i!2' S1Lann� by the �omplet��rth0!l0Inlal set EI!'�)_ EnC�(m, n = 1 , 2, ) . The scalar •

•

•

•

•

•

•

.

•

• For a single electron 62 is of coune always a constant of the motion, with the value f/i2, since both a and p and hence any vector of 4) x 6 are eigenfunctions of 61 with this eigenvalue.

1 16

27.

ADDITION OF ANGULAR MOMENTA

products in i:j are defined if we define them for the basic vectors Em( ') En(2) in terms of which any vector of i:j can be expanded. We define this scalar product as (1) ( �"P) �P), �/P) �,,(2») = ( Em(l), E,,(1») (�n(2), E.(2» = 8m,,8/IV' Operators which r��.I: to one sysJ�IP only, ��y to Ih operate�!1Jyjn aD -,,!J:
�l

i:j2'

9'(ri)

=

II9'(rl )'P(r2)f(rl)g(r2)dTldT2 I9'(rl)f(rl)dT.J'P(r2)g(r2)dT2' Operators referring to particle I only (e.g . rl or - gradl) are unit operators as far as particle 2 is concerned, e.g. (J(rl)g(r2), rl9'(rl )'P(r2» = (J(rl), rl9'(rl» (g(r2), 'P(r2» ; a typical interaction operator would be e2/ ( Irl -r2 /) representing the Coulomb repulsion between the electrons. The spin space @Sl x @S2 of two electrons, spanned by a l a2 , a 1P2, {JI a2, {Jl {J2, represents a second example of a product space. iii

Mter these preliminaries, we return to consider the addition of the angular momentum vectors of two systems Ii' (k = l , 2), described in function spaces iSi ' For each system by itself we define an angular momentum vector M(k) and the related operators A ± (k) = M. (k) ± iM, (k) (k = 1, 2) (2) as in section 25, and we assume that for each system commutation relations analogous to equations (25.6-8) and (25. 1 1-1 4) hold. Since the operators M{l) anLl\t:(2) refer to two diiferc;P-t. �y�te��__m.�>,­ com!!!�teLb��. (3) [M,(I), M,(2)] = 0 where M,(I) stands for any ��mponent ofM( I ) and similarly_for M,(2) . We define the to!�or resultal!L�I!lrnlar momentum vector of the combined .sys�em E bX M =M(l ) + M(2) (4) with analogous defi�itions for the cOIl}E9pents and for A ± . On account of (3) one can easily prove that the total angular momentum operator 1 17

27. ADDmON OF

ANGULAR

MOMENTA

(25.6-8)

(25.11-14) M2 (k), M�(k) have in general to be supplemented by a set of operators Q.,t to form a complete set of commuting observables for the system Ii' For the combined system, we could take the set of commuting operators Q.., �, M2(I), M2 (2), M�( l ) and M�(2) as a complete set. But since we are interested in the resultant angular momentum of I we prefer to replace the observables M�(l ) and MA2 ) by M2 and M�, and to take the . Since com lete set of observables Q.h �, M2 I , M2 2 , M2 and M satisfies the same commutation relations and as M(l) and separately (if Exercise VI.7). As in section the angular momentum operators

M(2) 25,

M

we are �!!J'_�l!lte£este �n !lle �J!g!!l�r l!!�mentl:!IIl p�Qpel'!i�s 9J�the system we shall for simplici!Y...Q.�L�� �perat0l! Q.l �J!d _Q2 �_d �eir -.cigenyalues, assuming that weu are_��il},s deali!tg_wi!h the same set of eigenvalues ql and _The state of the combined system can then or M20� be fl,lUy--.!!p���fi,�(;lJ)J' either M2 and Since the total angular momentum M satisfies all the conditions \\ stipulated in section the main result of that section holds : M� c:.a.n_ � only have the eigenvalues + i, I, L�!lc! these �re _@!)-fold ��gener�JC! _ cOf!"espgnding _ to klr. having �the eigenvalues \' Itm, - n, =- n�± IJ�' _'u ._n - l , n). Suppose that has the value n" for the system what are the allowed values of the resultant angular momentum and how are the corresponding eigenfunctions derived from those of the constituent systems ? 41 We shall not give the complete treatment of this problem which is straightforward but rather tedious. It can be found in many standard text·books (e.g. Condon and Shortley2 1 or van der Waerden23) . We shall only give enough of the theory to make the results appear plausible and we shall illustrate their applica­ tion to particular problelllS. Suppose the angular momentum which of 1:" has the value )-fold degenerate. The wave function of is then a vector is of if. spanned by lying in the )-dimensional manifold IDl"t using the notation of section but omit­ ting the eigenvalues (]I.. The possible states of the combined system I)-dimensional manifold then lie in the spanne!Lbyihe produc�e�unctions _

Q2_' 2 M (!)-,_�l!t':l2"-M«rnJ MA2),

M2(2),

M�._ 25,

(m =

1i�n(n 1 ), (n =0,

.

•

.

Ii' (k = I , 2), M

M(k)

ni M(k) I" (2n,l + 1 (k) (2n" + 1 25 P,,(ni' -n.), Pi(nj, nj), (2nJ + 1)(2n2 + IDlu.(1) xIDl'lt(2) (5) P1 (n., ml)P2 (n2, m2) (m,,= !l . . n,,) . Corresponding to this manifold, the total_�ngula!" Eloment!lm M2 assumes each of the values 1i2n(n +�, with (6) n=nl +n2, nl +n2 - I , . . . nl - n2 +-- I, nl -n- 2' •

•

•

-

•

is in

�, _ '

We ahaI1 again for brevity say that M(k) has the value lit when we mean that the system llD eigenstate of Ml(k) corresponding to the eigenvalue /i11lt(nt + 1).

118

27.

ADDITION OF ANGULAR MOMENTA

where for definiteness we have assumed that n l >n2' }!;!\ch eigenvalue 2 }i 1!(1! +JLis (2'! + I)-fold degener�te; the cori"espolldi!lg maIlif9ld 9R" �ll
decomposition of the product manifold 9R",( 1 ) x9Jl"J�) into��l!lanifo�s each of which ��!"responds to one -�fini!�.-e!genvt:llue::-oJ M;�sy:nib�Jicallx��x t4e eguation ml�, wliicharemutuciHy orthogonat,

"'+"1

9R",(I) X9R"1(2) �9R". =

n-n.-"l

( :/ "

,

(

\,: . ""

" " \; (7)

Though we shall not prove this result;-we can see that it is reasonable. Adding the dimensionalities of the manifolds 9R" occurring on the right­ hand side of (7) we obtain + [2 (nl + n2) + 1] [2 (nl - n2) + 1] + [2 (nl - n2 + + 1] + =t {[2(nJ - n2) + I] + [2(n, + n2) + In ( 2n2 + = (2n, + 1 ) (2n2 + I ) , agreeing, as required, with the dimensionality of 9R",(I) x9R"2 (2). Secondly, the allowed angular momentum values (6) are plausible by comparison with the corresponding classical result. Classically, angular momentum vectors M(l } and M(2) have a resultant M whose magnitude lies in the range (8) I M( I ) - M(2) 1 <M <M{l} + M(2) , as illustrated in Figure 10, for M(I) >M(2) .

1)

Figu" 10. Addition ofQllgu/Qr mommIum IMcIors; M( I» M(2)

5

+ Q..M.

119

•

.

.

I)

27.

ADDmON OF ANGULAR MOMENTA

Since all functions in ID'l.., (I) xBnIlz(2) are linear combinations of the product functions we can, in particular, write the !P(nh n2; n, m) in this way:

(5),

II, !P(nh n2; n, m) = 2:

liZ 2:c:.,DZmz !PI (nh ml) !P2(n2, m2)'

We can immediately simplify (10) . follows that

For since

M� = M,(I) + M,(2) it

M,!P1(n" ml) !P2(n2, mz) =A(ml + m2) !Pl (nh ml) !P2(n2, mz) ,

i.e. only those ter� Q.ccur lll. (101for_�l1ich We can hence write (10) as

(10)

m = ml + mI'

(11) (12)

(13)

!P(nh n2; n, m) = 2:Cn':.,lIzmz !PI (n., ml) !P2(n2, mz). m, +mz If we fix the normalization and phase factors in !pj;(n�: mj;), then the coefficients C;",IIZm are completely determined, apart from an 2 arbitrary phase factor, by the condition that Y'(n., n2; n, m) should be a nonned eigenfunction of M2 and satisfying These coeffi­ cients Cn�,IIZml defining the appropriate linear combinations of wave

M,

(9).

functions are known as Clebsch-Gordan coefficients or _alI_ Wigner coefficients. They can be calculated using the properties of angular momentum operators (see below) or using group theory. But in practice one does not usually have to calculate them as they have been extensively tabulated for the case where the wave functions Y'j;(nj;, mj;) in ( 13) are the angular momentum eigenfunctions as defined in section 25, equations (25.22) and (25.25) *. We now see why a con­ sistent choice of phase factors in the wave functions Y'j;(nj;, mj;) is important, as stated in section 25. We are now taking linear com­ binations of wave functions, and while an arbitrary phase factor is unimportant in the wave function as a whole, this is of course not the case for individual terms in a wave function. (CI Exercise VI.G, where this difference is illustrated.) We will now indicate in outline how some of the above results come about. The exact procedure will become clearer from the particular examples which follow. • The Wigner coefficients for nz = t, I, i, 2 arc tabulated by Condon and Short1cy11 , pp. 76-79; for nz =t, 3 they arc tabulated in the CatUlllian Joumid oj PIjysiu, vol. 30, p. 253 (May 1952). follows

as

In bOth these references the Wigner coefIicienta arc written in Dirac notation,

c:'m'"lml ... (nlnzmlmzlnln1'l'l)l .

van der Waerden23 gives tables for nz ""t, 1. He abbreviates the Wigner coefficient but the other subscripts and superscripts arc of course implied. A to notation which has many advantages is c:�:�.:. Instead of the C:�.:�:. one often uses defined by C':�::':: =C':::�D 8 111,"'1 since the Wiper coefficients vanish unless

c::'m,nzml C:,ml' c.:�:�n, ml + mz=m.

+m2'

120

(11 )

27.

ADOmON OF ANGULAR MOMENTA

states that !PI (n., ml) !P2(n2, m2) is an �ige�functio.�
m...ujmk =:.::-.1l.b�nk)�JQI' whic::lut!_=,-'!11 ±.t1l1."- _ The maximum eigenvalue of MI; is Ii(nl + n2) ; this eigenvalue is not degenerate ; the eigenfunction is !PI(n., nl) !P2 (n2, n2) . As in section 25, the condition

A-A+ !PI (n., nl) !P2 (n2, n2) = 0 gives the eigenvalue of M2 as 1; 2 (nl + n2) (nl + n2 + By successively operating on !PI (n., nl) !P2 (n2, n2) with A_ we generate the 2 (nl + n2) + eigenfunctions of MI; which span 9nnl+ltl. These eigenfunctions are

1) .

1

not yet normalized. The eigeny�lu�_�('-'l fE2.� D_ QfMl; is two-fold�erate; we have two linearly independent eigenfunctions !PI (n., nl) !P2(n2 , nl and �(nh nl - l)}l'z(nz,!i2Y aiid any-rmear comliiioiatf n of these is an eigenfunction of M& with eigenvalue li(nL+ n2 - Do!. One of these linear combinations corresponds to

- 1)

(14-)

A_ !PI (n., nl) !P2(n2, n2) a_nd belongs " to JDln, +nl (i.e. M has the eigenvalue _nl +_n2)-"---�h.c:.�ther linearly independent_combination is .orth.9gQI!3IJ9jJ1-) J!.nds.QI:t�PQn.dL to _I and, the eigenvalue (nl + n2 - I ) , i.e. .it belongs .to 9n1lI +" . 1_ " . M having . .

_

. .

_

_

in fact, is the largest possible eigenvalue of M,;jI!..tltis. ma.l!ifol� We obtain this linear combination from the condition that

0

A+{a!PI (n., nl) !P2(n2, n2 - 1 ) + b !PI (nl, nl - I ) !P2 (n2, n2)} = . ( 1 5) Starting from this eIgenfunction, we can again generate all 2 (nl + n2 - I ) + I eigenfunctions of MI; spanning 9n"I +"l - 1 by means of the operator A_ .

In this way we can continue to generate all eigenfunctions in the product manifold 9n"I ( I ) x 9n"2 (2) . I.!Ll.lsing_th�operators A:l; as we have.-cl�fiJll!.d _them,.Qlle_mustremember...1haUhcy..lead to eigenfilnctions which are not yet normed. In considering some particular examples we introduce the conven--· tional notation : I, s, j = 1 + s, are to denote the orbital, spin and resul­ tant angular momentum vectors of a single particle ; L, S. J =L + S are to denote corresponding vectors for a system of particles. We call states in which I has the values 0, 2, . . .• S-, p-, d-,J-, g-, h-, states, and states in which L has these values, S-, P-, D-, F-, G-, H-, states. We specify spin states by the value of 2S + I , rather than by S itself. 2S + I is known as the multiplicity of the spin state. For S = O, !. 1 , t, . . we talk of singlet, doublet, triplet, quartet states, etc. ----I -' Example I . The total angular momentum j of an electron with orbital angular momentum I can, according to (6), have the values I ±!. which have degeneracy (21 + 2) and 21 respectively. We con­ sidered this case at the end of section 26. On account of the spin-orbit

I, 3,

•

.

.

.

•

•

\

.

12 1

27.

ADDmON OF ANGULAR MOMENTA

interaction which is different in the two states each single energy level of the spin-free theory splits into two levels, a so-called doublet. In atoms, where the spin-orbit interaction is small (for not too heavy atoms at any rate), these two levels lie very close together. If a magnetic field is applied, these two levels split into (21 + and 21 levels respectively, corresponding to the different interaction energies of the total magnetic moment of the electron with the magnetic field. In the spin-free theory we should have expected a splitting into (21 + I ) levels. s-states (I = 0) differ from other states, in that, from (6), j can only have the one value j =t, corresponding to a two-fold degenerate level which splits into a doublet in a magnetic field. The spectra of the alkali metals, which are essentially due to a single electron ( the valency electron) in a spherically symmetric field ( the Coulomb field of the nucleus, screened by C closed shells ' of electrons), are in agree­ ment with the above interpretation. The two famous D lines of the sodium spectrum represent a typical example. They are due to radiation which the electron emits in falling from an excited p-state (1= 1) to the ground state, which is an s-state ( / = 0). Whereas the ground state is single (j =t), there are two excitedp-states with different energy levels, corresponding to j =t and j = t. The two observed spectral lines Dl and Dz are due to transitions from the j= t and j=f states respectively. Example As a second example we consider the resultant spin states of two electrons : St =Sz =t, hence the total spin can have the values S =O, 1 ; a singlet and a triplet state. The former is non­ degenerate, the latter three-fold degenerate, giving the four linearly independent spin states which span the 4-dimensional product spin space 51 x 5z. Equation we can now write as 5 1 x52 =IDlt +uno' We span the three-dimensional manifold unl corresponding to total spin S = by eigenfunctions of Sc. The eigenfunction corresponding to Sc = is al az: ( 16a) P(t, t; I, I ) = al a2' \ 1 17 Since·from equations (25.35) A_ ( at az) = Ii( alPZ + a2Pl) ,

2)

2.

(7)

1

1,

it follows that the normed eigenfunction of Sc with eigenvalue 0 is 1 10)

and similarly

I H '/

I

pet, ti 1 , 0) = -vI2 (aJPz + �1) '

P(t, t i 1 , - I ) = PIP2'

( 1 6b)

( 16c) We can find the vector defining the one-dimensional manifold IDlo easily. This vector must be of the form,

aaJ {J2 + ba2f1.,

1 22

27.

ADDITION OF ANGULAR MOMENTA

and it must be orthogonal to �(t, t; 1 , 0), equation ( 1 6b) . This gives at once a + b =0, i.e. I ( 1 6d) o 0 ,---' �H, t ; 0, 0) = v'2 ( aIP2 - alPI ) · -

I

We could equally well have obtained this vector from one of the con­ ditions A ± (aaIP2 + ba2Pl ) = 0, expressing the fact that IDlo contains no eigenfunctions of S, with eigenvalues equal to ±Ii. We summarize these exceedingly important results : the spin states of two particles of spin t divide into a triplet state, S = I , with spin wave functions

ala2

( 1 6a)

v'2( a lP2 + alPl )

( 16b)

I

( 1 6c) PIP2 (which are eigenfunctions of S, belonging to the eigenvalues + n, O, - Ii respectively), and a singlet state, S = O, with spin wave function I ( ) v'2 a1P2 - alPl .

( 1 6d)

In this second case, the resultant spin is necessarily zero : we can think of the individual spin vectors of the two particles as pointing in opposite directions : we say the spins are antiparalleI. We can think of the spins in the triplet states, in which S = as being parallel. One must of course not interpret this somewhat classical picture too literally. Whereas the triplet spin wave functions are symmetric in the two particles, the singlet spin function is antisymmetric. Example 3 . Finally we consider the combination of the orbital angular momenta of two electrons in p-states, i.e. II = 12 = I . The resultant product manifold

I,

IDll

(I )

x

IDl,

is spanned by the nine wave functions

(2)

�1 (1, ml ) �2 ( I , m2) (m. , m2 = - 1, 0, I ). The resultant orbital angular momentum L assumes, according to (6), the values L = 2, 0, i.e. we have a D-state (5-fold degenerate), a P-state (3-fold degenerate) and an S-state (non-degenerate) : <.

I,

(2) =IDl2 +IDll +IDlo. The manifold IDl2 of the L =2 states contains the vector �( I , I ; 2, 2) = �l ( I , I ) �2 ( 1 , I ). 123 IDl l ( 1 )

x IDll

(17a)

27. From it, one obtains by repeated application of A_, using equations (25.23) and (25.24) and normalizing the resultant wave functions, !P(I, I; 2, I ) = y1 2 {!Pt{I, 0) !P2{I, I ) + !PI (I, I) !P2{l, On ( 1 7b) !P(I, I ; 2, 0) = y1 6 {!Pt(I, - 1)!P2(I, I ) + 2 !Pt(I, 0)!P2 (I, 0) + !Pt(I, 1 ) !P2{I, - In ( 1 7c) 1 !P{ I, I; 2, - 1 ) = y2 {!Pt(I, - 1)!P2(I, O) + !Pt{l, O) !P2 (I, - In ( 1 7d) ( 1 7e) !P(I, 1 ; 2, - 2) = !Pt(1, - 1)!P2(I, - I ) . ADDITION OF

ANGULAR

MOMENTA

(The normalization of these wave functions is easily obtained, since the individual terms in each wave function are mutually orthogonal. Hence N ...

'P1 (n. , mI) 'P2(n2, m -mt)} = 2 la... 1 2.) The vector 'P(l, I ; I, 1) of the manifold IDlt is orthogonal to !P(I, I; 2, I ), equation ( 1 7b), and, furthermore, is the only other vector belonging to the eigenvalue L, = 1. Hence !P(I, I; 1, 1 ) = �2{'Pt(l, 0)'P2(I, I ) - !Pt(I, I )'P2 (I, O)}, ( I Ba) and applying A_ successively and normalizing, one obtains 'P(l, I ; 1, 0) = �2{!Pt(I , - 1)!P2 (1, I ) - !PI(I, 1 ) !P2( I, - I )} (18b) {2a "'I

"'I

and

!P{ I, I ; I, - I ) = �{!PI ( I , - I )!P2(1 , 0) - !PI (I, 0)!P2 (1, - I )}. (18c) (We could equally well have obtained !P(I, I ; I, I ), equation ( I Ba), by choosing a linear combination such that A+{a!Pt(1, 1 )!P2(I, 0) +b!PI(I, 0)!P2 (I, I )} =0 which at once gives a +b =0.) Finally we want the vector !P(I, I ; 0, 0) which defines the manifold by choosing numbers A, p, v such that IDlo• We most easily find either of the equations

this

A± {A!PI (I, 1 ) !P2 (1, - I ) +,u !Pt(I, 0)'P2(1, 0) + vWI (I, - 1 ) !P2(1, In =0 A + p=O, v + p =0 W( I, 1 ; 0, 0) = J3{WI (1, 1 ) !P2(1, - 1 ) - !PI (I, 0) !P2( I, 0) + !Pt( I, - 1 ) !P2(I, I )}. (19) 1 24

is satisfied. One at once finds that I.e.

28.

SYSTEMS OF IDENTICAL PARTICLES : PAULI PRINCIPLE

Using equations (25.28), (14.4) and (13.26) one easily proves that the wave function !P( I, 1 ; 0, 0) is proportional to the scalar product r, r2 of the position vectors r, and r2 of the two electrons, and is there­ fore invariant under rotations of axes. Thus the state !P( I, 1 ; 0, 0), corresponding to zero angular momentum (L =0), is spherically symmetric. We had an analogous result for the orbital angular momentum of a single particle, in section 14. In Chapter IX we shall see that this is a quite general result; a system with zero orbital angular momentum has an orbital wave function which is spherically symmetric. i.e. it is invariant under rotations of the coordinate axes. 28.

Systems of identical particles: Pauli principle Let us consider a system containing identical particles, say several electrons or protons or neutrons. This means that the two particles are physically indistinguishable. Two electrons, for example, are indistinguishable. We may be able to specify their position as being at two points Ph and P2, but it is meaningless to enquire which electron is at P, and which is at P2• Mathematically,_tbis jd�n.tiJrjs _expressed by the fact that the Ha�ltonian . of theJl�tem is _ symrn_e_tri�Jn _ th!! coordinates oLthe J )articles : in!ercha.nging the coordinate� QL�J}y !��Q identical particles leaves the Hamiltonian invariant. FQ.L�!l!}pJ!ci!y� we assume that we deal with two identical particles whose coordinates we label 1 and 2, then H( I, 2) = H(2, 1 ) . (1) We had a typical example of such a symmetric Hamiltonian in Exercise 11.2 : the Hamiltonian of a helium atom is symmetric in the two electrons. Let us now consider the eigenvalue problem of the Hamiltonian ( I ), H( l , 2) 'P (I, 2) = E'P ( I , 2) . From ( 1 ) it follows (if Exercise VI. l O) that if 'P ( 1 , 2) is an function belonging to the eigenvalue E, then 'P(2, 1 ) is also an function of the same eigenvalue, i.e. E is degenerate and any combination X ( l , 2) =o'P (I, 2) + 6 '1'(2, I )

(2) eigen­ eigen­ linear (3)

is also an eigenfunction belonging to E. Not all wave functions (3) represent physically permissible states. Since the two particles are indistinguishable, all physical predictions, such as expectation values and probabilities, must be symmetric in the coordinates of the two particles, i.e. they must be independent of which particle we label 1 and which 2. We require, in particular, that for any wave function 'P(I, 2), (3) should imply

IX ( I , 2) 1 2 = IX (2, 1 ) 1 2• 1 25

(4)

28. SYSTEMS OF mENTICAL PARTIOLES : PAULI PllINCIPLE One easily proves (if Exercise VI.l l ) that (4) implies a = ± h, i.e. the allowed wave functions are (5) X ( I , 2) = 11' ( 1 , 2) ± lp (2, I ) , i.e. they must be either symmetric or antisymmetric: in the former case, i!1te..cha!1ging the particles leaves the_ wave function invariant, in the l�tter ��e, interc4.�Jlgi!lg the particles inult1p!!ei_tli� �ave tun�t!o.!!�_ -'-= 1 )___ We can express this more elegantly if we introduce the particle exchange or permutation operator P12 defined by PI 2'1'( I , 2) =11'(2 , I ) . (6) Symmetric and antisymmetric wave functions then have the property that PI2X ( I, 2) = ± X ( I , 2) respectively, i.e. they are eigenfunctions of Plzwith the eigenvalues ± I respectively. It follows from the time�dependent Schrodinger equation that the symmetry character of a wave function persists throughout time. For

(7)

implies that

iA; lp, ( I , 2) =H( I , 2) '1', ( 1, 2) t

dt dt P12d'l',(I , 2) = liH(I , 2) '1', (2, I ) = ±JtH( I , 2)11', ( 1 , 2 ) = ± d'l', ( I , 2) ,

according as P12lp, ( I, 2) = ± 1p, (1 , 2 ), i.e. the wave function remains symmetric or antisymmetric throughout time : ·iiS-iY:inme!!:y- character is a constant of the motion. We can also see this directly from the operators : (8) [PI 2, H( I , 2) ] since for any wave function 11'(1, 2)

==0,

PaH( I , 2) '1' (1, 2) =H(2, 1 ) 11' (2, 1 ) .... H( I , 2)Pl zlp ( l , 2). The operator P12 exchanges the particles 1 and 2 , i.e. it exchanges all their coordinates, both spatial and spin. It is often convenient to use position and spin exchange operators P12'" and PIZtl which respec­ tively exchange the position coordinates and spin coordinates only, and clearly P12 =PIZIIPI 2" = P12t1PIZ1t. Thus if «P(rlJ S,(I); rz,s,(2» denotes a two-electron wave function then P12"'«P(rlJ s,II ) ; rz, Se(Z» = «P(rz , se(1) ; rlJ sP» , P12t1«P(rlJ S,II) ; rZ, $,(2» = «P(rlJ S,(2) ; rZ, S,(I» , and

(9)

126

28.

SYSTEMS OF IDENTICAL PARTIOLES : PAULI PRINCIPLE

SO fa.. �e have sho� tha! wave functions of identical particles are eith� symmetric -or antisymmetric.--In -pr�lctice onefindi that the­ w.�v�Junctions for a particular kind of elementary particle are either al��ys symmetric or always antisymmetric, i.e. only if we restrict the wave functions in this way do we obtain agreement with oDserv�tion. Thus the wave functions of particles of spin l sucl1 as electro�si­ trons, protons and neutrons are always antisymll!etri_<: and the wave functions of particles with spin zero (n-mesons) or wim l!ID!LJ (photons) are always symmetric. This distinction between particles whose wave functions are symmetric and antisymmetric is most funda­ mental and leads to different behaviour if we consider an assembly of particles, i.e. it leads to different statistics. Particles withintegralspin (sYIllJ!letric wave functions) satisfy Bose-Einstein s�tisti�L I>articles 'YiJ!t ' ha!f ()dd-::i ntegral' spin (antisymmet,!ic w�ve functiQBS) satisfy­ Fermi-Dirac statistics. Consequently- they are often called-bosons-and­ fermions respectively. The Hamiltonian of two non-interacting identical particles separates into two parts H( l , 2) =H( l ) + H(2) . If , ",,,(k), "'1 (k), "'2(k) , , (k = 2) are the eigenfunctions of H(k) , corresponding to eigenvalues Eh E2, E", , then the eigenfunction of H( I , 2), belonging to the eigenvalue Em + E" is __

•

•

•

.

•

4J1IIlI ( 1 , 2) =

•

•

•

•

•

•

1,

•

�2{",,,, ( l ),,,,,(2) ±", (2)",,,( 1 ) }. ..

(10)

If the particles are fermions, we must take the minus sign in ( 1 0). Hence for m = n, 4J",,,( l.-ID -O : it is not possible [Or two identical iermions p" .k be injtjentical slates. 1¥s_ is th_e__original formulation of Pauli's ex­ cl�on principle. However, if the particles interact, it is no longer possible to separate the Hamiltonian and the unsymmetrized eigen­ functions are no longer products of one-particle eigenfunctions. Hence we can then no longer formulate the Pauli principle in this way. But our original statement that the wave function must be antisymmetric is a generally valid formulation of Pauli's exclusion principle. Nev�,:, theless. even in the case of inJerac!i!!gJermio� it is often usefut to talk of individual particle states_and p_�oduct w
-� �"' /' r

�

28.

SYSTEMS OF IDENTICAL PARTICLES : PAULI PRINCIPLE

27

In section we derived the wave functions of systems containing several particles with definite angular momentum properties. We now illustrate, for some particular atomic and nuclear systems, how this procedure is to be modified if the particles satisfy the Pauli exclusion principle. To begin with, we remind the reader that the triplet spin wave functions of two particles of spin t are symmetric in the spin co­ ordinates, whereas the singlet spin wave function is antisymmetric [equations a-d)] :

(27.16

Example 1 . As first example we consider two atomic electrons, the spatial wave functions of which are so that the spatial and wave function of the combined system, apart from its symmetry properties, is Combining this wave function with the spin wave functions (27.16a-d) we can form four antisymmetric combina­ tions. We obtain the singlet state

9',,(1)

9'..(I )9'6(2}.

9'6(2),

(S=O) I [9',,(1 )9'6(2) +9',, (2)9'6 (1)] I ( aIPz - azPI) ( l l a) '\1'2 y'2 and the three triplet states (S = I ) J2 [9',,(I )9'6(2) - 9'..(2)9'6( 1 )] x )� aIPl + a,pI ) �:::� ( l id) PIPZ ' and these wave functions are normalized if 9'" and 9'6 are normalized. If the two electrons are in the same s-state (we call an s2-con­ figuration) , then 9'..(r) =9'6(r) and only the singlet state ( l l a) possible. We denote state by the symbol ISo, the general notation being 2S+ I Lj where 2S + I the spin multiplicity of the state and J

{

this

this

is

is

its resultant angular momentum. If we consider a p2-configuration we have all together 36 linearly independent product wave functions (three spatial and two spin, giving six total wave functions for each particle, and the total wave functions for the two particles can be combined in any order) . We easily enumerate the antisymmetric states. Firstly, we see that there are 1 5 such states only : the total wave function of the first electron can be chosen in ways, that of the second in only 5 ways ; this gives 30 possibilities; but the order of the electrons is immaterial since we anti­ symmetrize the wave function : hence there are 1 5 possible states. To classify these we remember that triplet spin wave functions are symmetric, singlet spin wave functions antisymmetric, and that the p2-configuration gives states with I , O. Of these the and states are symmetric, the L = I state antisymmetric (if section

6

L .... O

L = 2,

1 28

L=2

27,

28. SYSTEMS OF IDENTICAL PARTICLES : PAULI PRINCIPLE example 3 , particularly equations (2 7. 1 7-19)). Combining these results we find the possible antisymmetric states by combining symmetric spin states with antisymmetric spatial wave functions, and vice versa. In this way we obtain the following states :

( 12) We obtain three different P states, as the spin S = 1 can combine with the orbital angular momentum L = 1 to give resultants J = 0, 1 or 2 . One easily checks that the total number of states in ( 1 2) is 1 5 (2S+ I L J corresponds of course to 2J + 1 states) . It is not difficult to obtain the explicit wave functions of these states, using equations (2 7.1 7-19) and (2 7. 16), but we shall not write down these rather lengthy formulae. The reader should have no difficulty in deriving them if he so desires. Methods of finding the antisymmetric configurations of more com­ plicated systems have been given by van der Waerden 23• Example 2 . As second example we consider the ground state wave function of the helium nucleus or a-particle. The a-particle consists of two neutrons and two protons and the ground state is a ISO state, if one assumes the forces between the nucleons (protons and neutrons being the basic constituents of nuclei are called nucleons) to be central forces. Let 1 and 2 denote the protons and 3 and 4 the neutrons. Com­ bining the proton spin states we get a triplet and a singlet state, and similarly for the neutron spin states. To obtain a singlet state (S = 0) for the a-particle we must then oither combine the two singlet states giving the spin wave function

(13) �..( i2 j 34) = !- ( alP2 - a2Pl ) ( a3P4 - a4P3 ) , or we must combine the triplet spin wave functions analogously to the way we obtained an L = 0 state from a p2-configuration in equation (2 7.19) . The resulting spin function is - 1

{

E,(1 2 j 34) = 3 ala2P3P4 - !- (alP2 + a2Pl) (a3P4 + a4P3) .y + PI P2 a3 a4 '

}

(14)

4>(12 ; 34) = a9',( 1 2 j 34) E.. (12 ; 34) + b9'.. ( 12 ; 34) E, ( 1 2 j 34)

( 15)

The spin states (1 3) and ( 14) are respectively antisymmetric and symmetric in the like particles (we indicated this by the suffixes a and s, and by placing tildes and bars over pairs of particles in which the wave functions are antisymmctric and symmetric respectively). Hence the total wave function of the a-particle is of the form where 9', and 9',. are spatial wave functions with the indicated symmetry properties and a and b are constants specifying the mixture of symmetric

1 29

EX£RCISES VI

and antisymmetrie spin states.

i.e.

By comparing the calculated with the

experimentall)' observed binding energies orthe ground state, onc finds that

Ibl
in the ground state the protons and neutrons arc

predominantly paired off by themselves into singlet states.

EXERCISES VI

VI.I.

Prove equations (25.12) to (25.14).

V1.2.

Prove equation (25.2'�).

V1.3.

What is the component of thc Pauli spin "cctor of the unit vector n - (1110 11!, 1Il)?

�

Denoting the Pauli matrices by prove the anticotllmutation relations

a;,

a

in the direction

i - I, 2, 3, instead of Oz, 0"

a�,

OPl +olaj ... 28il·

C;� !�

Pl"Ove that any '2 x 2 Hermitian matrix call be written as a lincar ination of the 2 x 2 unit matrix and the Pauli matrices.

co

VI.6. (i) If a. is the component of a in the direction of thc uni t ,'cctor n'with components (sin 0 cos rp, sin 0 sin p. cos 0), find the eigenrunctions of a. n::reITcd to the vectors a and {J as basis {a,a -= a , 0,{J - -P. (a, a) - I ,

I

(P, Pl - I ) . (ii) If 1]1 -!PI(r) a + ljJl(r){J is a

Ilormed wave function, what are the prob· abilities of observing a component of spin ±1i/2 in the dire<;lions ± n respectively?

(iii) Ir wc arc not given the complete wavc function '.1', but only know that the systcm is in a stale in which s( has thc v:lluc.� ±li/2 with probabilities + - I), what nre the prob:lbilities of observing a and component of spin ±/i/2 in the directions ±n respectively?

1'21 2• (kI l l 1'21 2

1 '111

tvq. IfM -M(I) +M(2), and M(k). (/: - I , 2) satisry the commutation l?ir.Ons (25.6-7) and (25. 1 2- 1 4) , PI"O'"C that M also satisfies these relations.

re

. v.if Obtain the total allb'lilar momentum cigcnfunctions of the opcrators � 1lfj� of a single particle of spin 1 in a p-state. 1"1.9.

Givcn three particles of spin 1, Sl =S2 -sl-t, obt:lin the spin wavc � f the combined system which are eigenfunctions of S and S"

runctifu \

�15'J Provc th:lt if H(I, 2) - H(2, I) and H(I, 2)V,(I, 2) -EV,(I, 2), 42. \ ) is an cigenfunction of 1i(1, 2) belonging the same eigen­

then value

to

�) E.

Prove equation (28.5) from the indistinguishability condition (28.4).

Vl.12. The nucleus of the 'helium three ' isotope, written lHe, consists of two protons and one neutron. Its ground state, assuming central forces, is a LSI state. \Vrite the ground state wavc function in a rorm analogous to t.hc a-particle wave fUllction (28.15). 130

VII TIME-INDEPENDENT PERTURBATION THEORY

THERE exist several approximate methods of solution for eigenvalue problems which are too complicated to admit exact solutions. In this chapter, we shall consider one of the most important of these, per­ turbation theory. More particularly, we shall consider the application of perturbation theory to the calculation of the stationary energy levels and the corresponding energy eigenfunctions of systems (such as atoms or molecules) which are too complicated to permit a rigorous solution. 29. First-order perturbation: non-degenerate case The basic idea underlying the application of perturbation theory is the following. Corresponding to the operator H, whose eigenvalue problem we want to solve, we suppose that there exists an operator H(O) differing by only a small term from H, and that the complete solution of the eigenvalue problem of HCO) is known. We write

H = HIO)

+eV

(1)

and, for definiteness. we shall assume that V is of the same order of magnitude as H(OI and that is small compared to unity. The state of affairs here depicted exists in 110t too heavy atoms i the Coulomb repulsion between the electrons represents a small perturbation com­ -pared to the interaction of the electrons with the nucleus. Suppose

e

-

rpl (0), rp2(01,

EI (OI, E2(0),

•

•

•

•

•

•

rp,IO), E,(O),

•

•

•

•

•

•

}

(2)

are the complete sets of eigenfunctions and eigenvalues of the unper· turbed eigenvalue problem,

H(O) 'P(O) = E(O) 91(0),

(3 ) which are assumed known. On the basis of continuity arguments, it is then reasonable to assume that corresponding to each eigensolution rp,(O) and eigenvalue E,(O) of the unperturbed problem, there exists an eigenfunction rp, and eigenvalue E, of the perturbed (exact) eigenvalue problem,

(4)

which are nearly equal to the unperturbed quantities, differing only and the resulting set of by quantities of the order of magnitude

e,

131

29. FI RS1··OItOEII. I'ERTURDATIO;-.o : NO:-;·DECENERA"rE CASE eigenfunctions !Ph Pl '1'" . . is also complete *. Corresponding to a particular unperturbed eigenfunction 'PNIOI and eigenvalue E,,(O) we can then try to express p", and EN as power series in £ : •

.

.

.

.

'PN -tpNIOJ + r.rp./(I) +r.2ipNI21 + l!�\{_ENlO) + E£N(Il+S2E,,(2) +

• .

• •

(5 )

.

(6)

.

where e"rp..)�) and B"£.J�) arc the 11th order corrections to lhe unper· • turbed wave functions and eigenvalues. For the time being, we assume that all the unperturbed eigenvalues EI (OI, E210), • • • E,IO), arc non·degenerate, so that the corre· sponding eigenfunctions, assumed normed, arc completely determined apart from unimportant phase·factors. Substituting equations (5) and (6) into (4) we obtain (I-!10) -I-sV) (p",(O) + EIp",(l ) + e2PN(2) + ) (7) = (£",IO) -I- S£".11) + . . . )(p",IO) + Eip.....lI) + . . . ) .

.

.

.

•

•

.

(8a) (8b) (8e)

Equation (8a) is identically satisfied, as it must be, since i t represents the zero·order approximation of our perturbation problem. To obtain the first·order approximation we must take into account equation (8b) but neglect equation (Bc). We must determine the first·order corrections ip UJ and E.\,UJ to the wave function and the energy so that equation (8b) holds. For this purpose we expand 9" fIl in terms of the unperturbed eigenfunctions 'P,IO),

....

.....

•

ipNllJ _ 2a,p,fOJ.

(9)

Substituting this expansion into equation (8b), we obtain, usmg equation (3),

!a,.c�IO)rp,(O) + Vp,NlO) _EN(O)�a,!p,(O) + £N(1)p",IO).

,_1

,_1

(10)

Taking the inner product with rp,lOI, equation (10) gives a,E,(O) + (p,IOJ, Vpx(OI) _a,E",IOI +EN1U8,J{, (s - I , 2, . . . ). ( I I)

• For limplicity we h:l.I'e :ld.tpted our nOlation to Ihe ease where bolh H(O) and H have eomplelely di5Cl"ele lpec:tr.t ofeigen\Oalues. nut the following argumenu abo CO\OU the ClUe of:l Ih'rtly eontinuou� lpeetnn», by replacing lummadoru o\'cr cigem-:alues by integr.uioru ap lC:r, we are interested in O\'(:r appropriate r.l.lLges of ei�en\'atueso If, :u in Ihe pre.u:1II ch the Ihift due to the perturbauon of a given energy level £,1"1, it is of course implicit {hilt this level belongs to the dilcrete II)Cclruono

132

29.

FIRST-ORDER PERTURBATION : NON-DEGENERATE CASE

Abbreviating the matrix elements of the perturbation V by v" = (9',(0), V'P,(O» , equations ( 1 1 ) give for the cases s =N and s :l:N: EN (II = VNN and

( 1 2) ( 1 3) (14)

Equation (13) gives the first-order correction to the energy level! It is seen to be the expectation value of the perturbation operator in the unperturbed state lPN(O) and thus depends only on the zero-order wave functions. Equations ( 1 4) give the expansion coefficients a, of the first-order correction to the wave function, equation (9), except for the coefficient aN of the unperturbed wave function 'PN (O) of the state con­ sidered. This coefficient is determined from the normalization of the wave function (5) to terms of order e : ('PN, 'PN) = ('PN 10) + elPN( 1 ) + . . . , lPN (O) + elPN (I) + . . . ) = ('PN10), lPN(O» + e[('PN11), lPN(OI) + (lPN (O) , lPN (I})] + ( terms m • e 2, e3 , . . . ) . (IS) 9' Since ('PN(O), N(O» = I , it follows from (IS) and aN = (lPN 'O), lPN (I) that we require aN + aN = O! ( 1 6) i.e. aN is a purely imaginary number which we shall write aN = iYN (YN real). ( 1 7) We can hence write the eigenfunction lPN, correct to the first order in e, in the form V,N -.;;;: ' 'P,(0) , ( 18) lPN = lPN(0) ( I + ,.e YN) + e.L.,. _ 0 E , N ( ) E,(O) where denotes summation over all values of r, except r =N , Since to terms of the order of e, exp(iYNe) = I + iYNe, we can also write the first term in ( 1 8) as exp (iYNe) 'PN (OI. We now see that the choice of YN in (1 7), which we have so far left open, simply corresponds to a particular choice of phase factor of the perturbed wave function lPN and as such is unimportant. We can, without loss of generality, put aN = iYN =O, ( J9) and write the first-order wave function ( 18) as

Z'

(20) 133

30.

30.

FIRST�ORDER PERTURBATION ; DEGENERATE CASE

First-order perturbation I degenerate case We must now consider the case, which frequently arises in practice, that the unperturbed energy eigenvalue EN (O) , in which we are interested, is degenerate. The corresponding manifold of eigen­ vectors of H(O) belonging to the eigenvalue EN (O) can then be spanned in infinitely many ways and it is not apparent which linear combina­ tions are the correct zero-order wave functions. We can see in yet another way that the treatment of section 29 breaks down if EN (O) is degenerate. For if E,(O) -EN(O), for some then the corresponding denominators (ENIO) - E,(O) in equation (29.20) vanish, so that this equation becomes meaningless. But we can see from this equation how this difficulty is to be overcome. We must choose the zero order eigenfunctions corresponding to the eigenvalue EN(O) so that (1) whenever E,(O) = EN(O). The troublesome terms in (29.20) then vanish identically. Suppose the eigenvalue EN(O) is ( a + 1 )-fold degenerate : EN(O) -E(O (2) N+) l = = E(O N+) a' The corresponding eigenfunctions of HIO) define an ( a + 1 )-dimensional manifold 9n(O). We want to find a set of ( a + 1 ) orthonormal functions rpWta' spanning WlIO) such that ( 1 ) holds for them : rpN10), rpwtp 1, a) , VN+it NH = (rpW�j' Vq;W!�) =O, (3) " m ",, (0) 1" " rpN 10+a . a ) as baslS :,u" (0 ) I.e. ref1crred to rpN (0) , rpN+ N N V H. H IS diagonal matrix, a) , ( 4) k = O, I, VN+i. N;� = (rpW�j, VrpW��) = Vj" = v� 8jA, where (5) = 0, 1 , . . . a) . V� = (rpW!", VrpW!.), Suppose we initially choose an arbitrary set of ( a + 1 ) orthonormal eigenfunctions to span mz(O) . Our problem is then to find a unitary transformation U in Wl(O), i.e. an ( a + 1 ) x ( a + 1 ) unitary matrix, which transforms the basis into the correct linear to act as new basis in mIO) . combinations rpNIO), rpW�p rpW!a Apart from some slight changes in notation, this is exactly the eigen­ value problem which we solved in section 22 (d) : to diagonalize the operator V in the manifold 9Jl(O). Unlike the case in section 22, 9]l(0) now only forms a part of our function space. As a result, V� and rp���, 1, a) , are eigenvalues and eigenfunctions of V in mlO) onlY, but not, in general, of the whole function space. This follows from equation (4), which implies that (6) 1, . . . a) , VrpW!J: = vJ: rpW�� + XN+� where XN+k denotes a vector orthogonal to the whole manifold 9]l(0).

r #N,

•

•

•

•

•

•

(j #k; j, k =O,

•

•

•

•

•

•

(j,

(k

Eo, Ell

•

.

•

Ea, •

(k = O,

.

•

Eo, El l

•

•

•

•

•

.

(k = O,

- 0

Ea

•

30.

FIRST-ORDER PERTURBATION : DEGENERATE CASE

If we have solved the eigenvalue problem of V in Wl(O) by the methods of section 22 (d), we take the above linear combinations 9'�� t. (k = O, 1 , a) as zero-order wave functions corresponding to the eigenvalue EN(O) and proceed exactly as in section 29. Writing the first-order eigenfunctions and eigenvalues of H corresponding to the zero-order energy level EN(O) as .

•

.

9'N+t = 9'W!t + e9'�!t,

(k = O, 1 ,

ENH = EN(O) + eEJJ�k'

(k = O, 1 ,

. . . a) , .

•

•

(7)

a) ,

(8)

we obtain, analogously to equations (29. 1 3) and (29.20).

E��, = VNH. N+k = Vk,

L:'E( OV),,NHE (019',(0),

9'

- (0) 9'NH - N+' + e

where

2: '

(k = O, 1,

N+k

r

_

r

• • •

a) ,

(k = O, 1 ,

•

now denotes summation over all values of

r

(9) .

.

T

a) ,

( 1 0)

except T = N,

N + 1, N + a. Equation (9) gives the first-order corrections to the energy eigen­ values, (11) •

•

•

The first-order corrections again only depend on the zero-order wave functions and not on the first-order corrections to these. In general, Va of V in Wl(O) will not all also be degenerate, the eigenvalues vo, Vb so that some (even if not all) of the V, will be different from each other. As a result, the ( a + 1 )-fold degenerate zero-order energy level EN(O) is, in general, split into several [at most ( a + I )] distinct levels, whose positions are given by equation ( 1 1 ) . Equation ( 1 0) gives the corre­ sponding first-order eigenfunctions. One easily verifies that, as in the non-degenerate case, these wave functions are correctly normalized. In Chapter IX we shall return to the determination of the correct zero-order wave functions for perturbation calculations of degenerate levels and splitting of these levels by perturbations. We shall find that these problems can be treated using general qualitative arguments depending on the symmetry properties of the unperturbed Hamiltonian H(O) and the perturbation e V. Furthermore, many of these results will be true rigorously, unlike the approximate results of this chapter. The first-order perturbation theory which we have developed in the last two sections represents those parts of the theory which are most frequently required in practice. One can derive expressions for the higher order perturbations but these become increasingly com­ plicated and are therefore often of little use in practice. We shall consider second-order perturbation theory in Exercise VilA. •

•

•

135

31.

31.

SIMPLE EXAMPLES ON PERTURBATION THEORY

Simple examples on perturbation theory

We shall now illustrate by some examples the perturbation theory which has been developed. We shall only consider very simple examples, as the practical problems usually lead to very heavy analysis and computation.

Example 1 : GTound slate of the helium atom In Chapter III (Exercise IlI.6) we found

the wave function of the

ground state of helium to be

(1) where rl and r are the position vectors of the two electrons relative 2 to the nucleus, provided the nucleus is taken as infinitely heavy (i.e. we neglect its motion) and provided the Coulomb repUlsion

e2 V= - (T1 2 = Irl - r2 / ) '12 between the electrons is neglected. ground state is

(2)

The corresponding energy of the

(3)

E(OI = - 4e2 a,

a-I =1i2jme2

being the Bohr radius of the hydrogen atom. We now of the ground want to calculate the correction to the energy level state due to the Coulomb interaction (2), using first-order perturbation theory. From section 29, the corrected energy eigenvalue is where

(3)

E = E(O) + (9'(0), V9'(O)

t-e-4a('I+'z) e2 drldr2 = -e2a.

6 -6 (9'(0), V9'(O) = -4a n2

T1 2

5

4-

(4) (5)

(This six-fold integral has been evaluated by Pauling and Wilson 1 1 Appendix V.) Hence first-order perturbation theory gives the energy of the ground state as

E = - 2 ·75e2a.

This represents a considerable improvement on the zero-order approxi­ mation, equation since the experimental value is - 2·90e2 a. We have been able to neglect spin completely in this analysis, since we have assumed the unperturbed Hamiltonian and the perturbation spin-independent. The antisymmetric zero-order wave function, including spin,

(3),

is

(6) but the spin function does not alter the value of the matrix element (5).

136

31.

SIMPLE EXAMPLES ON PERTURBATION THEORY

Example 2 : Two identical particles We consider the more general case of two identical particles (position vectors rl and r2) , described by the Schrodinger equation ( 7) (H(O) + V) cp (rh r2) =Ecp (rh r2) , where (8) is the interaction between the particles which we assume to depend on rl and r2 only ; on account of the identity of the particles it must be symmetric in rl and r2 ' We shall treat V as a perturbation of the unperturbed Hamilton H(O) which is separable, with

H(O) = H( l ) + H(2)

(9)

H(k) = H(rl;, - iii grad,,) , (k = I , 2),

(10)

describing the motion of the kth particle in an external field (e.g. two electrons in an atom ; in this case V is their mutual Coulomb repulsion) . If u,, (k) , en, (n = l , 2 , 3, . . . ), are the eigenfunctions and eigenvalues of H(k) , i.e.

H(k)u. (k) = e..u,,{k) , then

(!:!: �, 3,

• • .

),

( I I)

E(O) = e.. + e", (m :;en, m, n = l , 2, 3, . . . )

( 12) is a two-fold degenerate eigenvalue of the unperturbed Hamiltonian (9), with eigenfunctions ( 1 3) �I {rh r2) = u",(I)u,,(2) , �2{rh r2) =u.,(2)un{I) . To calculate the effect of the interaction V on the level E(O), in first­ order perturbation, we must diagonalize the matrix ( 1 4) [ Vii] = ( Ei, VEl), (i, j = 1 , 2 ) , in the two-dimensional manifold spanned by �1 and E2• On account of the symmetry of V, equation (8), one easily shows that VIl = V22 and VI2 = V2h so that the matrix [ Vii] h�s the particularly symmetric form ( IS) [ Vii] = V1 1 V12 V12 Vl1 The diagonalization of this matrix is exactly the problem treated in Exercise V.S. The eigenfunctions, the correct zero-order wave functions for a perturbation calculation, are the symmetric and anti­ symmetric combinations 1 9'1(0) = (16a) {u.. { I )u,,{2) + um(2)un( 1)}

[

.y2 9'2(0) = I {u.. .y2

]

•

{I)u.(2) - u",(2)u,,(I)} 137

( 1 6b)

31.

SIMPLE EXAMPLES ON PERTURBATION THEORY

and the corresponding energy levels are given by with

E; =E (O) + Ej(l} :::I (e.. + e.,) + EP}, EP' = (rp,l0), Vrp;(O»

=

(i = 1, 2)

(17)

(i = 1, 2), symmetric ( i = 1 )

( 1 8)

Vll ± V1 2,

the + and - signs corresponding to the and anti. symmetric (i = 2) wave functions respectively. We see that the interaction V splits the unperturbed level E(Ol, equation (12), into two levels. We have shown these levels in

Figrul 11. splitting QIId displacement oj degm4rat, eTIII'KY leoel oJ two identical parliciu by their interaction, in ./irslorder perturbation theory

11 for a repulsive interaction, V(I, 2) >0, so that Vll >0. the purpose of illustration we have also assumed Vl l > V12 >0. The case m =n represents an exception. The energy level

For

Figure

E(O' = 2e"

(19)

is not degenerate, the only eigenfunction being

( 20)

El = u.,( I)u/I(2).

The effect of the perturbation V is to shift the energy level

E = EIO) + E(I} = E (O) + (Eh VEl) =E(O) + Vlh

as shown i n Figure 12, again for a repulsive interaction.

(19)

to

(21 )

Figurl l2. Displacement oj non-tiegmerale trletlJ level oj two iclmtical particles by their interaction, in jirst-order pertrubation theory

The matrix element Vll represents the mean interaction of the two earticles in the states ".. and "". As such, we expect it to supplement the zero-order energy of the system. The occurrence of the additiona! t� in 18 on the other hand cannot be understood in this classical manner. It is a direct consequence 0 e m st10gws a 1 ty or t1le two particles and represents a quantum-tnechamcal exchan�

138

3 1.

SIMPLE EXAMPLES ON PERTURBATION THEORY

interaction. Hence. terms of this type are often called exchange The occurrence of these exchange interaction terms is of very great importance, particularly for the understanding of the homopolar or covalent chemical bond . We now take the two identical particles, which we are considering, to be electrons. In that case we know from section that the cor­ rectly antisymmetrized zero-order wave functions, allowing for spin, which are derived from are

i�

28

( 16)

9'1(0)_y'21_ ( alP2 -a2Pl ) I I = y' {Um(1)u.. (2 ) +um(2)u,,(I)} y' (a1P2 - a1Pl ), (22a) 2 2 and

9'l�{�2( :: } ��2 {u.(1)u.(2) .,

.,p, )

'

- ••

}

{

(2)u.( 1 ») �2 ( .,P:;:,p, ) . (22b) P1P2

Since HIO) and V are spin�inde endent, introducin the s in wave unc ons oes no a ect t e position or sp tllmg of the perturbed energy levels, but it does affect the multipliciry of the levels. The levels EI and E [Figure 12 and equation [Figure 11 and equation arising from symmetric spatial wave functions are singlet levels, as the level E [Figure J 1 and equation given by equation arises from an antisymmetric spatial wave function and hence leads to a three-fold degenerate triplet level, as given by eq uation We consider the particular case of the first excited state of helium, which has a configuration, i.e. one electron is in the ground state, and one electron is in the first excited state, On account of the Coulomb repulsion between the electrons, this 1 2-fold degenerate level splits into two levels; a 3-fold degenerate singlet level, specified and derived from the symmetric orbital functions by S = = I, J = and (if sections

(22a);

Is,

( 17 )]

2

(22b) .

( Is, 2p)

(2 1) ] ( 17) ]

2p.

0, L I 25 27, ) �I(m) = �2 {�I(O)Y'2( I , m) + �2(0)�1 ( I , m)},

(m =

- 1, 0, 1), (23)

the total wave functions being

�l (m)�2 ( alP2 -a2Pl) ;

and a 9-fold degenerate triplet level specified by S =

1 39

=

I and L I (with

31.

SIMPLE EXAMPLES ON PERTURBATION THEORY

all possible resultants J = 0, I or 2) derived by combining the anti­ symmetric orbital functions

�

UIII) = 2 {Y'1 (0) Y'2 (1, m) - Y'2 (0) Y'1 (1, m) } , (m = - 1 , 0, I), (24)

in all possible ways with the triplet spin functions al a2, etc. That the energy levels with m = - 1, 0, 1 are degenerate follows from the fact that we are considering a problem which is spherically symmetric whereas the value of m depends on the orientation of the coordinate axes. But for a spherically symmetric problem this orientation can­ not be of physical significance. In Chapter IX we shall consider this rather general type of argument in greater detail.

Example 3 : Normal Zeeman eJJect In section 26 we mentioned the normal Zeeman effect, restricting our­ selves to a single electron. We shall now consider an atom containing N electrons whose spin we shall neglect completely. Placing the atom in a homogeneous magnetic field HI; acting in the z-direction, gives rise to an interaction energy V, analogous to equation (26. 1),

e V=2pcH,.LI;,

(25)

where LI; is the z-component of the total orbital angular momentum L of the atom. We know from section 25 that, if the Hamiltonian H(O) of the atom in the absence of the magnetic field HI; commutes with 4 and L,., then the stationary states of the atom are eigenfunctions ofL 2 with the eigenvalUes li2n(n + I), (n = O, 1, 2 ) *. The eigenvalue liln(n + l) is (2n + I )-fold degenerate, corresponding to the (2n + I) eigenstates n) of L,. whose corresponding eigen­ Y'(n, m), (m = - n, - n + I, values are lim, (m = - n, n). We now want to calculate the effect ofplacing the atom in a magnetic field H,. on such a ( 2n + 1 )-fold degenerate energy level EIO). We do not have to use p erturbation theory in this particular case. For the wave functions Y'(n, m) are exact eigenfunctions not only of the un­ perturbed Hamiltonian HIO) but also of the perturbation V:

L",

.

•

•

(HIO) + V) Y'(n, m) •

=

•

.

•

.

•

•

(E(O) + ;;cHl;m) Y'(n, m),

(m = - n,

•

.

•

n) , (26)

In Chapter IX we shall see that the conditions

[HIOl, Ls] '" [HIO), 1..,] ., [JIIOl, lt] = 0

imply that the Hamilton HIO) of the atom, in the absence or the magnetic field, is spherically symmetric, i.e. it is invariant under rotations. This is certainly the case, if the potential energy term, enteringH(Ol, depends only on the distances between the electrons, and between the electrons and the nucleus. (Cj the footnote referring to equation (25.9), where an analogow case is discwscd.)

140

EXERCISE.:i VII

i.e. the magnetic field H� splits lhe (211 + I }-fold degenerate level E(O) into (211 + equidistantly spaccd levels, the separation between two neighbouring Icvels being (tli/211C) H.. The qu an ti ty (eli/2,lc), which is thc natural unit of mngnctic moment to usc in atomic problems, is called the Bohr mngneton. Thc quantum nu mber TIl is orten known as the mngnctic qunntum number ns it was first encountcrcd in the abovc problem and is on 1)' of significance if the sphcrical symmctry of lhe system is disturbcd by some external agency such as an applied magnetic field.

I)

EXERCISES VII

V I L l . Carry out the analysis, analogous to that of section 29, which leads 10 cCJu:l.lions (30.9) and (30.10).

VII.2. The Schrodinger equation or a ollr dimensional oscillator s i given in sec tion I I , cquation ( 1 1 .3). Calculate the first-order pertutha tions i applied. of the cnc'l.'Y levels if a pertutbation lI(x) _ /l.t l s -

VII.3. Obtain the ene'l.'Y levels, accotding to first-order perturbation theory, of n one dimensional oscillator, the perturbation being lI(x) -/lX�. -

VIlA. Ohtain the eigenfunctiOlu and eigenvalucs of the perturbation problem defined in section 29 to the second approximation, i.e. including terms of the order of magnitude of e1, �Ulning the eige nvalucs of the un­ perturbed Hamiltonian to be non-dcgener:l.te.

Y-II�

If H is Ihe diagonal matrix with eigenvalues EL, Ez and EJ (/b ; I !'E'l=FEl =#:EL) , use pertutbation lheory to find the eigenvalues and eigenvectors of the matrix (f1 +£11) to first order in the small quantity E, II being a 3 x 3 matrix. Consider the same problem in the degenerate case E\ -E1=FEl.

1+1

VIII COLLISION PROCESSES

Wl!: now conte to consider collision processes, i.e. procesSes in which an incident particle (such as an electron, proton or neutron) strikes a target (e.g. an atom or nucleus) which scatters it. The target may remain in its initial state (elastic collisions) or it may be ' knocked ' into an excited state, emitting other particles if the excitation is 'sufficiently high (inelastic processes ). The theoretical analysis of such collision phenomena is of funda­ mental importance. For scattering experiments represent one of the best methods of studying the properties of atoms and nuclei, and the interactions of elementary particles with each other and with atomic and nuclear systems. Only a very few collision problems are capable of exact analytic solutions in closed form. In general, these problems are very com­ plicated and require special approximation methods suitable for the problem considered ; even these often lead to very heavy computational analysis. It is, of course, not our object to discuss these special methods of treatfug collision problems ·. We shall deal only with some of the general principles underlying the theory of scattering and shall derive some of the simplest and most important results which often form the starting point for more elaborate approximation methods. 32. Elastic scattering by a fixed centre of force We begin by considering the idealized problem of a beam of particles of mass I' travelling in the z-direction with velocity v, being scattered by a fixed scattering centre placed at the origin of coordinates, r =0, and acting primarily in the neighbourhood of the origin. We describe this scattering centre by a potential V(r}. As a result of this potential field, the incident beam of particles experiences a force and is deflected. The scattering centre in an actual experiment will be an atom or a nucleus (or, rather, a whole group of atoms, etc, but for the purpose of analysis we need consider only one scattering centre) . The potential which the incident particle experiences is then very much more com­ plicated, depending not only on the separation r but also on the internal coordinates of the target. Nevertheless, one can in many cases represent the atomic or nuclear scattering centre approximately •

Thele questions arc treated comprehensiVely in the book by Mott and Massey:U and by Wentzel2" and Ma.ssey2.5.

the articles

1 42

32.

ELASTIC SCATTERING fly A PIXED CENTRE OF FORCE

by such a potential. We have slated above that the scattering centre is fixed. This would correspond to an infinitely heavy atom or nucleus. But by interpreting It as the reduced mass (if Exercise 111.3) and v as lhe velocity of the incident panicle relative to the scattering centre, our considerations apply equally to a scattering centre of finite mass. We shall come back to this point in seclion 35. Mathematically, the problem we have 10 consider is the Schrodinger equation

[

-

?

t. '

-I'

-

17 ' +

1

(I)

VCr) �(r) -E�(r).

The potential VCr), describing the scattering centre, will be most effective ncar the origin, but at large distances from the origin its effect will be small. More precisely, we shall assume that (2) ,VCr) _ 0 ('-Irl)

Under these conditions the tolal energy as r _ 00, in any direction·. E of the particle is also equal to its kinetic energy at large distances from the scattering centre where VCr) ... 0, i,t. (3) It is convenient to measure the pOlential in units of ,,2/2p, i.t. define 2� U(r) - " , VCr), U) and to pul (5) " -IW/", i.t. k is the magnitude of the wave vector of the incident particle. Equation ( I ) then becomes (6) On account of our assumptions about the pOlential, equation (2), equation (6) has a continuous spectrum of eigenvalues for E>O, i.e. the kinetic energy of the incident pilrticle (at large distances.j,e. outside catteringfield) mayassumea�y posi tivevalw::. thes 'We now require a solution 1p{r) of (6) whieh satisfies two conditions. Firstly, we want Ip(r) __ ei.l( as V(r) _ 0 (7) otcntial, p g scatterin stren the decrease we if .t. solution the the of th g j should tend 10 the planc wave solution representing a monochromatic beam of particles. ?econdly, we want a solution representing the ... incident beam of particles, given by the �nc waye __

•

'n,i$

ei!�

-

condition is nOI s:lIisfied (or 'the Coulomb potenli;,1 lIT which is ;, 'long·r;,nge' I'lenee our eotnider;'lions do not ;,pply in Ihis c:uo:, But in practice we do not potenti�L ulu;,lly de;,1 with .'I pure Coulomb field, but with .'I screened Coulomh ficId, (or which .he potenlial dccreaSl!'! sufficiently rapidly. For example, the Coulomb ficId of the atomic nucleus is ,creened by .he dce.ronlsurrounding it.

143

32.

ELASTIC

SCATTERING

BY A FIXED CENTRE OF FORCE

(which we have nonned SG that the incident beam of particles contains one particle per unit volume), together with outgoing waves which have been scattered from the target. From equation (1 2.12) we expect equation (6) to possess two asymptotic solutions 1 (8) ,e±ibf(8, 9'), representing outgoing and ingoing spherical waves of particles respec­ tively, with the same energy E = 1i2k2/21' as the incident beam, con­ firming, as is physically obvious, that we are dealing with elastic scattering. Only the former solution is physically acceptable for our problem, so that the total asymptotic form of the solution of (6) which we require

is

(9)

the first term representing the incident wave, the second the scattered wave. ({8, p) is called the scattering amplitude of the procCS§. To specify the angular distribution of the particles which have been scattered, we consider the passage of these particles through a large sphere of radius whose centre is at the target. Consider an element of surface d!1 being an element of solid angle, on the sphere subtending the polar angles (8, rp) at the target. The number of scattered particles crossing tiS per unit time or, as we shall more briefly sa the current of articles scattered across tiS, is given by the product' . nSI 0 scattere artic es at x ve oCl 0 scattere artic es x (element of area dS), i.e. by

R tiS=R2df),

(Current of particles) = 1 ! eURIi(O, rp'1 2(�)R2df) scattered across R JI ) I' = (�) If(8, rp) 1 2d.Q. , (10) tiS

This elq)ression gives the relative probabilities of a particle being scattered into two different directions. The absolute probability, how­ ever, still depends on the flux of the incident beam of particles striking the target. The current of particles scattered across tiS is clearly proportional to the flux of incident particles, i.e. the number of par­ ticles incident per unit area per unit time. Since we have taken the density of the incident beam to be unity, the flux is simply equal to the Velocity We then define the differential cross-section a{O, p)d!J by the eguation

(lik/I')'

C1

i.e.

(0, rp) d.Q _

( Current of particles scattered across dS) (Flux of incident beam) a(O, rp) dQ = If(8, rp) 1 2d.Q.

144

(11)

33.

THE INTEGRAL EQ.UATION OF THE SCATTERING PROBLEM

We see that the cross-section has the dimensions of an area. It is the effective area which the target presents to the incident beam of articles for the rocess considered. From the angular istrlbution or differential cross-section ( 1 1 ) we obtain the total cross-section by integrating over all angles, alOI =

r

2

1 1(0, 9') 1 d.Q. J unll apbere

( 12)

To find the scattering amplitude 1(0, 9'), i.e. to solve equation (6), , it is most convenient to convert it first to an integral equation. We shall not give a mathematically rigorous derivation of this integral equation, but only a plausible one which brings out its physical significance. 33.

The integral equadoD of the scattering probleDl Before considering the wave equation (32.6), let us consider an electro­ static problem with which the reader may be more familiar, namely, to obtain the potential ",(r) due to a distribution of electric charge given by the density function e (r) . This potential is the solution of Poisson's equation (1) V� (r) = - 4ne(r) . I f we consider a unit charge at the point r =r', this equation reduces to (2) V2", (r) = - 4n8(r - r') , where 8 (r - r') is the three-dimensional Dirac 8-function defined in equation (4. 18) *. The solution of this equation is well known to be the Coulomb potential 1 " , (3) , r'P (r) = -r as one can also easily verifY from equation (2). To obtain the poten­ tial due to a general charge distribution we go back to equation ( 1 ). • For the benefit of readen who omitted section 4 we briefly state the relevant properties of the Dirac 8·function 8(x). It is defined by (I') 3(x) -0

(ii) Lim 3(c)-..co O

c...

if x:;>!:O

in such a way that f 3(x)"", i

for any range of integration·enclosing the origin. From this definition its most important property follows at once : if a
f& J(x)3(x _ �)" = {J d

(�), a<�b.

The three-dimellllional 3·function 3(r) is defined by

3(r) -3(x)3(y)3(�)

and it follows at once that for any volume of integration Y: r _ u = { f(ro) , if the point ro lies Inside the volume Y 0 ro .. outalde .. Jvf(r)8(r ro) .. .. .. •

1 45

33. THE INTEGRAL EQUATION OF THE SCA'ITERINO PROBLE)'! Since this equation is linear we expect the potential it generates to be :l. linear superposition or the potentials due to the individual clements or charge o(r')ar' in the different volume clements dr',

f

e(r')dr' ( tp r) - Ir r'l

i.e.

(4)

the integral being over all space, and this exprcssion represents the complete solution or our electrostatic problem, \Vc now treat the inhomogeneous wave equation

(1I'+k')�(r) �F(r)

(5)

in the same way as equation ( I ) . Whereas solutions o r the homo· geneous wave equation represent waves in the absence of the scattering tar et, wc can think or F r in 5 as a source term which enerates or absorbs waves., The solution or equation 5) thcn consIsts or two parts. t consists or waves generated by the source term F(r}, to which may be added any solution or the homogeneous wave equation

(II' +k')y(r) -0,

,

representing waves generated and absorbed at infinity (i.e. at vcry large distances) but not affected by lhe sourcc term F{r). �ain start by considering a unit sourcc at the origin. so thai (5) becomes.,

(6) We easily veriry by direct substitution that equation (6) possesses the rollowing two soilltions, which arc rllllctions or tile radial coordinate r only, (7)

(if Exercise VIII. l ) , corresponding 10 outgoing spherical waves generated by the unit source at the origin and to incoming spherical waves absorbed by the unit sink at the origin respectively. As we arc intcrested in outgoing spherical waves only, in connection with scat· tering problems, we shall only require the solution containing the positive exponential function. Analogously to the treatment or Poisson's equation, we can now at once write down a particular integral or tlle W;1\'C equation (5) con· taining a general source term F{r), as a linear superposition or waves produced by different clements or the source. Thc solmion corre· sponding to outgoing wavcs only is given by

�(r)

_

-

, cfl1r-r1 J

I -IF(r')dr'. , - ';'I r - r'

(8)

To this particular integral or the wave equation we can still add any solut.ion or the homogencous wave equation,

'46

33.

THE INTEGRAL E Q.PATION OF TilE SCATTERING PROBLEM

We can apply these results to the scattering problem of section 32. From the discussion leading to equation (32.9), we see that the solu­ tion of the homogeneous wave equation which we must add to (8) is above, we see eik(. Furthermore, comparing equations (32.6) and that i.e. we finally obtain the following integral equation representing our scattering problem

(5)

F(r) = U(r)1p(r),

&lr-r'1

1p(r) = eLb: - 1 fIr _ r" U(r')1p(r')dr', , �

(9)

the integration being over all space. The great advantage in express­ ing the scattering problem by this integral equation lies in the fact that this equation already incorporates all our boundary conditions. We introduce the unit vector D in the direction of the scattered particle, defined by the polar angles 0, qJ, i.e. D has the components (sin ° cos qJ, sin ° sin qJ, cos 0), and the wave vector of the scattered particle, so that ( 1 0)

k

k =kn.

To obtain the differential cross-section, we require the asymptotic form, for large values of r, of the wave function (9) . Since the poten­ tial in (9) vanishes for large r', we can, for sufficiently large values of r, i.e. r� r', put 1 I � r - ar', (1 1 ) -r' Substituting (1 1 ) into (9) leads to the asymptotic wave function for large r,

U(r')

Ir -r'l

Ir -r'l

""

1p(r) � eik - �fe-iIa'U(r')1p(r')dr' i&

( 1 2)

i.e. the wave function has the required asymptotic form as given by equation (32.9), with the scattering amplitude/CO, qJ) being defined by f(O, qJ)

= - .infe-iIa'U(r')1p(r')dr',

(1 3)

Substituting this equation into (32 . 1 1 ) for the differential cross-section we obtain, using (32.4), a (O, qJ) dQ

= (�) 2 \ fe-'lu' V(r)1p(r)dr I 2dD = (�rl(eikr,V(r)1p(r» 1 2dD.

( 1 4)

Equation ( 1 4) is a rigorous expression for the differential cross-section. The number of particles scattered into states with momentum is seen to be fully determined by the matrix element of the scattering potential between the actual stationary state of the system and the particular scattered state eikr in which we are interested.

1p(r)

147

lik

33.

THE 1r.-n:GRAL EQ.UATION OF THE. SCATTERING PRonLBf

From equations ( 14) and (32.10--1 1), we obtain the current of particles scattered across dS, as Current or particlCS pkdn I(cil
(

)

(eU'Tcnt or p"";cI"') _ 2� scattered across as

Ii

J

k" dk'dQ ( " " )I (r) (r» I'o( E' C (2:1V I ' 'P "

_

E) (15)

the integration beingover J:' only, but not over lhe angular coordinates o and p. Here Eco/;lk2/211 is the energy of an incident p:miclc wilh wave veclor k _ (0, 0, kl, as before, and E' _All/2/2fl is the energy of a scattered particle with wave vector k' = (k' sin 0 cos p, k' sin 0 sin p, t' cos 0). 'We seem to have ascribed a wave vector of magnitude k' to the scattered panicle, different to that of the incident particle, but the conservation of energy has been ensured through the 8·funclion 8(E' -E) in equation (15), on account of which only those scattering processes arc possible for which E' =E. Equation (15) is easily verified 10 be equivalent to ( 1 4), by carrying om the integration ovcr 1;'. '''Ire can now interpret (15) as follows. In a scaltering experiment one uses counters to count the number of events of a particular kindj in our cnse, the number of particles scattered into a definite clement of solid angle d!} with a given energy E', i.e. the number ofscallered particles with a wave vector in Ihe illlerval (k', It' +dk'). Though we nrc dealingwitl! a stationary state, wccnn in a certain sense think of the scatlering centre as causing transitions to the group of stales with wave vectors in the interval (k', k' +dlt') in whieh we arc interested. Now /;" dk'dQ dk' (2�) 1 - (2�)' represents the number of stau:s with momentum veclors in the interval (k', k'+dk'), ns follows from equation (10.13) *. Hence we can interpret (16) as thetransitioll IIrobabilit),(Jr.r IIl1it lime to the group of slates with wnve vector in theinlerval (It', It' + dl,') , The fact that we arc dealing with a stationary stale, finds its expression in the fact that the transition probability per unit time is independent of time. This result for transition probabilities pcr unit time, equation (16). can be generalized greatly, ns we shnU sec in section 37 wherc we shall consider a different approach to collision problems. lU

• Tu section 10 we had nornl3liu:d thc waw.: functions in a volumc V :unl found (L/2:rpdk thc UlHllber nr.tatC$ with momentum \'cctor in (k, k +dk). In this section wc h:l.\·c taken

L-1.

14B

340

34.

BORN APPROXIMATION

Born approxilnatioD

The expression which we derived for the differential cross-section in the last section, equation (33 .14), does of course not yet represent a solution of the scattering problem, because it still contains the unknown wave function We will consider the special case where the scattering potential is weak. It follows from equation (32.7) that the actual wave function is in that case not much different from the incident plane wave as a small perturba­ ew.. We can think of the scattering potential tion which slightly distorts the plane wave. Under these circumstances, we can, as a first approximation, replace by the plane wave elk in the matrix the actual wave function element in (33. 1 4) . This gives the very important first Born ap­ proximation, in which the differential cross-section (33.14) becomes replaced by

?per).

VCr)

'P(r)

V(r)

'P(r)

a(O,

where

rp)d[J = (2:n2) 2 1 fV(r)eiCko-k)rdrI2dfJ = (�r l (e''lu, V(r) e'"kor) 1 2d.Q,

(1)

Ito = (0, 0, k) is the wave vector of the incident particle so that kor =k<:.

If we look at the integral equation of the scattering problem, equation (33.9), we see that the first Born approximation corresponds to the first approximation of a solution by iteration, in which one replaces the true wave function by the plane wave eUe under the integral sign. In other words, we assume that to a first approximation the wave scattered from the volume clement is the plane incident wave, unmodified by any scattered waves. In the Born approxima­ tion, the transition probability, equation (33. 16), assumes a particu­ is replaced by the larly clear meaning. The true wave function « unperturbed ' plane wave eW. and we can think of the scattering potential as causing transitions to other unperturbed states. There exist also higher Born approximations corresponding to iterative solutions of (33.9) of higher order, but these are usually too compli­ cated to be of much practical use. The first Born approximation, however, is exceedingly simple and therefore very useful. The momentum vector which occurs in the matrix element of the Born approximation, equation ( 1 ), represents the momentum (in units of Ii) which the incident particle has transferred to the scat­ tering target as a result of the collision. The matrix element itself is seen to be simply the Fourier transform (if Chapter I, Exercise 1.6) of the scattering potential.

'P(r)

dr'

'P(r)

VCr)

(Ito - k)

149

34. The Born approximation (I) assumes a particularly simple form in the important case where the scattering potential is a function of the radial coordinate r only, V = VCr}. We take the momentum transfer vector K = ko -k as polar axis of a new set of polar coordinates with polar angles a and p. Referred to these new coordinates, the vector has rectangular components (r sin cos p, r sin a sin p, r cos (X) and the vector K the components (0, 0, K), so that Kr=Krcos The matrix element in (I) then becomes 2n Jo{II)V(r)r2d/Jo dpJofnelA"rcat sin a da = 2nJOfIl)V(r}r2drJ-l1 elll"dt f'"V(r)rsin (Kr}dr. (2) = 4n K Jo The scattering amplitude, equation (33.13), is now independent of cp and is only a function of 0, the angle through which the incident particle has been scattered. Using (2) the scattering amplitude becomes 2 f(O) = - � fll)V(r)rsin (Kr}dr (3) li Jo and the cross-section is then obtained from equation (32.IO). The only variable p eter entering the scattering amplitude f(8), and therefore the cross�section, in the first Born approximation is the magnitude of the momentum transfer liK. Since K2=(ko-k) 2 =ko2 +k2 -2kok =2k2(I -cos 0) =4k2 sin2 : , (4) BORN APPROXIMATION

r

a

a.

n

aram

i.e.

(5)

and the cross-section depend on the angle of scattering 8 and the momentum of the incident particle lik through the combination (k sin 8/2) only. The validity of the Born approximation has been discussed by Mott and Massey 22. They have shown that if one considers a scattering p-otential limited to a region of linear dimension 6 (say <6) and whose strength is of the order of Vo, then the Born approximation is valid, provided that

f(6)

r

(6) v

being the velocity of the incident particle. 150

34. BORN APPROXIMATION

It follows from equation (6) that the Born approximation improves with increasing energy of the bombarding particle. For scattering of electrons by hydrogen or helium, the Born approximation is valid for bombarding energies above approximately 100 eV, for heavier atoms (with larger Vo) this minimum energy increases to about 1 keV. For the scattering of neutrons or protons by nuclei, Born approximation begins to give moderately adequate results at 100 to 150 MeV, but is not really satisfactory below 200 MeV. Equation (6) is a sufficient condition for the validity of the Born approximation, but often it is too severe a restriction, i.e. the Born approximation gives good results even though (6) fails to hold. In particular, the Born approximation may give small angle scattering quite well, but break down for large angle scattering. For small angle scattering is due to particles moving past the scatterer at great dis­ tances, whereas for large angle deflections the projectile must penetrate deeply into the scattering centre. We consider two simple examples of the Born approximation. Example 1 : Scattering by a square well potential Take the scattering potential VCr) to be VCr) = - Vo, ra then equation (3) gives at once

"

I

2", Vo 1( 6) = 1i K 2

2", Vo . . ( ) Kr dr = 1i 3 {sm (aK) - aK cos (aK) } , 2K

o r sm

(7)

where 6 and K are related by equation (5),

;

K=2k sin ,

and the differential cross-section is given by (32. 1 1 ) , a(8, '1') =a(8) = 1/ ( 8) 1 2• Example 2 : Scattering by a screened Coulomb potential As the second example we will take the potential Ze2 V r) = - -e- ....

C

r

(8)

corresponding to the scattering of an electron by an atom of nuclear charge Ze. The potential (8) represents the Coulomb field of the nucleus, being screened by the atomic electrons surrounding it. The parameter a determining the screening is approximately given by me2Z! a = ""/i2 . 6 + Q..M.

151

34.

BORN APPROXIMATION

We obtain for the scattering amplitude

f( 8) = 2Pze2 1i2K '"e

a 8)

J

0

"" 11

sm •

2pZe2 , (Kr)dr = 1i2(K2 + a2)

(9)

for the differential cross-section

If(8) 1 2= Cp"ie2Y(K2 � a2)2 ' and for the total cross-section, on account "of (4), = 2�f:a(8) sin 8 d8 (

(10)

=

C11O\

-2{a(O}�,! KdK = 2�k2 (2pze2) 2J:U0 (K2+a2)2 16�p2Z2e4 A2

(11) = 1i4a2( a2 + 4k2)' we consider the case of large momentum transfers, K>a, w e can neglect a2 compared with Kl in equations (9) and (10). Equation (10) for the differential cross-section case becomes, with = (12) C1,(8) (2�:2r cosec4 ; , which is Rutherford's celebrated formula for the scattering by a pure limiting formula Coulomb potential - Zelr. That we obtain If

=

in this

'lk

this

pv,

agrees with the fact that for large momentum transfers the electron must penetrate well inside the atom so that the screening is no longer effective. The differential cross-section for Coulomb scattering in the forward direction becomes infinite. This is due to the fact that the Coulomb potential is a long range potential which decreases only slowly with distance. Hence electrons passing even at a great distance from the atom experience small angle scattering. To apply the validity criterion to the Coulomb scattering, we can take the range and strength of the potential to be of the order of

(8 = 0)

(12)

(6)

a!:::!. l /a, Ze2e-aal = -Zela/e, a so that the validity criterion (6) becomes V"'"

a-I/o

�:2< 1. 152

35.

LABORATORY AND CENTRE OF MASS COORDINATE SYSTEMS

Since (e2/lic) is dimensionless and has the value 1 / 1 37 very nearly we can write this criterion as

� (�)

1 1 7 < . This criterion also applies in the relativistic case, when vlc� 1 (particle velocity approaches the velocity of light) . It then only fails for the heaviest atoms, and even for these Z/137< 1 . This application of the validity criterion is of course in no way rigorous and is only meant to make the results (which are correct) appear plausible. 35-

Laboratory and centre of !Dass coordinate syste!Ds For the purpose of describing an experiment, one chooses a coordinate system corresponding as closely as possible to the experimental set-up, i.e. one considers the target initially at rest. This coordinate system, which we shall specify more fully in a moment, is known as the laboratory coordinate system. A theoretically much simpler description is obtained if we choose a coordinate system in which the centre of mass of the whole system (target + incident particle) is at rest. For in this system, known as the centre-of-mass coordinate system, the total momentum is always zero (both before and after collision) . We must now show how these two modes of description are related to each other. In the laboratory coordinate system, we consider a particle of mass m which is shot at a target with velocity v. The target particle of mass m ' is initially at rest, v' = O. As a result of the collision, the incident particle is scattered into a direction with polar angles 0" f/Jl with velocity w, the target experiencing a recoil, in the direction 0/, f/J/ with velocity w'. Figure 13 (a) illustrates the laboratory coordinate system. The laws of conservation of energy and momentum give us the following four equations tmv2 =tmw2 + tm'w'2 mv = mw cos O, + m'w' cos Or' mw sin O, = m'w' sin 0/ f/J/ =f/J, + n The last of these equations simply states that a single process occurs entirely in one plane (so that we can arbitrarily put f/J, = O, f/J/ = n) . The other three equations enable us to solve the scattering problem completely, given one of the final coordinates. For example, given 0, (in addition to m, m', V; and v ' = O) we can determine 0/, w and w'. To determine (), itself as well, we would require the law of force governing the scattering process. Hence we can define a cross-section in the laboratory system, (11 ( 0" f/J,) dD, = (1,( 0" f/J,) sin OldO,df/J" 1 53

35.

LABORATORY AND CENTRE OF MASS COORDINATE SYSTEMS w

111 V , m_ _ _ 0--------.

,

,,/� ;":1�--7 -- ' ,

III '

,

,

,

,

,

Oz.?,

,

,

,

8;.9'/

,

,

,

,

Co.A/.

/ Vo

) * (--....,� .

---

,

,

,

\

w'

Figure IS.

Description of scattering process in

(a) Itlboratol')' coordinate systlm,

(b) """,-of-mass eoorditftlk SYltem.

1 54

- - - _ z-ar/t

35.

LABORATORY AND CENTRE OF MASS COORDINATE SYSTEMS

for scattering leading to a definite final state, analogously to the treat­ ment in section 32 * . Just as in classical mechanics, so we can, in quantum mechanics too, impose a uniform velocity on our coordinate system (or the physical system), and these different coordinate systems are all equally good and must all give consistent descriptions of our system. We will now go over to the centre-of-mass coordinate system in which the centre of mass (C.o.M.) of the system is permanently at rest. In the laboratory coordinate system, the centre of mass moves with a uniform velocity

mv Vo = --, m +m '

We go over to the C.o.M. coordinate system by imposing a velocity, equal and opposite to vo, on the system. We show this in Figure 13 (b) . In the C.o.M. system, the scattering process can be thought of as two beams of particles travelling towards the C.o.M., itself at rest, being scattered through the same angle O. and moving off in opposite directions. The velocities of ' projectile ' m and ' target' m ' (which are now treated on an exactly equal footing) before collision are

m'v u =v - vo = m + m'

U

"

mv

--" = V - Vo = m +m

(and mu + m'u' =0, as required) . The velocities after scattering through the angle O. are unchanged. The velocities in the two coordi­ nate systems are related by these equations, which are evident from Figure 13 (b) :

rp, = rp., w sin 0, = U sin 0,

cos 0/ = U cos 0, + vo. From the last two equations, we obtain sin O. tan 0/ =0 cos 0. + a ' where w

}

(1)

(2)

Vo m a = u = m"

We can now define a differential cross-section in the C.o.M. system 0'. ( 0., lP.)dD. 1:0 0'. ( 0., rp,) sin ().dO. drpc for the colliding particles to be scattered through an angle 0., into the element of solid angle dD.. Now a particular element of solid angle • In general the cross-section dependJ only on 0, and is independent of 9',. For it to depend on Ph the scattering potential must not be axially symmetric about the {-axil.

155

35.

LABORATORY AND CENTRE OF MASS COORDINATE SYSTEMS

tID, in the laboratory system will subtend a corresponding solid angle tIDl in the C.o.M. system, in such a way that (3) a,( 0" rp,)tID, = a,( 0" rp,)dD., for either side of this equation gives the current of particles through the same surface element per unit flux of incident beam. We must now find the element of solid angle tID, in the laboratory system corresponding to a particular element of solid angle dD, in the C.o.M. system. From ( I ) we have that (4-) drp, = drp" and from (2) we find that a + cos 0, cos 8 = ' ( 1 + a2 + 2 a cos O,)! ' so that + a cos (1) • 0,dOI (5 ) sm S1D O,dO, = ( 1 + a2 + 2a COS Ol)t Equation (3) then gives for the relation of the cross-sections in labora­ tory and C.o.M. systems ( 1 + a2 + 2a cos 0,)' (6) a,(O" rp,) =a,(OI, rp,) II + a cos 0,1 ' Though the differential cross-sections differ in the two cases, the total cross-section derived by integrating over all directions in either coordinate system is of course the same. For the total cross-section is proportional to the total number of scattering events irrespective of direction of scattering. We obtain a particularly simple important case by considering two particles of equal mass : m' = m, a = l Equation (2) then gives

.

(I

.

0, = 0,/2 ;

0, goes from 0 to n/2, 01 goes from 0 to n. We obtain, with 81=20, = 20 and rp, = rp, =rp, a, (O, rp) sin 20d(28) = = 4 cos 0, sin OdO a,(O, rp)

as

so that

a,(O, rp) = 4 cos 8 a,(8, rp), as follows also from (6). It follows from equation (2) for a < l , i.e. the projectile is lighter than the target, that 8, varies from 0 to n, as 8, varies from 0 to n. For a = 1, i.e. for two particles of equal mass, we obtain the special case just considered. If a > l , i.e. the projectile is heavier than the target, conservation of energy and momentum considerations restrict the possible scattering angles 0, in the laboratory system. As 01 varies from 0 to COS-I ( - I/a), 8, increases from 0 to its maximum value 156

35.

LABORATORY AND CENTRE OF MASS COORDINATE SYSTEMS

sin-I (I/a) which is always less than n12. As 0, increases further to n, 8, decreases again to O. We also see tIlls from equation (5) : if a >l , ( 1 + a cos 0.) and hence dO,ldO, are negative for O. >cos-l ( - I /a) . In equation (6) we have introduced a modulus sign in the denominator as cross-sections are essentially positive quantities. The values of 0 that occur for a given 0 can be distinguished by the energy of the scattered particle, !mw2 = !m (vo2 + u2 + 2voU cos 0.), the smaller angle corresponding to the larger energy. In the limiting case where one of the scattering particles is infinitely heavy, m' - co (in practice this means m<¢n'), this particle remains at rest all the time and coincides with the C.o.M. The two coordinate systems then become identical, as follows also at once from the above analysis. We considered this case in section 32 (where we wrote I' for the mass of the projectile which we now denote by m, and 0 for the scattering angle) . But we also saw in section 32 that we can interpret I' as the reduced mass of the incident particle with respect to the scattering centre, and we must now see how the cross-section derived in this way is related to the laboratory or C.o.M. cross-sections. We saw in section 32 that, if one uses the reduced mass, one at the same time uses coordinates relative to the scattering centre, i.e. one uses a coordinate system (called the relative coordinate system) • fixed ' to the target particle, so that one calculates as though the target were infinitely heavy and consequently remained at rest. One easily sees that in this relative coordinate system the scattering angles 0, and fIJ, are identical with the corresponding angles in the C.o.M. system, 8, = 8" fIJ, = fIJ.. In order to see this, we reduce the target m' to rest in the C.o.M. system [Figure 13 (b)] by imposing the appropriate velocity (namely, minus that ofm' in the C.o.M. system) on the system, after the collision. This does not alter the direction but only the magnitude of the velocity of the scattered particle, so that follows. Hence the cross-section calculated in the relative coordinate system is the C.o.M. cross-section. In using these three coordinate systems one must use the appropriate energies. The energy in the laboratory system is E, =!mv2, the energy associated with the motion of the centre of gravity being m Eo = ! (m + m')vo2 = ,E,. m +m The kinetic energy of the two particles in the C.o.M. system is � E. =�mu 2 + �m'u/ 2 = m + m"E, "2" "2"

(7)

(7)

157

36.

INELASTIC SCATTERINO PROCY..5SES

so that

E,-Eo + E" as required. For the energy of the incident particle, m the relative coordinate system, we have

m' E, -"i,uv1 -i111mnl' + 111.v1---,E 1II + m 1 -E.' --

36. Inelastic scattering processes Wc now come to consider inelastic scattering processes i n which the state of the target is changed by the collision. The same stationary theory which was developed in sections 32 to 34 can easily be adapted to this case, so that we need only consider it briefly. At first sight it may seem surprising that a time·independent ap· proach can be ttsed to describe these inelastic processes in which a collision changes the state of the target. What we really consider is, of course, the statistics of a succession of individual collisions between a single incident panicle and a single atomic or nuclear scattering centre. We can best visualize this in the C.o.M. coordinate system, where we are dealing with two streams of particles (the ' incident ' particles and the ' target' systems) colliding and flying off in opposite directions again, the scaltered target being in a new state. (This treatment does of course not allow for such processes as double col. lisions of incident particles with the target. Correspondingly one must use a thin target, or modify the theory.) ..5.upposc the target system (i.e. the atom ot' nucleus) by itself is described b the Hamiltonian If standin for a whole set of co males cscn IIlg the target, and assume the complete sets of .cigcnfunctions and eigenvalues

p,(q),p, (q), . . . p,,(q),

.

•

.

of I1(q) are known. {For simplicity, we have here assumed a djscrcte spectnlln of elgCilV:llues, but the generalization to the continuous or partly continuous case is easy.} If,u is the reduced mass of the incident particle P with posilion vector r - (x,y, z} relative to the C.o.M. of the scatterer, and VCr, q} is the interaction potential of P with the scatterer, the total system is described by the wave equation

{ -}'n'V'+ H(q) + V(r, q) �(r, q) _E�(r, q). 2 }

(I)

rVer, q) -+ 0 as r -+ 00,

(2)

We assume that the scattering potential is a short I'ange potelllial, so that, analogously to equation (32.2),

158

36.

INELASTIC SOATIERING PROCESSES

k

in any direction. If o = (0, 0, ko) is the wave vector of the incident particle, and 9'o (q rthe state of the scatterer before collision, then 1 E = 1;,2ko2 + Eo. (3) 2p We again impose two conditions on the solutions of ( 1 ) we seek. Firstly, we require, analogously to equation ( 32.7), that 1JI (r, q) - el'kor9'o(q) as VCr, q) - 0. For in the limit of vanishing interaction, we want the solution to describe a definite state of the target system, together with a plane wave, describing the uniform motion of the incident particles. Secondly, we require that the only incident waves are those corre­ sponding to the beam of incident particles, all other waves being out­ going waves corresponding to scattered waves. This leads to a restriction on the asymptotic form of the solution analogous to (32.9) but more general, allowing for inelastic collisions. We can expand 1JI(r, q) in terms of the complete set 9'", (q), the coefficients being functions of r of' course, (5) 1JI (r, q) = �9'm (q) 1JIm (r) .

(4)

•

Substitution in ( 1 ) gives

with

'"

�9'm (q) {V2 + � (E - Em) }'PIII (r) = U(r, q) 1JI(r, q) 21' U(r, q ) = 1;,2 VCr, q),

(6) (7)

and taking the inner product with 9',(q), just as far as the coordinates q of the scattercr are concerned, gives

f

(V2 + k,2) 1JI,(r) = dqtp, (q) U(r, q) 1JI (r, q) =F,(r), ($ = 0, 1 , 2, . . . ), (8) where

21' k,2 = 1;,2 (E - E,) ,

(9)

($ = 0, 1 , 2, . . . ) .

Equations (8) and (9) describe collisions in which the target system collision is in the state 9',(q) . For $ = 0, these are the elastic they repre­ processes already considered in sections 32 to 34; for sent inelastic processes in which the scattered particle is described by a wave vector = (k, sin () cos 9', k, sin () sin 9', k, cos () , whose magni­ tude is defined by the conservation of energy equation (9) . Since the kinetic energy of the scattered particle must be positive, collisions leading to the state 9',( q) are only possible if 1i2 E,<E =Eo + ko2. 2p after

s=l=O,

k,

159

36.

INELASTIC SCATTERING PROCESSES

The methods of section 33 at once give the set of integral equations equivalent to the set of equations (8) :

f ....,F, d '

I eil.,rv),(r) = eik�roo' - 4;T rr=r'j (r' ) r ,

(s =O, I,

•

.

.

).

(10)

Here the second term represents the scattered outgoing wa� es (there are no terms with negative exponentials, since there are no incoming spherical waves). The first term is only different from zero for s = O and corresponds to the plane incident wave which occurs i n the solu­ tion of the clastic scattcring problem [if equation (33.9)]. The asymBtotic solution of(10) is ($ ... 0, I, . . .

)

(I I)

giving the scattering amplitude

-

J,(O, p) ... - 'inI c-ilc, F,(r')dr' ....

c�Rondin to the as

III

($-0, I, . . . )

(12)

)(otic wavc form [if cquation (32.9)] ( 1 3)

Equations (12) and (13) refer to proccsses in which the target is left in the state after the collision. Analogously to section 32, we find for the differcntial cross-section for these processes

p,(q)

k'R2dn /..!.ci.t , Jl J, , lli p) (O k .(O, PJI'dD, .,(O, p)dD_ R (�) I' -(fo)IJ (,-0, I,

.

. . ).

(1'1)

(k,lko)

Apart from the f.1ctor this agrees with equation (32. 1 1 ) . By procedures entirely analogous to those of sections 33 and 34 we could consider transition probabilities and Born approximation. However, the reader should by this stage be well able to carry these investigations out for himself. The validity criterion for Born ap­ proximation, equation (34.6), must now be replaced by two analogous criteria for both incident and scattered velocities of the incident particle

V.,a <:: 1 . .,-IW,

(15)

Amongst the many important problems in collision theory, which we have not disclissed, that of collisions involving identical particles is particularly important. For the clastic scattering between lWO 160

37. TIME·DEI'ENDENT APPROACH TO COLLISION PROBLE�'S identical particles, one can easily show that the differential cross· section in the C.o.M. system is givcn by (16) 0(0, �) - If(O, p) ±f(� - 0, p +�) I' ± where the signs correspond 10 stales which are respectively symmetric and antisymmelric i n the spatial coordinates. In the case of inelastic scattering, it is less easy to allow for the identity of particles. In general, .here exists some ambiguity as to how the wave functions should be symmetrized or antisymmetrized * . Bul in many co\lision problems involving fast particles, our above treatment is adequate; e.g. we can think of a f.-ut electron, scaltered by an atom so as to leave the atom with a high energy, as having retained its idenlily-il did not spend sufficiently long in the atom for e1eclron J exchange to have taken place. -----.J 37·

I

Tirnc.dcpcndcnt approach to collision problems

So far wc have treated collision problems by time·indeptndent methods, i.e. we have looked for stationary states of
il!0/� � fl.

(2)

for the wave function 'I'(q, t), given its value :1.t some instant 1 = 0 : .(q, 0). Suppose the unperturbed Hamiltonian liD possesses the complete orthonormal sets of eigenfuncuons Po(q), 911(q), . . . tp.(q), . . . and eigenvalues Eo, Ell . . . E" . . . , i.e.

(3) • For a ful! disclluion, sec Masser and nurliop, E lt!lrrm;c 111111 Ionic Impc.d Phnwmma, Oxford, Clarendon I'res.!l, t952, p. 1-12,1 Slq.

161

37.

TIMIH)F.PENDF.NT APPROACH TO COLLISION PROIIU:MS

'We can then expand 'P(q, 0) in terms of this set •

,'(q, 0) - 2>.(0)�.(q) where we have denoted the consume expansion coefficients by c,(O), for reasons which will shortly be apparent·. \Vc can expand the solution of the time-dependent problem (2) in terms of the time­ dependent eigenfunctions 'P,e-iEtl,'t of lfO :

(5) �(q, 1) -2.,.(I)P.(q).-"" j, and we mllst now determine the time-dependent expansion coefficients c.(t) . The h sieal inter fetation of these coefficients is that c t 2 gives the probability of finding the system in the state !p.(q)e-iH,., at time t. Substituting (5) iOlo (2) we obtain the following infinite system of equations for the coefficients c,(/) : il/;; _'L.V,/OC/O(t) ei(E.-Ii/OJI/J where

,

(v-O, 1 ,

.

J

.

•

)

V., - V.,(I) - (�.(q), V(q, I)p,(q)) - dqp,(q) V(q, I)�,(q) .

(6) (7)

We can solve the system of equations (6) by a method of successive approximations. We shall now assume that the perturbing potential V does not depend on the lime c.xplicitIy. (The general case can also be easily derived.) We obtain the first approximation to the solution of (6) by putting ,.(1) -,.(0) (,-0, I, . . . ) s ide p�fquation (6), This gives at once on the right-hand ,""' L�""''---' �\ It!" ei(F..-E/O)I/A I (8) ,.(1) -,.(0) - 2.1' ."",(0) (E E ) . , . _

_

,

We shall, in particular, assume that initially, at time 1-0, the system is in an eigenstate of the unperturbed Hamiltonian HO, which we take to be Po(q), so lhal (9) ,.(0) - I",. (The general case, given by (4), can be deduced from this special case, which is, however, adequate for most applications.) Equntions (8) then become d{E�-EolIIA 1 ( 1 0) '.(1) - 1", - 1',0 (E, Eo) , (,,-0, 1 , . . . ) . _

• \Ve :tn: writing eigenfunctiOil exp:uuionl 111 suml ovcr diJcretc !t:tlel; if the problem h:ts continuoul spectnllll these lumm:ttiolll :tfe of COII� 10 be replnccd by the appropriate jutegr.>b.

II.

162

37.

T1�tE'DEPENDENT APPROACH TO COLLISION

PRODLE�!S

Substituting equations (10) in the right-hand side of (6) we can obtain a second approximation. 13ut for the prescnl we shall COIl­ sider the first approximation ani". This will be adequate if the per­ turbingpotcntial V is weak or at any rate has acted for not too long a timc intcrval t. Since, with Ollr assumptions. the system is initialJ}' in a definite eigenstate fPoof the unperturbed Hamiltonian no,wecannow inlcrp.r.ct k.(t)12 : as theprobabilityfor a transition to the state Po to have occurrcd in the time I. and we can think of Vas causing these transitions. In many problems, particularly in collision theory, we arc dealing with continuous sets of states. \\le arc then not interested in one individual state, but in groups of neighbouring states. The parameter I' now varics continuously. We shall accordingly write E=-E{v) instead of E.. We 5h:1I1 then consider the group of stales for which the energy E lies in the interval (II) E - LlE <E(••) <E+ LlE. The probability for a transition to one of these states to have occurred in time t is then

(

where

'

)

,;n '[(E- Eo)1/21,] ( 13) iii n(E Eo)" /2;' . We considered a similar function in Exercise 1.6. differs only appreciably from zeroin the narrowjlllcc),al ( 1 4) IE - Eo l reciable if ' I' ""'" I.e if I r is conserved, within Ihe limitatiolls ojtIlt III/certaintyprinciplt. Furthermore D E - Eo,

f"

(

') If"

D E -Eo, 2h dE

-",

-

-

f

I _;

_'"

"

[

�l dE l(E - Eo) 221fi1

;n E E ) ' ' ( - o

�

.

sin2 X � dx

-.

-I, independently of the value or 1/21i. Hence, if (dl'ldE) I V.ol� is a slowly varying function of E in the very narrow interval (14), compared with

163

37.

TIME-DEPENDENT APPROACH TO COLLISION I'RODLEMS

the variation of D, then the function D in the integrand of ( 1 2 ) acts simply as the 8-function 8{E- Eo), and we can write ( 1 2 ) in the form (15) which is correct for any interval of integration. The transition prob­ ability ( 1 5 ) is proporlional lo the time. Hence we can define a transi­ tion probability per unit time, w, which is constant,

f

2n d, w � T. dE W"I'S(E - E,), dE

( 16)

or, carrying out the integration, ( 1 7)

(dv/dE) gives the number of E + dE) per unit energy range.

states with energy

III

the range

(E,

The form which our result takes in the most general case is now apparent. Suppose we label the eigen­ functions and eigenvalues of 110 by the parameters IIl J 112, II, which only assume discrete values (say 0, 1 , 2, . . . ) , and by the continuous parameters VI> 1'2, • • • v" i.e. we write these eigenfunctions and eigen­ values as 9'1(111) ' ' ' TI,; "10 . . . v.) and E=E(III> ' " II,; VI> • • • I'.) . Consider a particular initial state 91o=tp(aJ> . . . fl.; ah a.) with energy Eo=E(ah . . . fl.; ah al) and a group of final states for which the discrete parameters have the values Ill> 112 • • • • 1/" and the continuous parameters lie in the intervals (Vi' "i+ dl'i), (i ... 1 , 2, . . . f). Let e(nlJ ' " II,; VI> V,)dl'ldl'2 . . . dl'. be the number of states belonging to this group. Then the probability per unit time for a transition from the given initial state 910 to the group of final states considered, is •

•

•

•

•

•

•

•

•

•

.

•

(18) Equation ( 1 8 ) represents the most general form o f our result. From it, any other case can be deduced by summing over appropriate ranges of the discrete parameters and integration over appropriate ranges of I'L' • • • "r' In particular, the transition probabilit.ies obtained in section 33 ancl in Exercise VIII.2 arc special cases of equation ( 1 8 ) . W e must now discuss the physical meaning of the approximations we have made and sec how r:'lr they are fulfilled in collision processes. We assumed that, initially, at time t = O, the system was i n an eigen­ state f!1. of the unperturbed Hamiltonian NO, i.e. we treated the system as though the perturbation was not acting before t ... O, and was only switched on at this instant. We have seen that an actual scattering

37.

T1�lE-DEPE"DENT APPROACH TO

COLUSION PROBLEMS

experiment docs not correspond to a truly stationary state j nor, how· ever, docs i t correspond accurately to the type or time-dependent process we have considered. However, we made the assumption that l';PIi/Eo. This means that we arc considering intervals or time suffi­ ciently long ror 'end effects' (essentially, transients due to the sudden switching on or II at 1 = 0 when the system is not in an eigenstate or J-/O + V) to be unimportant. In practice, the condjtion 1»Ii/Eo docs not introduce any limitations, on account or the smallness or Ii . Secondly, we used first-order perturbation theory. This leads to transition probabilities proportional to the time, i.t. to stationary solutions (apart rrom the initial transients) . It is also responsible ror the matrix clements

V,o - (p" V.o), between the zero-order wave runctions, occurring i n equation (18). This is essentially equivalent to t.he first Born approximation (i.t. re­ placing the exact wave function YI(r) in (33.16) by the zero-order wave runclion eil!} for the general case under consideration. Hence we � l1eCl(IS) to be valid under the same conditions. i.e. the potential V sh.o.uld be weak or the time t for which it actsshoulCl be sufficiently short. In collision problems. it again implies sufficiently high bombardingenergies. Weshall not consider the higher ap"p-roximatio.ns, but only state the result for thecase where all the matrix clements V.n in (1 8) vanish. In-this case transitions from the inilial slate 10 the grollQ or final st:u.cs..oLintcr.esU llayJ!il1 be Rossiblc in the second approximation. TIle transition nrobabili is then still given b>:: e ualion (181. with _tl'l.CJTla.lri . ill. m c.nLE".o.J:£giaccd by

VOl' V,..

L (Eo - E,,) /.

,

(19)

t!u:....summatmrLb.cing oyer the c.o.mplc.te sel_oc... cig.CJlrllnctionL or 110. .t.rans i Lions.J:an... Qc.c.urj!LSecQncho.r:der... approxjmation._iL .!.h \'�c.c...a.t... tJler.u."{i�s 0/,. ror which V/4l and VI" are both different rrom zero. SuclLstates. ar.c_callec lintcun.c.dia.tut
165

38.

PARTIAL WAVES

r::�8. Partial waves I We now come to discuss briefly the general principles of a completely

different approach to scattering problems which is or great importance in applications·, We discuss the same problem as in section 32 : the scatteringof a particle with mass. neglectingspin (i.e. either theparticle is Slinlcss or the interaction causin the scatterin is s in-indc endent , y a spherically symmetric potential V(r) which has a finite range fl, i.e, for r>a,V(,) -0. (1) This condition can be relaxed to include'potentials which tend to zero faster than lIT as r _ co. In section 32 we considered a plane wave scattered by this potential. Now we shall analyse this plane wave into spherical waves and examine the distortion. due to the potential. of each separate partial spherical wave. For this purpose we write the scattering equation (32.6) in spherical polar coordinatcs. For a spherically symmetric potential and spinless particlcs the problem is axially symmetric about the direction of motion of the incident particles which is taken as z-axis. Hence lhe solutions are independent of the azimuthal angle p. Separating variables, as in sections 12 and 13, one finds solutions to equation (32.6) given by (2) ( )P, (co, 0), (1-0, 1, 2 . . .) where GtCr} is a solution of equation (12.7): � '! + U(,) _ 1(1 + 1 ) C,(,) -0, 2 dr r2 r dr (i-O, 1, . . . , for the interval Oa, this equation reduces to the free particle equation which we shall consider first of all , : .!. !! 1(1 + 1 + 1 -0, .\·2 dt d:< .t2 (4) (l =0, I , . . . ), ,/ with

C, , (,'dC,(,)) [kl

]

_

(X'dF,(x)) [

_

x=kr.

)

)] F,(x)

}

This is a second-order differential equation for each value of I and so possesses two linearlyindependent solutions 111( 11 and h,(z)(x).known as spherical Hankel functions of order l and of the first and second kind. The spherical Hankel functions have been extensively investigated. We shall only state those properties which we requiret .

(x)

• For comprehensive treatments if MOil :md Mrusey21 and Massey2J.

useful sections in Molt nnd SneddonlO nnd Schifflz.

There are also

t For a ooHection of propcrliC3 cfSchiIT!2; an cxccllent mathcmnticnl treatment of th� functions is given by SOZ1lmerfdd�. The standnrd treatise s i G. N. Watson's Tht()r} ofBuSt! funtli ()1I1, Cambridge Unil'crsity Press, 19014, and Macmillan, Ncw York, 19-14.

166

,--

38.

PARTiAl. WAVES

The spherical Hankel functions of order I are related to the Hankel functions of order l +t by

hl') (h) �

7t

Ur

!

(,)

lJ,.,(h) ,

(,- 1 , 2) .

(5)

For real values of l and :0: (and we of course restrict ourselvcs to these), II/I )(X) and hP)(.t) are complex conjugates of cach other:

[ft}U(x)] = IIP)(x). (6) This follows in fact since equation (4) is a differential equation with real coefficients, so that ifh/ I) (:0:) is a solution so is its complex conjugate II,W(x). yoI' large x, the 11,(')(:0:) behave asymptotically like

::';:� cxp {{t - (I + I );]) hP)(x) ::,;: � exp { - i [x - (l + I ) ;]}

(7a)

11/(1)(:0:)

.

(7b)

From equation (33 . 7) we see that they behave like spherical outgoing and incoming waves, respectively. This suggests forming linear COIll­ binations which behave like standing waves, i.t. like sine and cosine functions. These functions arc known as spherical Bessel functions and spherical Nculllann functions (they arc also called spherical Bcssel functions oflhc first and sccond kinds) and arc respectively defincd by jl:o:) =

�[hl(I)(.\') + h/2J(.\.)] ,

(8a) (8b)

It follows from equation (6) that these functions arc real for real values of x and t. From equations (7) one obtains the asymptotic behaviour, for large x : j/(x) ::';:

.� cos [

x

( ) ::';: � sin

lI/ X

.

.•

)�] - (l + I )�]' . - (l + I

,

(9a)

(9b)

For small x (i.t. as x -:>- 0), the spherical Bessel and Neumann functions behave likc . x' ( lOa) x ),( ) " 1 . 3 . . . (21 + I ) ' 1I/ (X) ::';:

( - 1 ) 1·3 . . . (21 - 1 ) �+ I

( lOb)

From c uation lOa we see that the s)herical Bessel function is that 167

38.

PARTIAL WAVES

s olution of the radial wave equation ('q which is regular at the origin ; all othersolutionshave a sin�y there... The general solution of the free particle equation can now be written as a superposition •

L: [A,h'<"(h) + B,h,<" (k,))P,(cos 0).

1-0

(II)

In particular, we saw in section 32 that it possesses a plane wave solu tion (12)

which is regular everywhere. Hence it will conlain only Bessel fune· tions in its expansion which can be shown to bc·

'0 - c;" - � (2/ + 1 ) iJ,(h)P,(cos O). /. 0 �� \.?r('/Wc can decompose 1po into'_olltgoing and incoming wavcs where

V1o -lJIoW + V:o(2)

( 1 3)

(Ha)

•

V'o'" - L: (21 + l )i'-!h'<" (h)P,(cos 0).

( 1 4b)

'- 0

'tVe return to the scattering equation (3). Outside the range of the for r>a, this equation is idemical with the free particle equation, so that for r>a, the general solution is still givcn by cqua· tion ( l l). For comparison with the plane wave expansion, equa· tions (13) and ( 1 4), it will be conveni ent 10 write the general solution, equation ( 1 I ), as potential, i.t.

•

, -L:(21 + l )i'![A,h,"'(h) + B,h,'"(h)]P.(cos '_0

0).

(,>a).

( 1 Ia)

Inside the range of the potential, i.t. for r
168

We :wume the

38.

PARTIAL WAVES

Only the bounded solutions are physically admissible. this follows from the fact that

For 1> 0

f,'I'pdr, I'

representing the probability of finding the particle in a volume V, must be finite. For a small region V surrounding the origin, where the irregular solution behaves like ,-'- 1, this integral diverges if 1>0. We shall now prove that for 1 = 0 too, only the regular bounded solution is admissible, i.e. ( np) ,..o must be zero (cJ Diracls, pp. 1 55156) . We have

( 1 5) and it is the singular behaviour of these expressions at r = O which causes the trouble. We can write the radial equation (3 ) for [ = 0 as

1 d2 { rGo (r) } + [k2 - U(r)] Go(r) = 0. r dr2

( 1 6)

Consider first the special case of a free particle of zero energy :

k2 = U = 0. Equation ( 16) then becomes I d2 { rGo(r) } =0 r dr2

( 1 7)

and has the irregular solution

Go{r) = ! . r The general scattering equation (32.6) : [V2 + k2 - U(r)]'I' = O, in our special case with k2 = U =0 and 'I' = Go (r) reads

V2Go{r) = 0.

(18)

(19)

(20)

But the solution ( 1 8) of equation ( 1 7) is not a solution of the specialized scattering equation (20) , since

1 V 2.: = - 4n 8(r) ; r i.e. it is a solution everywhere except at the origin r = O. Similarly in the general case k2:;60:;!: U(r), if equation ( 16) possesses a solution 1 69

38.

PARTIAL WAVES

Go(r) which behaves like I {r �lt the origin, this is not a solution of the scattering equation (19). For

{

I d' I d' -,"Go - - -(,Go) + - - (,Go) + (k' - U]Go r dr2 r dr2

}

and if Co behaves like I/r ncar the origin, this is equal to 1 'V'- _ - 4,,8 (r) ,

ncar the origin and hence is not a solution of the scattering equation at [he origin. This shows that we must always consider only those solutions of the radial scattering equation (3) for which np -+ 0 as r -+ 0, i.e. the bounded solutions. From the required asymptotic form (32.9) it follows that '1' - eil• must contain only outgoing waves. From equations (14) and ( I l a) this implies (21) The physical interpretation of this is of course that the incomingpart ofthe plane wave which originates at large distances from the origin i§ not a ff ected b y the scattering centre. This only affects the outgoing waves, distorting them from what they would have been in the free particle case. We can now rewrite equation ( 1 1 a) using equations (21) and (8) as 'P

• =

2:(2l + 1 )iIJ[A/ll1)(kr} +hp)(kr}]P/(cos 0),

'·0 •

(r>a),

-2(21 + i );'H(A, + I )j,(h) + ;(A, - 1 )",(k,)]P,(co, 0), '.0

(,>a). (22)

we have seen above, 'P is to be a regular solution. For each (-value, the regular solution CI(r) of the scaltering equation (3) iii completely determined apart from a constant multiplicative factor. It behaves like r' at the origin. By a suitable choice of the multiplicative factor, we can make the first non-vanishing derivative at the origin, (d'C,(r){dr').�o, real. Then CI(r) is real for all valucs of r, since the scattering equation (3) is a real differential equation, i.t. I, U{r) and k2

As

170

38.

PARTIAL WAVES

arc rcal. Since �'. equation (22), is a regular solution, it rollows that, ror all values or r>(I,

where C, is a complex number determined by the normalization orthe real runction G,(r). Consequently the ratio .A, - l 11, + 1

, --

is completely dctermined and is real sincej,{kr) and II/{kr) are real. It rollows at once that A, is a phase ractor or modulus unity which we can write (25)

where 3, is real. Thus we finally oluain ror thewave (unctionwhich represents the solution or our scnttcringproblem •

� - ::E(21 + i);'![cxp(2;0,)h,'''(h) +h,'"(h)JP,(co, 0), (r>a), ,.,

(26)

Equation (25) allows a vely simple physical interpretation. Each pnrtial wnvc with a given I-value corresponds to particles or angular momentum I. \Vc are treating elastic sC
-::E(21 + Wt"'<"(kr)[c"p(2;0,) - IJP,(co, 0), ".

(r>a),

(27)

which has lhe asymptotic rorm, ror large values orr, c,',Ir

�

V'" '" 2 " , ::E (2 1 + 1 ) [cxp(2;8,) - I J P,(co, 0), 'M 1 _ 0

171

(20)

38.

PARTIAL WAVES

Comparison with equation (32.9) gives the scattering amplitude

� = 1 � (21 + l ) exp (i8,) sin3,P, (cos 8).

1 � ( 8) = 2ik (21 + 1 ) [.exp(2i8,) - 1]P, (cos 8) '-0

(29)

}

Equations (29) represent the basic result of the partial wave analysis. The scattering amplitude (and hence the differential and total cross­ sections) are expressed in terms of the phase shifts. These form an infinite set. We shall see presently that often only a few phases make a significant contriBUtion to the cross-section, i.e. the remaining phase shifts 3, are very small. It is in these cases that the p,artial wave analysis is so useful. The differential cross-section is now given by equations (29) as a finite polynomial in cos 8 whose coefficients are known functions of the phase shifts. By fitting these expressions to the experimentally determined differential cross-sections one can try to determine the phase shifts. From equations (�9), (32. 1 1) and (32. 12) one easily obtains the total cross-section, using the orthonormality of the Legendre polynomials [equations (13.16) and (13. 1 7 )]. One finds CTtot(k) =

�� (21 + I ) sin28, (k), '-0

(30)

where we have written 3, (k) and CTtot (k) to stress the energy dependence of phase shifts and cross-sections. One easily proves the optical theorem from equations (29) and (30) which relates the imaginary part of the forward scattering amplitude (i.e. 8 = 0) to the total cross-section [using the fact that P,(I) = 1] : 4n (3 1 ) «1tot = k Imj(O),

(if Exercises VIII.6-7). . To find the phases for a givenpotential is in general a difficult problem. For a few potentials one can find the phases from an exact solution of the scattering problem. In general, one has to solve the scattering problem by a numerical integration of the radial wave equation, going outwards from the centre of the potential, matching the logarithmic derivative of the wave function at the surface of the potential (if Exercise III.5) , and proceeding to large distances to find the asymptotic behaviour and hence the phase shifts. Under certain circumstances various special methods exist for determining the phases. These matters are fully discussed in the references given at the beginning of this section. Here we only wish to discuss a few points. We stated above that, in general, only a few partial waves will be

1 72

38.

PARTIAL WAVES

of importance. Classically, one can easily see this. A particle of momentum ftk and angular momentum iii will have an impact para­ meter b defined by (3 2 ) 111 = 1Ikb and this particle will only be scattered if the particle comes within the range of the potential, i.e. if the impact parameter b is less than the range of the potential a : b
\

for the lth partial wave to be scattered. A similar result can be derived quantum-mechanically, but it is not easy to give a rigorous proof of general validity. We shall only give a plausibility argument. The radial equation for the Ith partial wave of a free particle, equation (4), contains the term - /(1 + 1 ) fr2. This term represents a repulsive potential, of increasing strength as I in­ creases. We can interpret it as a centrifugal barrier which prevents a particle of high angular momentum from penetrating to small distances r. This interpretation is justified by the fact that the solution jl (kr) of equation (4) is steeply increasing for small (kr) . From equation ( l Oa) it is proportional to (kr) l and one can show that it has its first maximum near kr� l. Hence there is only a very small probability of particles penetrating to distances (34) rka, agreeing with the classical criterion, equation (33). We can think of this criterion as follows. For a given potential, we fix our attention on a definite I-wave. ALs.ufficiently low ener� this partial waye will in general not experience appreciable scattering. As the energy is increased, i.e. as k is increased sufficiently, the scattering of the partial wave gradually becomes important. The region where it is no longer negligible can be estimated from equation (33) . .At sufficiently low energies, only s-wave scattering (1 = 0) will occur.. As the energy is increased. the p-waye contributioD to the scattering wm become important and so on. .. From equations (29) w e see that the angular distribution undergoes a definite change as successive partial waves come into play: if only partial waves with I-values less or equal to L contribute significantly to the scattering, the angular distribution will be given by a polynomial of degree not greater than 2L in cos 9. In particular it is isotropic if there is only s-wave scattering.

1 73

38.

PARTIAL WAVES

'Ve shall next derive an exact expression for the phase shifts in terms of the wave function. Defining (35) equation (4) becomes

[

d211/ _1 + k2 (1 dr2 + r2

I)] ", �O'

(36)

and from equation (9a), II, behaves asymptotically like I/,(k,) �sill (kr - tl:t).

(37)

Similarly writing

v,(r) �krG,(r), equation (3) becomes

[

d2v, k2 - U(r) .,..:; nr2 +

/(/ + ,2

1 )]

- -_

V,_

0

•

(38)

From equations (23), (25) and (9), we see that v, behavcs asymptotically like (39) v,(r) �sin (k, - -![a + 8,). From equations (36) and (38) one obtains d2u/ d2vj v,- -111- - - U(r)u/v,. dr2 dfl

(10)

We integrate this equation from r - O to r. The contribution to the left-hand side from the lower limit vanishes, since /l/{O) -v,(O) -0. [11,(0) -0 follows dircccly from equations (35) and ( lOa). v,(O) - O follows since G,(r) is t o be the bounded solution of the scattering equation.] Hence

;'

,

;'

v,(r) u,(kr) - ",(kr) ,,(r) -

-f U(r')",(kr')v,(r')dr'.

(+1)

,

If we now let r lend to infinity and substitute on the left-hand side asymptotic forms (37) and (39), we obtain k sin 8,-

fU(r)u,(kr)v,(r)dr.

tbe

•

-

,

(+2)

In the energy region where a given 8, is small, due 1'0 the centrifugal barrier as discussed above, wc can calculate 8, in first order perturba­ tion theory, i.e. use the unperturbed wave function in equation (42). 174

EXERCISES VIII

Using equations (32.4) and (32.5) and writing 8, ror sin 8" we find

This equation shows that, in the conditions under which it is valid, an attractive potential (i.e. V(r) <0 ror all r) leads to positive phases, a repulsive potential (i.e. J/(r» O ror all r) to negative phases. This result can be proved generally (if Exercise VIIL8). Using equation (43) ror all partial waves and assuming the 8, small is equivalent 10 usi ng Born approximation. To see th is, we substitute it into equation (29), replacing exp (2i8/) - 1 by 2i8, :\I1d use the identity [if Watson, loco cit., p. 363, eq. (3)]: sin Kr � - .<,(21 + ---, n r ,- '-0

. 1)1',(c05 O)[l,(kr)]'.

where K "",2k sin (0/2) as in equation (3'kS). In this way one again obtains the Born approximation (3'k3) for the scattering amplitude. The above considerations must be modified if one deals with particles with spin. The detailed analysis becomes considerably more compli� cated although the basic ideas arc not changed. One may now no longer simply consider states of given orbital angular momentum t. Instead one must decompose thc wave function into states of definitc total angular momentum and consider these separately. This is, for c.xample, what has to be done in nucleon-nucleon scallcring.

EXERCISE.S VIII

VIII . ] . Verify that equation (33.7) is an exaet solution of eq ation (3 di � '. !).k2 · Obtain, for the inelastic scallenng process considered in sec Ion 36, the transition probability per unit time to a state in which the

�

u

scallerer is in the state rp,(q)

�Obtain

;;-< '-;; -;::

the scattered particle

has a

momentum

the differential eross-scction for the inelastic sC:lItcring

",,'dS"" in section

1�

and

interval (Ilk" lik, +MI,,).

36, in first Born approximation.

. the differential scallering cros.<;-scc!.ion, equ:llion (36.16) ,� �tic s ttering of two identical particl . '@ Calculate the differential and total cross-sections ror elastic

fo

ca

es

scallenng or fast electrons by atomic hydrogen (ifE.xercise

I I 1.6),

!It:glecting

exchange cffects due to the identity of incoming and atomic eleCtrons.

1 15

EX.ERCISES V1II

�.y6ivc equation (38.30).

�

Derive the optical theorem, equation (30.31).

� Prove that a repulsive potential leads to negative phase shifts � rnctivc potential to positive phase shifts.

P

rVIII.

Jo for s-wavc SC"lt1cring if raj

Obtain an e.xprcssion for the phase shift

by a sp lcric."ll square-well potential:

JI(r)=

-

Vo

(170) 0). Show t1mt, though 80 _ 0 as the bombarding energy tends to 7.cro, the cross-section tends, in general, to a finite limit.

176

IX INTRODUCTION TO GROUP-THEORETICAL IDEAS

IN earlier chapters of this book, we have repeatedly dealt willI angular momCIIlUI11, parity and symmetry properties of wave functiOlls. These play a fundamental role in many problems, particularly in the C:lSC or complicated systems where exact, quantitative calculations arc difficult and where it is important to gain as much information as possible nbOllt the system by general methods. We shall now consider these symmetry properties morc systematically. For this purpose, we shall introduce the basic ideas of group theory. These arc exceedingly simple, and, as we shall sec, h.we a vcry beautiful unifying cncct, allowing many apparently diverse questions to be :LIlswcrcd by csscmially the same methods. These melilOds have been developed into an elaborate mathematical theory enabling one to solve very complicated problems. We shall of course restrict our� selves to the fundamental concepts underlying these methods, which should not only serve as an introduction to more specialized books on group theory· but should enable the reader to treat many important problems in atomic and nuclear physics. 39.

Groups of transformations

Consider a system of,Atidentical particle sofm ass.n lwhos eHamiltonian.. rcfcn:cd. Jo a given... system of rectangular coordinates C is fI -

L2�" p,' + L fI([r; - r;[).

(I)

i<j

If we consider transformations to olher coordinate systems. wefind that some of these leave the Hamiltonian ( l ) unehang�. We shall say that H is invariant or symmetric under these transf ormations. These particulnr transformations include; (I) Permutations of the }{ identical particles; (If) RcOection of the coordinate system C in the ongill. i.e. the transformation r -. r, going from a right-handed to a left-hande d coordinate system andviceyersa' -r:tfIj Sna ia! rotations of the coordinate system about the fixed origin; (IV) Rotations of the coordinate system about an axis fixed in space. (We shall usually take this to be the {-axis.) .. A p3rticularly lucid and simple tre:nment WaerdenU.

177

....ilI

be: found in the book

by

"an dec

39.

GROUPS 01' TRANSFORMATIONS

The invariance of ( I ) with respect to permutations of particles follows from the ident.ity of the particlcs; permuting them simply implies a rearrangement of terms in (I). The invariance of (I) with respect to 'the three other sets of transformations follows from the [.'1ct Ihat (1) only dep'ends on the magnitude of certain vectors, but not on UlCjrsense or oricnta1ion. Applying a homogeneous magnetic ficid in the z·direction to the atom, removes the spherical symmetry (111), but the system will sLill be axially symmetric abom the z·axis, like the magnetic interaction term, equation (31.25), which must now be added to the Hamil· Ionian ( I ) ·. If we supplement each of the sets of transformations (D to (ID by the identity transfol'm;ujoD, which !cayes the coor dinale system Ill)· a a I

we denote in these two the by t and the systems by q and q' (standing for q.. Q2, . . . and ql ', q2', . . . ), so that I : q _ q' =Iq

(2)

(i.e. t operaLing on q transforms it into Q' = Iq) . We can define a group of coordinate transformations as follows. A set of transformations {s, /, Il, } forms a group. if (i) any two transformations s and I applied successively arc a trans· formation of the set; we ",rile this transformation Is and call it the product transformationj i.e. if s :q=t'1' =rq, and I'q' =kq- FIq', then (ls):9 _q- -tsqj (ii) to every transformation J of the set, there exists an inverse transformation, denoted by I- I, such that if 1 :'1_'1' =19, then 1 l :q' _ q =J Iq'. It follows from Ui)..Jllat the unit clement or identity e which leaves the coordinate system 1II1chan�d, e : q _ q, has the properties that /I- I _ t I/=e, and le-el_l. QUI' notation Iq and terminology 'product', 'unit clement', elc must not be interpreted too naively as multiplication elc as I stands for a quite general transformaLion . Consider lhe group ffi, of a.xi�1 rotations about tlle ,,-axis, exampic (IV) abovc (!Hz as we arc dealing with two-dimensional rotations). A rotation of the coordinate system through an angle a (we shall denote tlIe corresponding transformation by l�) only changes the azimutha1 angle rp of the polar coordinates •

•

•

• For the moment we are neglecting Ipin. which we _hall only introduce in section

178

42.

39.

GROUPS OF TRANSI'OIHIATIONS

(r, O, rp) of a point P ; I,, : rp -+ ({I' - /aP-({I - a, as can be seen from One casily sees that (/0) - 1 -La, that the identity trans­ formation is e - 'o (i.e. 0 "", 0, no rotation) and the product of to. and I� has the properly

Figllre 14.

(3) This last equation states the geometrically obvious fact that two suc­ cessive rotations about the same axis carried out in cither order lead to the same final configuration. We sa the transformations of the a..xial rOlation group cOinmUlej it is called an Abelian grQ!!p. TIe

reflection or inversion group @t �ple (I I) above), consisting of only the two transformations e : r -+ r and i : r -+ - r is also Abelian. y

y'

Figure U.

Axial ralaliOIl

, , , , , , , , , , , , , , , , , , ,

of

coordinate .J)'slem aboul ;:-nxj .J IIlrough (lIIgle

u

..

.

, , , ,

--

----

z'

" �----L

_ _ _

z

The group of spatial rotations alJ, or as we shall say morc briefly, the rolation grollp »"1) [example above], however, is non-commuta­ tive: two finite rotations, carricd out in either order, in general lead to two diflCrcnt configurations. Similarly, the permutation group 6N of N objects [example (1) above] is non-commutative for N>2. We now consider the effect of the transformation I, equation on a functionf(q). It is equivalent to a !ijlbstitlltion of new coordi­

(Ill)

(2),

nates:

f(,,) -f('-' q') -J'(q').

(4)

Given the transformation I, equation (4) defines a new function f' (q) for any given fUl1ction f(q) , i.e. i t defines an operator T: T:f(q)

�

f(q) -J(t-' q) . j' (q) - T

(5)

The 0 erator T transforms the function f(1) into that new function w IIC 1 las t Ie same va lie at e same omt rc errc '10 t Ie new coordinates. To give a simple example, suppose f(1) sin q, and q' _lq_ q1., so that 1 Iq _ ,lq. Thenf'(q} =f(/-lq} = sin ";q.

179

'10.

ROl'ATIO:"< OPERATORS

.Q..! mechanics it is of For the applications of group theorY...!.uantum fundamental imnortance tllat i( the set oC !;{t QU1i. natc....tIilllsNrJD.a.tiolls­ Is, I, 1/, } forms a g !:.Q1!j2 . so docs theset o(operatofS {� .I Jl.--,-,-,,, }. . This is easily proved. Suppose q' =sq, q- =Iq' and q- =lIq, so that II = ts. \'Ve have, from equation (5), that Sj(q') =j(q) , and putting �(q) �Sf(q), that T�(q ) ��(q'), ;.,. TSf(q') �f(q). But we also have UJ(q") =J(q), and, since J(q) is an arbitrary function, it follows that U = TS. One similarly proves that if r l is the coordinate trans· (ormation q=r l q', then the operator associated with this transforma­ tion is S-I :j(q) --+J(q') =S-lj(q), i.t. the inverse operator to S. Hence the operators {S, T, ·U• . . . . } have exactly the same laws of combina­ } . and hence form tion as the coordinate transformations {s, t, II, a group which differs in 110 essentials from the group {s, I, II, } *. The functions j(q) that occur in quantum mechanics arc of two k inds. J(q) may be an operator (in which case the transformation I must of course also be applied to any differential operators, such as - ili0j8q, that may occur). Alternatively, it may be a wave function describing the system under consideration. so that T becomes an operator in the corresponding function space. ''''c can then think of the coordinate transformation t in configuration space as inducing a transformation in the function space, by means of lhe operator T. The operators T occurring in quantum mechanics arc always unitary. For the transformation groups that occur, in particular the four examples we gave, leave the inner product invariant. They correspond simply to a change of the variables of integration. Con­ sider a spatial rotation t : r --+ r' =tr. Since dr' =dr, it follows from equation (4) that for any N-panicle wave functionsj and g, (g,j) =Jdrl . . . drxg(rlJ . . . rN) ](rl > . . . rJ{) =Jdrl' . . . drJ/ g'(rl" . . . r)/)J'(rl' • . . . r,v' ) - (g',!') � ( Tg, Tf), (6) i.e. the operator T is unitary [if equation (18.15)]. In lhe same way one proves this result for other transformation groups. •

•

•

'

•

•

•

•

•

•

40.

Rotation operator5 ''''e consider the spatial rotations of a system of N particles, with position coordinates (x;,),;. <"i), i = I , 2, . . . N. If l� denotes a co­ ordinate transformation rotating the coordinate system through an angle aa about the z-axis, then X( = I-:-'(i=X;+ OX;, (i= l , 2, . . . N), etc where, as one easily sees by going over to spherical polar coordinates, ox; =) ;o a, 0)';= - x,·oa, OZi = O . '

} are sol1UJrphi" i i.t. to every dement Thul the group propcrlia of two s i omorphic groups are the same. We h:l\"e merely labelled the elementl differentl y . • In f:tel, the grouP! {s,

/, u,

• • •

} and IS, T, U,

• • •

S coTTC3ponds exactly One element S and vice vensa, and if sl-u, then ST-U.

180

'H.

INVARIANTS OF GROUPS OF TRANSFORMATIONS

vVriling p(:rj,y,<, �i) for P(Xh ' (;,.,,) , one finds, on account of equations (39.4-5), that q/(;fi'Yi' �i) ... T.p(x;,Y;, �i) =p(t�-IXi' t�-�'i' t�- I.ti) = p(x; - OX;,)'; - oY;, �j - o�;) .

""' W{·'(j,)'i, .ti)

-�(x" y"

(

-

<0) -

•

x

2: (:�ox; + . . . + . . . ) i-I

X

8a2:( y,;, - x,�}(x" )'" <,) i-I

- I - �:L�)!p (X;'Y;' .ti), so that

( I) where L� is the (;-eomponent of the total orbital angular momentum operator of the system (Exercise 11.1). Since analogous expressions hold for the operators T� and �, corresponding to infinitesimal rol'a­ lions through an angle 00 about the .'( and Y axes, it follows that the angular momentum operators completely determine thc infinitcsim:ll rotation operators Tn 7" T� . But one can further prove that the transformation propcrties of a wave function under a finite rotation arc completely determjned by its behaviour undcr thc infinitesimal rotations. For since the rotations form a group, thc opcrator of a finitc rotation can be built from the infinitesimal rotation opcrators by can)'ing out successive infinitesimal rotations, i.e. by a process of intcgration. lience the angular momentum properties of a systcm (i.e. the effect of the operators L", L" L, on its wave function) com­ pletely determine its properties under spatial rotations.

11.

Invariants of groups of transformations

A functionf(q) (which again may be a wave function or an operator) is invariant under a group of transformations {so f. II, • . • } iffor every transformation t of thc group (I) f'(q) - Tf(q) �f('- ' q) �f(q) · For example, the spatial rotation group D1) leavcs the magnitudes of vectors, or functions depending only on the magnitudes of vectors, invariant. A very gcneral Hamiltonian of an N-parlic1e system, which is invariant under rotations, i.e. spherically symmetric, is A'

Ho = - ""z1i.2 L ;_1

111;

'Vl + V(rl> r2,

•

.

•

rN; r12, ro, . . . )

(2)

where the interaction potential V is an arbitrary function of the distances '; = J r; J and 'ij .", Ir; - rj J .

IBI

41. INVARIANTS OF GnoUf'S OF TRANSFORMATIONS

,9lle can casjlysee that if an operator C(1j) is.lnvari.:!nt under a }, then it commutes withevery transformation group {SI t, U, opcrato rof the correSJloDdinggroup{Sor.u,.. .1 : (r(q), S) -0, ,/c. (3) For it follows from (I) for any wave function 9'(q), that Sr(q)�(q) _ F'(q)�'(q) _r(q)�'(q) - r(q)S�(q), i.e. (3) holds. Conversely, if p(g) is an arbitrary function and we dcfine 'P(g) -S-Ip(q), then equation (3), which must in particular hold for 'P(q) (since it holds generally), gives (,�) (r'(q) - r(q)} �(q) -0, and since p(q) is an arbitrary function, equation ( I ) follows. The results we have just obtained can be applied with advantage to ule diSCUSSIon 01't he constants ofthe motion (or the good guantum numbers, as it isoftenPllt) ofa system. We give two examples. (I) A spherically symmetric Hamiltonian 11 [e.g. equation (2)] is invariant under spatial rotations. Hence it commutes with all rota­ tion operators, in particular with the infinitcsimal rotation operators, equation (40.1), and hence with lhe total orbital angular momentum operators: [fl, L..] "",0, etc. (5) Since L2 is also sphencally symmetric and V· =L/· +L,2 +L.'- we also obtain •

(H, £2] -0

•

•

[£2, L..] "",0, etc.

(6)

Iience, fol' a spherically symmetric Hamiltonian, f1, LZ and one of the compollenlS, say Lu arc constants of the motion that can be fixed. (Since the three components L�, 4, Lc do not commute with each other, we can only fix one of them.) We have here obtained one of the main results about angular momentum (already derived in Chapters n, IV and VI), with very little labour. Furthermore, the significance of equations (5), which in section 25 we introduced some­ what artificially, is now apparent. (2) Consider the reflection or inversion group GI,. If J is the oncrator corresponding to the inversion of the coordinate system, 1 ;( _ r, then (7) f Qrwave (unctions of even and odd r.an res ecuvel . If the Hamil­ tonian If of a system s invariam under the reflection group, then If commutes with I, and so parity is a good quantum number. Tn applying lhe inversion transformation, j : r ---+ -r, one must care­ fully distinguish between vectors and pseudo-vectors which transform in the same way under rotations, but differently under reflection. If a(u) is a vector and a(plJ) is a pseudo-vector, then (8) ;a(v) - - a(v), ;a(pv) -a(pv). i

182

'�2.

SPIN

It follows from the defmition of inversion thnt such quantities :IS

momentum,

p =-mt, and force arc Yectprs. whereas angular momentum._ ..

rAp, is a pseudo-yector :

I

42.

i:l=ri\p _ ( - r) A ( -p)

Spin

...

1.

(9)

We now gener:liize our methods to be applicable to particles with spin t.

For simplicity we shall consider a single particle but the generaliza­

tions to several panicles will be obvious. As in section

wave function being

a

26,

we specify a particle of spin t by a two-camponcllI

vcctor in the product space .\l

x

space will now generallv be functions of

I'(r, 0). {s, t, Il, . . }

I'

.

(I )

6.

r

The operators in this and the Pauli vector

o .

We consider the eflcct of a group of transformations

on such operators r and wave fUllclions (j).

Applied to a wave function, :I coordinate transformation

t

now

induces transformations both in the orbital space .'iJ and the spin

space G.

Though our treatment and results :Ire generally valid, we

shall consider the rot:l.lion group, i n order to be able to interprct our rcsults physically at every stage.

/:r

Consider thc sp:ltial rotation

C to C'. As 'Pj (t- I r') .... 'P/ (r'), as systcm

_

r' =/r

from the coordinate

a rcsult lhe orbital wave functions in section

39.

'Pj(r)

become

nut referred to the new coordi­

nates the spin of lhe particle will point in a dincrent dircclion, thc

spin wave functions

a

and {J will be transformcd into !incar combina­

tions or the new spin wave functions

a = clu' + C2fJ',

(I'

and

f3' :

P = c3u' + c4f3'·

(2)

"Ve shall not derive this linem transformation, but merely slate that for a given lransrormation ( il is completely determincd, that it is a unitary transformation.

Cj = Cj(t),

and

Combining these results. we have that

'PI (r) a + 'P2(r)f3 = 'PI (i-I r') {CI a' + C2f3'} + 'P2(t- I r') {cJa' + c4P'}. (3)

As in section

39, we can express this by saying that t induces a trans­ T i n the function space .\1 x 6 defined by 11'PI (r) a + 'Pz(r)f3] =!Pl (t- I r) {el a + C2f3} + 'P2( t-I r) {cJ a + c4fJ}. (4a) We can decompose the operator T into two operators T� and T"

formation

operating in orbital and spin space respectively. defined by

T"�(r) �,, (t- 'r) P[�, (r)a + p,(rW] -p, (r) {e, a H,,8) + p,(r) {e,a H,fi) .

(4b) (4c)

and one easily shows that

(5) 183

42. SPIN P1 2, Pu" and pua which we introduced in

The permutation operators, section 28, are, of course, particular cases of T, T" and P correspond­ ing to the symmetric group (52' The operators T" are of course identical with the operators T which we had in the spinless theory (section 41 ). W e can now apply the operators T, T" and Ta to a n operator r(r, a) and obtain

T"r(r, a) = r(t-1r, a),

Tr(r, a) = r(t-1r, t-1a),

Tar(r, a) = r(r, 1- 1 a) .

(6)

Analogously to section 41, we can now define the invariance of the operator r(r, a under the group {s, t, . . . } , considered to induce transformations in the total, orbital or spin spaces respectively, by requiring that for all transformations t of the group :

)

Tr(r, a) = r(r, a) , and, as in section respectively imply

41,

[ T, r(r, a) ] = 0,

Tar(r, a) = r(r, a), (7)

T"r(r, a) .... r(r, a)

one easily proves that these three equations

[ T", r(r, a) ] = 0,

[ Ta, r(r, a) ] = 0

(8)

for all operators of the groups {S, T, • • • } , {S", T", • • • } and {sa, Ta, . . . } respectively, and vice versa. All the above considerations are easily generalized to systems of particles. An operator r(r, a) becomes replaced by r(r., rl, . . . j a" a2, • • • ) and a wave function {l)(r" r2, • • • ; sP), si2) ) takes the place of {l)(r, s�) . We consider the spatial rotation group atl• If t is such a rotation in configuration space, we shall call T", Ta and T rotations in orbital, spin and total function space. In section 40 we obtained the in­ finitesimal rotation operators in orbital space : .

.

•

(9a) [equation (40. 1) ] , L being the total orbital angular momentum vector. For infinitesimal rotations in spin space one expects by analogy (and one can prove this to be correct) that

T a = (1 "

8

;:S.),

etc

(9b)

being the total spin vector of the system,

8=

;2:a(,) . i

) 84

(1 0)

42.

SPIN

From equations (5), (9a) and (9b) one obtains for the infinitesimal rotation operators in the total space (9c) where J = L + S is the total angular momentum of the system. We next consider the inversion group &1' The inversion i: r -r' = - r again induces operations /', ]" and ]=/']" =]"]" in the function spaces of), 6 and of) x 6. Since the Pauli vector 0 represents an angular momentum and is therefore a pseudo-vector [if equation (41 .9) ] , inversion leaves it invariant, so that ( 1 1) ]"r(r, 0) = r(r, io) = r(r, 0) . One easily sees that ( 1 2) a' =]"a = a, P' =]"P = P, in other words, inversion does not affect spin space at all, ]" is simply the unit operator and ] = 1". For if (x,y, .t) is a coordinate system and (x',y', .t ') the inverted system, so that x ' = - x, etc, then U(' = u( since 0 is a pseudo-vector. Hence, uJ:. a = a implies Ur: a = a. But uJ:.. a' = a', and we know that the eigenfunctions of UJ:. are non-degenerate, so that a' = a. It similarly follows that P' = P, proving equation ( 1 2) . We illustrate these results for the X-particle Hamiltonian

HI = Ho + 2: Vy(rlt i<j

•

•

•

rN; r12,

•

•

•

rN-l. N) OjOj

( 1 3)

where Ho is the spin-independent Hamiltonian defined in equation (41 .2) and the second term represents a spin-dependent interaction. The functions Vy again are quite general functions of the arguments ,\Ve see that both parts of the Hamiltonian are , r12, rl t invariant under separate rotations in orbital and spin space, since O;Oj being a scalar product of two vectors is invariant under rotations, and of course, also under joint rotations. Furthermore, this Hamiltonian is also invariant under inversion. . Hence. the operators L2, LI;' S2, S(, A.Jc and parity are constants of the motion which can be determined simultaneously. The interaction ( 1 3) (with a much less sophisticated spatial dependence) is one often assumed in nuclear physics. With it, the ground states of the deuteron, of 3He and of 4He are 3SI t 2Si and ISo states respectively. One knows that the forces between protons and neutrons are not adequately represented by the central potential in the Hamiltonian Hi t equation (13) . One must supplement it by a tensor interaction which contributes an additional term to the Hamiltonian which now becomes •

•

•

•

•

•

N H2 =H1 + 2:WiiSiJ i
( 1 4)

where the Wy are arbitrary operators which are invariant under 1 85

43.

TilE REPRESEr."TATION OF GROUPS

separate rOlations in o.-bilal and spin space, and S, is the tensor operator s..=3(airij)(aJrij) - (aao) (is) I , II rij1 with rij = ri - rj' Under separate rotations in arbiml and spin space, the operator Sij, and hence the Hamiltonian N1, is no longer invariant. For the scalar products (air,;) are, of course, only invariant if we rot:lIe axes for both vectors. I-Ience, with the Hamiltonian H2, the operators LZ, L�. S2 and S, arc no longer constants of the motion. The case of a (wo-particle system is an cxception to the last result. In this ease S2 is a good quantum number cvcn when using tensor forces. For thc total Hamiltonian Hz is now symmetric in the spin of (he two particles, i.e. [N2• I'l l"] - D. Hence lhe wave function must be either symmetric or antisymmetric in the spin coordinates, i.e. the spin wave functions must correspond 10 either a triplet or singlet st'ate for which S = 1 and 0 respectively_ Though S is a good quantum number in this case, S. is not. We obtain anOlher example of a Hamiltonian which is only in­ variant under lOlaI rotntiolls, from the field of atomic physics, where the cenlral force Hamiltonian Ho. equation (1· 1 .2), s i supplemcnted by a term representing the spin-orbit interaction. This term is usually taken proportional to N LS or �)iSI (i6) ,.. I

and is clearly not invnriant under separalC orbital and spin space rotations. 43.

The representation of groups Suppose r is a Hermitian operator, corresponding to a physical observable, which is invariant uncler a group of transformations. It follows from section 1·2, that r commules with the group of operators (jj _ IS, T, }, (1) [ T, J'] - 0, elc. Suppose y is nn N·fold degenernte eigenvalue of r, i.e. there c."{ists an N-dimensional manifold rut in Hilbert space such that all vectors of9.l1 are eigenvectors of r belonging to the eigenvalue y. If rp is any \'eCIOI' ofIDI and T an operator of thc grollp(]. it follows from equation (1) that F(Tp) _ TFp- T,p _y(Tp) (2) i.t_ (Tp) is also an eigenvector of r belonging to the same eigenvalue Y. i.e. it also lies in \))1 . Hence 1J,ll is an i,warirml manifold of the I;(QUP. & : 0lum/ors 01& {mlls fonll L'ulors 0['»1 ;1110 veelors 0[,))1. If we choose any orthonormal set of vectors 'I . E2• . . . ';N to span 9.11. we can form the matrices (3) T,; - (", TO;) .

.

0

i86

43.

TilE

IlEI'RESENTATION

OF

GROUI'S

corresponding to the operators of Ihe group Q.I, and we cal! this set of N x N matrices, an N-dimensional matri:( representation Ll of the group Q.I. It has the property that to every operator T of the group corre­ sponds a matrix Ti;. and to the product operator ST corresponds the product matrix IS,l Ttj• Corresponding to a given group W, there will of course in general be several different matrix representntions. \Ve saw in section 22, that we can choose many different onho­ normal bases in WI, and two such bases arc related by a unitary trans­ formation U slLch that f";j =(U*TU)ij [if equation (22.13)]. The representations referred to two snch bases will of course appear dif­ ferent, but thC}' are not really significantly different: the effect of an operator T Oil a vector rp of,)}1 is independent of the choice of basis. We call two such representations, diflcring only in the choice of bases, eqllillnlC/lt ma trix representa Iions. If a vector rp belongs to Ihe invariant manifold ml, in which a repre­ scntation Ll of the group OJ occurs, wc shall say that rp belongs to or transforms according to the representation Ll. The stress is here on the manifold W1, independently of the choice of basis, that is, we treat equivalcnt representations essentially as a single representation. Supposc the invariant N-dimensional manifold IJJ1 can be subdivided into two manifolds 9Jl1 and 1))/2, of di mell Sionali ty III and 11 respectively,

(4) and each of these two manifolds, by ilsclG is invariant under the group (�, i.e. operators of�j tramferm vectors of mIl into vectors oflJJl h and si.mil�w:...l.9J .. [foor a group of unitary operators, one can show that ifmll is an invariant subspace ofIDl, so is mlz _1JJ1 _9Jl .. if E",crcise 1X-5.] Tfsuch a decomposition of WI is possible we call �l a rrdurible manifold, and the eorrcsP-Qndin� rcprcsentatio n LI reducibk, otherwise We can we call the manifold m1 and the representation LI jrmlucible. now try and reducc, i.e. decompose into invaria m subspaces, both the manifolds mIl and mlz. We can continue this process of reduction until IDl is completely reduced, i.e. in the decomposition ml."ml\ +ID12 +

+911, (5) all the sub-manifolds m1l, . . . IJJI, are irreducible. Suppose we take a vector rp of an invariant manifold 1]/ and operate on it with all the operators {S, T, . . . } of the group. ]n this way we generate a manifold 911', spanned by the set of vcctors {Srp, Tp, . . . } . HIDI is irreducible, then wr=�m, for al!y vector rp of ml with which we start. Ifml is reducible, thcn for some vect-ors rp of<JJ1, 9J1' will be a subspace ofrol. We call put this diOCrcntlyandsaythjl!only;1'211 isjrrcchlciblc_ will the group(lJ generate the whole manifold WI from (lin' vector p. Suppose we span the reducible manifold 1]1, cqw\lion (4), by the orthonormal veeton; �h . . . �s which we choose so that the vectors 1 B7

.

.

•

43.

THE REPRESENTATION OF GROUPS

Eh Em and E 1t E" span the manifolds ml and m2 by them­ selves. Referredm+to this basis, any operator T of QJ will have the matrix •

•

•

•

Tu T2 1

. Tm! .

•

•

.

•

.

•

•

•

.

.

.

•

•

•

.

•

Tim I

•

.

.

.

.

.

.

•

•

.

•

•

...............................-........_ .

since for 1
I

0

I:

Tmm. . ,� ..............................................................

.

.

O

I Tm+1. m+ 1 •

i i

• .

.

.. ! T", m+l :

.

.

•

.

.

.

.

.

. .

•

•

•

.

•

.

•

.

.

.

.

.

m " TEi = LE4 Ta = LE4 Ta, i.e. TEi 4-1 4-1 m+ 1
•

(6)

Tm+ 1, "

•

.

•

T""

lies in m.,

lies in m2• Hence a reducible representation is of the form (6), with a suitable choice of basis inm. However, a unitary transformation to a basis in which the basis vectors are linear combinations of the Elt E with the Em+1 E" will spoil this appearance. We shall illustrate our results for some of the groups introduced in section 39. Let us consider the inversion group QJ/ consisting of only two trans­ formations, the identity e r r and the inversion i r - r. The corresponding operators E, [ applied to a wave function 'P(r) usually generate a two-dimensional manifold gn spanned by (7 ) E'P(r) = 'P(r), and ['P(r) = '1'( r). This manifold is, however, always reducible into two one-dimensional (and hence necessarily irreducible) subspaces. For we can always write 'P(r) as the sum of a function of even and a function of odd parity (or parity ± I respectively), (8) tp (r) = H'P(r) + tp ( - r)} +H'P(r) - 'P( - r)} , and a functionf(r) of definite parity ( _ 1)n defines a one-dimensional manifold under reflection: (9) Ef(r) =f(r), .(f(r) =f( - r) = ( - l)�(r). So we obtain two irreducible one-dimensional representations of the inversion group, the symmetric and the antisymmetric representations, for which [= ± 1 respectively j we shall call them the ( + 1) and ( - 1) representations. 188 •

•

.

•

•

•

:

_

:

-

-+

..

43 .

THE REPRESENTATION OF GROUPS

Exactly the same two irreducible representations are obtained for the permutation group 62 of two objects, as the reader will easily verify, the permutation operator P1 2 taking the place of the inversion . ��_ L We next consider the rotation group which, as we know, is closely related to the angular momentum properties of a system, and one easily sees that the (2n + 1 )-dimensional manifold IDl,,, spanned by the eigen­ functions n, - n), n, n) (if section 25) of the angular momentum operators is invariant under the rotation group. For we saw, in section 25, that this manifold IDln is invariant under the in­ finitesimal rotations (in fact, we generated it by repeated application of the infinitesimal rotation operators) and any finite rotation can be built up from these. One can also prove that the manifolds IDln, n = 0, !, 1 , . . . , (and the corresponding representations, usually denoted by '1)"), are irreducible and that there are no other irreducible representa­ tions of the rotation group. For n an integer, the wave functions n, m) are essentially the spherical harmonics 'rnm(O, 9') [if equation (25.26)]. The representation �o is the identical one-dimen­ sional representation corresponding to a wave function 'roo which is independent of the angles (0, 9') and hence invariant under rotations. The manifold IDlI of the three-dimensional representation '1:>1 is spanned by 'rlm(O, 9') , m = - 1 , 0, I , i.e. by X,Y and � [if equation ( 1 3 .26)]. The addition of angular momenta in section 27 affords a useful example of the concept of reducibility. Equation (27.7),

'P(q,

•

•

•

'l'(q,

'l'(q,

IDl"I ( I )

nl+nJ

x IDl"J (2) = L: IDl",

(n l >n2)

(10)

in particular, corresponds to the reduction of the product manifold (which is itselfinvariant under 913 and furnishes the product representa­ tion 'DDI x 'Il"l) into its irreducible sub-manifolds. One also often writes equation ( 1 0) for the representations themselves 'Dnl

"1 +"2

x 'IlnJ = L:'D".

( 1 1)

An analogous situation holds generally. Corresponding to two representations Ltl and .1 2 of a group, one can form a product repre­ sentation .1 1 x .1 2 which is generally reducible. For example, the orbital wave function of two particles with orbital angular momentum nl and n2 lies in the manifold ( 1 0) . Under inversions, such a wave function transforms according to the product representation ( - 1 ) "1 X ( - 1 ) "1, i.e. according to the irreducible representation ( - 1 ) "1+"1. Lastly, we obtain the irreducible representations of the axial rotation group. This is an Abelian group [if equation (39.3)] and as such 189

43.

Till!. REPRESENTATION OF CROtJPS

its irreduciblc representations Excrcise lX.6)

i.e.

\ ; rC neccss.1rily

onc�dimcnsional

to each rotation of axes through an angle

(I

denOled this transformation by I" in section 39) corresponds a I matrix,

i.e.

a single (complex) number ?J( a) .

(if (we x

I

From equation (39.3)

;'IIld the propcrties of a rcpresentation it then follows that ( 12) The only continuous solutions of this functional equation are 91(a) -exp (ca) ,

(c = a constant)

(13 )

and if this represcntation is to bc singlc�valucd, as we require in practice, we must have ?J(O) -tp(2;1),

C _ IIII,

i.t.

(111 -0, ± I , ±2, . . . ) .

(14)

Hence, the axial rotation group possesses thc following irreducible representations (and one can show that they arc tbe only ones)

p.. (a) -exp (jllla),

(m-O, ± l , ± 2 ,

. . . ).

( 1 5)

So far we have only considered the rcpresentations of groups in terms of wave functions. transformations {S,

T,

.

.

operator

{of,

t, . . .

But we have seen that a group of coordinate }

induces

. } i n operators as well.

A

a

group

of

transrormations

Hence we can also talk of an

transrorming according to a particular representation

LJA

of a group
;p

belongs.

LIA x LI, LI

For exampic,

the

being thc rcprescntation to which

intcraction

betwcen

two idemical

particles is nccessarily unchanged if we interchange thc particles,

i.e.

this intcraction transforms nccording to the ( +

rcprescntation of the permutation group.

e1/r.. 1.

is an example of this.

I)

or symmetric

The Coulomb repulsion

Thc 'clectric dipole operator' (which we

shall mcet again in connection with selection rules) of a system of electric charges ii. with position coordinates forms according to the antisymllletric or (

rcnection group
- I)

(16) ri,

trans�

representation of the

Under rotations in spin�sp:lcc 6, the electric

dipole operator is invariant,

i.e.

i t transforms according to

'Do;

under

rotations in orbital space V, it transforms according to <;D I> since it tr;\Ilsrorms in the same way as the wavc function T, O{O, p)oc�.

Applicd to a wavc rUllction V' of orbital angular momentum L, D�'P transforms according to the product reprcscntation

ir L - O , irL > I ,

190

( 1 7)

44.

'H. I'ERTUIWATIO:-'S

Perturbations

In Chapter V I I we considered timc·independelll perturbation t.heory and, in particular, ils application to the calculation of the shifting and splitting of energy levels by perturbing potcntials . The group. theoretical methods, which we have so far dc\'cioped, allow scveral quile fundamental deductions to be m:lde in this field. Suppose lhe Hamiltonian [-/() is invariant under a group (1]. I f this Hamiltonian possesses an eigcnvalue E() which is TI()-fold degenerate, the corresponding eigenfunctions span an Il()·dimensional manifold 1))10 in which a representation of the groupru takes place, i.c. IJJ!O is invariant undcr the group(lj. Let LIS consider Ihe effect of a perturbation V which is also invariant under OJ. We val"y the T-Iamiltoni:tn and considcr thc Hamiltonian (I)

H=!1() + e V,

obtained from 11 0 b y continuously incrcasing the paramcter e from £-0. As a result of val" , in the Hamiltonian the dc cnerale ener y level EO WI I )c dis )laced anc! in cneral s )lit into sevcral lcvels and tllc maIU 0 ( �Io WI c transformed into a new manifold

comprisin the con·cs )ondin dllllenslOna Ity ol"I)JI, then

ertmbed ci renfullclion. 11 =1I(),

If

(2) II is the (3)

L by . . .in.o:casjllg e for we cannOt produce a discontinuous change il1� continuousl}' from c = O. Since we can in this way make I; as I::trge as wc le:\se it follows that eql!?tion (3) is true generally for an arbitrarily large perturbation . . In general a perturbation splits a degenerate level. However, by means ofthc group-theoretical ideas which we have developed, we can obtain some restrictions on tJle number oflevcls into which a degenerate level c:\n be split by a perturbation with the same symmetry propertics as the unperturbed Hamiltonian. Wc shall firstl),nrove the iml!2I.!.i!!! t result that if�lo is ilTeducible. so is�l. For suppose thaI for some small positive valuc of £, �l is reducible. The matrices of lhe operators of(lj, referred to a suitable basis, are then of the form

[

-

0]

A(,) . . . .. o B(,) ..

...

.

_.. ..

.

.

(4)

whcre A{ll) and B(e) arc submatriccs whose clements depend of course

i*

1 91

44.

p�IlTURnATIONS

on e. But if we proceed to the limit " -+ 0, a matrix of the form (4) always relains this form, i,e. in lhe limit i t is of the form

[ 'llO) I-Ei:; ]

(5)

,

i.e. mlO must also be reducible, contrary to hypothesis. As before, we can make the perturbation arbitrarily large by increasing £ continuously. We then have . Theorem J. A perturbation V with the same iI/variance properties as tht g....!!!. lin utuThed Hamiltonian flo (on mUtT s lit a de meTale [IUti ,orrts� ), he. an meduel t rt restnta/ioll however lar t Ihe ((turbalion ma The perturbation will of course in general e1isp acc IliCCllcrgr. level. c find the displaced energy' level, 10 the first order in _e, from c( uation 29.13 lo be £-EO +£('2. Vp) (6) where 'P is an)' nonncd function lYing In the manifold IDlo, We can verify that the matrix element ('I', V1/I) has the same value for any vector 1p of �mo (and so again prove theorem I ) by considering two different normed functions 'P and rp of !lJlo, Since !DlO is irreducible, there exists a unitary operator T of the group OJ stich that rp ... Ttp and [T, 1'] -0_ Hence (�, V�) - ( T�, TV�) - ( T" , VT�) - (�, V�). (7) The degeneracy associated with an irreducible manifold IS called melliial. We must now consider the case where the manifold 9)10 is reducible. In this case we decompose the manirold !lJ!0 into its irreducible mani­ rolds (8) where IDli (i = I, . , . s) is an irreducible space of dimensionality IIi (II-nl +nz+ . . . +11.), Theorem I tllen a t once gives Theorem 2. If the perturbatjon V has the same invariallceproperties as t�e 11Il/!ertllrbtd Hamiltonian HO, alld the manifold 9)lO of the Im/Jull/rom level EO call be comp/tle{y reduced, accordi1lg to (8), Ihell the pufllrbatioll V. however large, call at most split the level EO illto s leuels E E . , . E () de t1e 9' -1lJ. 'It. , ' , n, resfl!f1n!1.!)'. We call this degeneracy. which can be sPolit by a gctturbatio!1 V with the same symmetry as HO. accidellial. To find the energy level E,- to the first order in 6, we must decom­ pose 13,110 into its irreducible subspaccs, equation (8). We obtain Ej, as before, by taking allY normed wave function 'Pi belonging to the manifold roli ; tllen, E, _E° +e(�" V�,), (i - I, . . . s). (9) •

192

44.

PERTURBATIONS

Equation (9) gives the energy level Ej correctly only if the manifold IDli is inequivalent to all the other manifolds occurring in the decom­ Hosition (8), as is shown below ·. [By a slight generalization of section 43, we call two invariant mani­ folds IDl, and IDlJ equivalent if they have the same dimensionality nj = nj, and we can span them with orthonormal vectors Eh E2 , En, and 1}h 1}2, • . . 1}", such that the matrices of the operators {S, T, U, . . . } referred to these bases are the same (E" TEk) = (1}h T1}k) , (k, l = l, nj). •

•

•

•

•

•

If this is not possible, we call the manifolds, and the corresponding representations, inequivalent. Suppose the invariant manifolds IDlj and IDlJ are equivalent. In this case the further decomposition of the manifold IDl, + IDlJ is not com­ pletely determined. In fact, we can reduce it into two irreducible mutually orthogonal manifolds in infinitely many ways. Any two sets of functions hi = a ,. + b1}k ( 1 0) n,) (k = 1 , 2, 1.2 = eE. + d1}!

}

.

.

•

which are mutually orthogonal, (I." 1.2) = 0, define two such mani­ folds. We must then use the methods of section 30 to determine the correct linear combinations of1. 1 andh2 such that the matrix (fki, Vfl;/) becomes diagonal. But we need only do this for any one value of k ; for we see from ( 10), that the same linear combinations then hold (or all k. By choosing the linear combinations ( 10) , we have thus only to diagonalize a (2 x 2) matrix, instead of a (2nj x 2nj)-matrix. If, more generally, there are v equivalent manifolds IDlj, IDli+1 IDli+r- l , we have to diagonalize a v x v matrix.] Let us illustrate the above results by some simple examples. (1) Consider an atom with Z electrons. Neglecting the mutual interaction of the electrons, the atom will be described by a product wave function [if equation (25.26)] •

1p(Qh Ih ml)

•

•

•

•

•

1p(Q�, I�, m�)

and will hence have the parity ( - 1 ) '1+11+ . . . . . +I,. Introducing an arbitrarily large interaction V, which is invariant under inversion (e.g. the Coulomb repulsion '2,e2jrij) , does not change this parity, even j< i

though the atomic wave function is no longer just a product of one­ electron wave functions. For introducing the perturbation con­ tinuously cannot produce a discontinuous change by a factor ( - I ) in the parity. • The following di!ICussion, in parenthCJis, will not be needed, and may be omitted if the reader so dCJirCJ.

1 93

'H. PERTURBATIONS

(2) For the problem of two identical particles, which we have repeatedly treated, any perturbation V must of course also be inWlriant under the permlllation of the particles. Hence it cannot split the exchange degeneracy. The correct linear combinations for calcu­ lating the perturbation energy arc the symmetric and antisymmctric wave functions which each ' span ' a onc-dimension:ll irreducible mani­ fold. Hence, the first-order corrections to the energy levels are given by

'«'" (I )",(2) H, (2)",( I)], 1'[",(1 )",(2) ± '" (2)",( I)]) V" ± V", agreeing with our previous result, equation (31.IS) , but obtained with much less trouble. (3) Consider two atomic electrons with orbital angular moment:l I, and 11.( :lnd obtain (2/2 + I) 2(2L+ 1 ) ­ dimensional inv:lri:lnt manifolds 9J/L of m), with L=/I - 12, • • •, I, + /2, The manifold roIL corresponds to states of rcsuilalll orbital angular momentum L, and it is, of COUl'se, not irreducible. It is spanned by the wave functions �

�

q)i� = .z: C{:':ll"" ,v 2;:: {�jfl (II ' 11I1) lj/2 (12, IT/2) ± YI2(l1. 1111) tj/l (l2, 1//2)} , ..,_M

m!+

(M� - L, . . . L),

(II)

where lhe C's arc Wigner coefficients [if equ:llion (27.13) ] . Since there exists only Olle irreducible representation of 91), corresponding to wave functions of angular momentum L, namely ':DI., it follows that !!lli. must consist of two such equivalent manifolds. However, if we now consider the reduction of the manifold 9J1l. under the symmetric group 62, we at once obtain two (2L + I )-dimensional manifolds mIL", ' Each of these is irreducible under 9l) and 62, transforming according to the �L and ( ± I ) representations of lhese two groups respectively, and being spanned by the wave functions rp/:� (M= -L, . . . L), respectively. The Coulomb interaction then can at most split the 2(211 + 1) (212 + I )-fold level EO into 2 (2/'2 + I ) levels (and, in general, will do so), given by El.j:-Eo + (rpt�, J!91£�), (L=II - /2, · · · 'I + 12), ( 1 2) The level ELl: is (2L + I )-fold degenerate, and we can take any one of the values of M( = - L, . . . L) in calculating it'. We can sec the .M-independenee of the matrix element (rp£�. Vrpi�) in yet another way. The interaction V is invariant under �otatiolls 194

�· 5.

SELECTIO:-l RULES

and so independent of the azinllilhal angles PI and fpZ of the electrons. These angles enter the wave function 'Pt� only through exponentials like exp i[(IIII'1'I) + (M -ml)9'2], i.e. the matrix clement in (12) through terllls like exp i[(1111 - IIII') (IPI -11'2)]' These arc independent of M, and furthermore vanish unless 111 1 - 1111', i.e. lhe ' cross·tcrms' do not contribute to the matrix clement. 45. Selection rules I n Chapter VIn we considered collision problems and obtained as the general result certain transition probabilities w which were pro· portional to the square of the modulus of matrix clements of the form (1)

'

Here 'P represents some 'final' state in which we arc particularly interested, sllch as one in which a panicle is scatlered into a given direction, A is essentially the interaction which causes the scattering, and rp is the actual wave function of the system. In general 11' is not known and we approximate it somehow, for example, by a wave function of the unperturbed Hamiltonian, i.e. disregarding the inter­ action A. To calculate a transition probability one must of course evaluate the matrix element ( I ) . But in many cases it is possible 10 deduce by general symmetry arguments that a particular interaction A cannot cause transitions to a particular final state q;': the matrix clement ( I ) must vanish. Such rules, forbidding certain transitions, arc known as stlution rules. Though one can of couI'Se obtain such selection rules without inU'oducing any group theory explicitly. this procedure docs bring out the essential ideas common to all selection rules. To obtain selection rules fol' a particular process, we consider lhe symmetry groups of the system. For angular momentum selection rules, this is the rotation group, for parity selection I'lIles it is the reflection group. Consider a transformation group QL In general it will be possible to split the function space 0:, in which our system is described quantum mechanically, into mutually orthogonal subspaces �1Jl 1 > 1)]/2, 1)]/;, , which, together, span the whole function space, •

•

•

•

(2)

•

and these subspaccs �D1 . . IDl2' . . . l)]li, have the properly of being invariant and irreducible under the group Liz, . . . LI", . . . of the group
195

.

•

.-J

45.

SELEC110N RULES

We can expand any wave function V' into components belonging to the different manifolds IDl., IDl2, occurring in (2), i.e. into com­ ponents transforming according to the irreducible representations L1., LIz, , (3) •

•

•

•

•

•

where Ej lies in IDlj and is uniquely determined. Under the group <M, the wave functions '1', '1" and the operator A will transform according to some representations L1, LI ' and L1,c of this group, respectively. In general, these representations will of course be reducible. The wave function will

X = Arp

(4)

transform according to the product representation

(5) L1,c x Li which, of course, will also in general be reducible. We now expand the wave functions X and '1" into components belonging to different irreducible manifolds IDlb IDlz, , as indicated in equation (3). It then follows that the matrix element •

•

•

(6)

must vanish, if the expansions of X and '1" contain no common mani­ fold !Dl;, for in this case the two wave functions are necessarily ortho­ gonal. Only if these expansions contain at least one common mani­ fold !Ul; can the matrix element be different from zero. We can express this differently by saying that the matrix element (6) vanishes, if the complete reductions of the representations LI,c x LI and L1 ' contain no irreducible representation L1j in common. We can interpret this result as follows. The matrix element (6), corresponding to a physi­ cally observable transition probability, must be independent of the coordinate system chosen, that is, it must be invariant under the group of transformations <M. Hence, only those parts of the wave functions X and '1" can give a non-vanishing contribution to the matrix element, which transform in the same manner (i.e. according to the same irre­ ducible representations) . The matrix elements of parts of the wave functions which transform differently must vanish identically, as other­ wise they would give contributions to the matrix element which change under transformations of the group. The application of these methods is greatly simplified in many cases by the fact that the representations L1,c, LI and LI', according to which A, rp and rp' transform, are irreducible. We need, then, only investigate whether the product representation L1,c x LI contains the representation L1 '. It is in general only in this case that enough transitions are forbidden for any useful selection rules to arise. To deal with parity selection rules, we consider the inversion group 1 96

45. SELECTION RULES

G.l,.

If rp, rp' and A transform according to the irreducible representa­ tions ( - I )", ( - I ) . and ( - I )-, then Af}' transforms according to the irrcducible represcntation ( - l)·+�, and transitions can only occm if ( - 1)'+' - ( - I)" that is a + n +11' = cvcn. ( 7) To obtain angular momentum selection rules wc considcr the rota­ tion group. If, under rotations in spin space, p. rp' and A transform according to the irreducible representations 'l)s, '!Js' and Tls", then Arp transforms according to (8) '!Js" x'Ds-'D1.I',,_.I'I + . . . + 'Ds,,+.I' and the matrix clement (6) vanishes unless S' is equal to one of the numbers ISA -SI, . . ., SA +S. In particular, if A transforms according to 'Do. i.e. is invariant under spin rotations, then tr;:\nsitions can only occur if S =S'. Since (2S + I ) and (2S' + I ) aTe the spin multiplicities of initial and final states, corresponding to resuilant spins S and S', S=S' means that the spin multiplicity has not been changed by the transliJOn. Any operator 11 which only operates on the orbital part of the wave function has of course this property. In exactly the same way one obtains selection rules for orbital :md total angular momenta, provided the initial and final states and the operator II transform according to irreducible representations of the orbital and total rotation groups. The methods which we have given for obtaining selection rules arc very useful and can be applied to a great variety of problems: radiative processes in atoms and nuelci, fJ-deeay and nuclear reactions generally. The reader wilt find most of these discussed in the books by Sommer­ feld I l and Blatt and WeisskopPO. \Ve shall only give a few illustrative examples. (1) The line spectra of atoms arc due to radiative transitions, in which an atom in an excited state rp, with energy E, ' drops' to a leS!! excited state rp', with energy E'<E. At the same time a quantum of radiation of frequency , - (E - E')/I, is emitted. In the quantum theory of radiation it is shown that the transition probability of this process is proportional to the modulus squared of the matrix element (9) where D� is the electric dipole operator of the atom ,

<

D, -,L.r. l:"-" "-"-"'-Q ' ( 10) i_I (Z =number of atomic electrons, ri _ position vector of ith electron). 197

+5.

SEI.ECTION RULES

The dipole operator D� is the component of a ,'ector. Hence it transforms according to the ( - I) representation of the inversion group. It is independent of spin and hence transforms according to 'ilo under spin rotations. Under orbital rotations it transforms according to 'n i l being the component or a vee lor, and under 10lal ' rotations according to 'ill X 'Do ",, 'D I• Ir rp and p have total angular momenta] and]' and parity ( - 1)- and ( - I)·', then + . . . S)IJ_II. ir J >t = = (II) . ..... . x ....]_ if] ... O. '.VII Hence equation (7) gives the parity selection rule 1I' -n + l ( 1 2) i.�. : \11 eleClric dipole transition changes the parity of the atomic state. This is known as Laporl�'s mle. The angular momentum sclct:tion rules, obtained from ( I I ) , are, if \\Ie define iJ] J' -]. given by �J-O, ± I ("0, 0 _ 0) (13) with the transition ] -0 to J' -0 excluded as follo\\ls from equation ( 1 1), as we indicated in brackets in equation (13). 1r the total spin and tot,,1 orbilal angubr momenta by themselves are good quantulll numbers so thllt we can ascribe definite values S, L and S', L' to them in tbe two st:!.tes, we obtain the further selection rules (M=S' -S, LlL=L' -L), M-O ( If ·) ( 1 5) iJL-O, ± I (lIot O � O). Equ:!.lion (J4.) states that there lire no spectral lines connccting st"tcs of diffcrent spin multiplicities (known as intcrcombination lines). Such lines are not observed for light atoms, such as helium, but do occur for hcavy atoms. This is in :!.grecmcnt with the r...ct that Ihe spin-orbit intcraction is only important for hcavy atoms, and it is of course a consequcncc or spin-orbit interaction th:!.t L :!.l1d S arc no longer good quantum numbers [if equation ('J.2 . 1 6)]. lfone considers radi:!.tive tmllsitions, due to an electric dipole intcr­ action an:!.logous to (10), in nuclei, one obt"ins thc s"me selection rules as :!.bove. Selection rules ( 1 +) and (15) now only apply if we usc central rorccs and neglect tensor forces [qequalion (,1·2 .15)]. (2) In addition to these radiative transitions due to the elcctric dipole opemtor (10), other types of radiative transitions can occur. Thcse various types or transitions differ in the angular momentum and p:!.rity propcrties of the emitted radiation_ Howcver, these other kinds of mdiative Imnsitions are very weak compared with elcctric dipole rndiation. Consequcntly, they only becomc important for tmnsitions for which the matrix clement of the elcctric dipole operator vanishes, on account of a selection rule. Thcy lIrc almost exclusively of interest in nuclear rather than atomic physics.

{'.VJ+I

198

45.

SEI.RCTION RULES

In nuclear physics, magnetic dipole transitions arc of imporlance. For a nucleus consisting of Z prOlons and N neutrons, whose coordi­ nates we label with the numbcrs 1 , 2, 3, . . . Z, and Z + I , Z + 2, . . . Z +N-A, respectively, the mngnetie dipole transitions from a nuclear slate cp to a stale 'P' are associated with the matrix clement ( 16) where ( k) nnd oAk) arc the <-components of the orbital angular momentum nnd spin operators of the kth particle :lIld PI is its intrinsic magnetic moment measured in units of t/./2.Uc (mass of neutron - mass of protOIl - AI; t e1ectric charge of prOlon; /1, of course assumes only IWO values). One cnsily obtains Ihe selection rules for magnetic dipole transitions (if Exercise IX.7). (3) It is of interest 10 indicate briefly how the interaction operators (IO) and (16) come about. For simplicity, we consider the particular case of a single spinlcss particle of charge e. For :l radi:uive transition in which the panicle changes from a state rp to a sl:lte cp', emilting or absorbing n quantum of r:ldialion of momentum lik (i.t. w:lvc!ength 2:rlk) in the z-direction with a polarization veclor pointing in the x-direction, the matrix clement, derived from quanlUm electro­ dynamics, is ....

( 17) where p�- - ilia/a...: is the oper:ttor of the x-componelll of momentum and the plus and minus signs respectively apply to Pl"OCCSSCS in which a photon is :lbsorbed or created. To obtain parity and angular momentum selection rules relating the initial and final states cp and ql' in equation (17), we must nnalyse the exponential function into ttrms which themselves have definite parity and angular momcntum properties. I t can be shown that these arc in fact just the properties of the photon involved in the par­ ticular Pl"OCCSS considered, so that the selection rules merely Slale the conservation of :lngular momentum and parity for the system as a whole, i.e. panicle + radiation. 1n most applications, either or both of the wave funClions rp and 1" will refer to states in which the particle is restricted to a definite region; it may be an electron in an atom, or a prolan or neutron in a nucleus. Thc wave functions rp and p' are then only large in a region r
45.

SELECTION RULES

A� I to 10 X 10-9 cm our arguments do not apply. For nuclei, R�2 to 8 x 10-13 em, so kR is small even for photons with several MeV. For a 3 MeV ,,-ray and R�6 x 1 0-13 em, corresponding to a nucleus like copper, kR�O· l .) We can then expand the exponential in ( 1 7) into a power series and successive terms will decrease by a factor l /kR approximately : (18) Substituting the first of these terms in ( 1 7) we obtain a term propor­ tional to the matrix element (if Exercise IX.8)

f�I(r)(ex)tp(r)dr

( 1 9)

of the electric dipole interaction, equation ( 1 0) . Hence, if our con­ dition kR« 1 is fulfilled, radiative transitions are largely due to the electric dipole interaction. If on account of selection rules, the matrix element ( 1 9) vanishes, one must consider the second term in ( 1 8) giving the matrix element

(P,,�)

iekf�I (r)(p,,�)tp(r)dr.

(20)

The operator in this matrix element has a definite parity but does not belong to one irreducible representation of the rotation group, so we must decompose it suitably. If we consider two vectors r and their 9-dimensional product space is spanned by all cross-products

p,

xp.,yp", z:.Pn xp"

•

•

.

,

z:.p"

�l

and forms an invariant manifold transforming according to '.1)1 X under rotations. One easily finds the irreducible sub­ spaces transforming according to and '.1)1 ' The invariant quantity one can form from two vectors is the scalar product

= '.1)2 +



The first of these expressions i s simply they-component I, of the orbital angular momentum, and taking this term by itselfleads to the magnetic dipole transitions (if equation (16) : we are now considering a spinless particle, hence the second term of equation (16) drops out) . The second term in (2 1 ) transforms according to under rotations and leads to what are known as electric quadrupole transitions. In this way one can continue and obtain the selection rules for higher order transitions.

�2

200

EXERCISF.5 IX

The method which we have here developed is also exceedi ngly useful Ihe theory of p-decay. i.t. the disintegration of a neutron (usually in a nucleus) into a pro ton and an electron. (In addition a very light particle of spin -t. called a neutrino, must be produced in these dis­ in

integrations to ensure conservation of energy and angular momentum.) The matrix c1emellt for

such a

transition is of the form (22)

where p is the initial wave fUIlClion of the neutrOll, p' the final wave function of the proton and IiK the momentum transferred to the electron and neutrino. V represents an interaction potential which is not fully known . General considerations show that for slow neutrons and protons (so that they can be described non-relativistically) this interact.ion can have only two forms, either

0'

'

V _g

(23.)

V - g'a

(23b)

where g and g are constants and a is the spin vector of the nucleon (i.t. the particle which starts life as a neutron and transforms to a proton). Again, the wave functions rp and tp' are only appreciable ins id e the n ucl eus, r
sponding to the first term

(I)

p-decays corre­

arc known as allowed transitions, SlIC­

cessive terms as first, second, . . . forbidden transitions.

The

reader

should have no difficulty in deriving the selection rules for these cases.

(A

shorter and less complcte account of p-decay. than in 13Iatt and

'VeisskopPO, containing

Bethc and Morrison 19.)

the essentials however.

will

be found

in

EXERCISES IX IX.I. Show that the permutation group of n objects, G", is Abelian, if -2, and is not Abelian if II >2. lX.2. If G"-{s, I, fl, } s i the group of permutations of /I objec ts, prove that the corresponding group of operators {S, T, U, . . . } consists of unitary operators. IX.3. Prove that each of equations ('�2.7) implies the corresponding equation (42.8) and vice versa. IXA. Given that the ground state of the deuteron is a JS1 state, under the assumption of central forces, how is this state modified if we assume n

•

•

•

some tensor force acting?

( lX� If� ill an invariant manifold of a group of unitary operators ;' {S, T, . . . } and llll is an invariant subspace of ml, prove that the

-Q.I

201

subspace of "lll orthogonal to the group

m.

EXERCISES

I!}lL>

IX.

i.t. Il!tl -IDl-9Jlj,

is

also IIlvariant under

IX.G.

Prove that the irreducible repre.senw . tions of an Abelian group of

U:�

Obtain the selection rnlcs for magnetic dipole radiation, the

lII� operator.! nrc one-dimensional.

ma tr'x clement for which is given by equation ('1-5.16).

xy

Show that the matrix element of the clcctric dipole transilioll,

equation (45.19), follows from equation (45.17) if Olle approximately pulS

cxp{ik�} lX.9.

..

I , i.e. takes

the first term in equation (45 . 1 8 ) .

The matri.,. clement for

eiet:lric quadrupole transitions from a

nucleus, of charge .(e and mass number ii, is, by generalization from equations (45.20-2 1 ) , found to be proportional to

�

rp'[.o:: , PAk) +x,p.(J:)]trdr, . . . drA i _ '1'

where rA :\1\d p(k) are the position and momcntl1m vectors of the kth proton and

ncutron for

I


selection rules for electric quadrupole tramitions.

202

Obtain thc

X RELATIVISTIC T]-IEOR Y OF THE ELECTRON

46. The Dirac equation for a free particle The qU:Ullum mechanics which has so far been developed in this book docs not satisfy the requirements of the special theory of relativity. We sec this, for example, by looking at the Schrodingcr equation which was derived from a non-relativistic '-bmiltonian. I I is a first-order differential equation in time but second-order in the space coordinates: time and space coordinates do not enter it in a symmclI;c form-on :1Il equal footing as we may say-as required by the special theory of relativity. These remarks suggest two ways of obtaining relativistic wave equations. The first of these was already proposed by Schrodingcr in his originnl papers on wave mechanics. Instead of the non·relativistic Hamiltonian for a free particle, equation (5.7), he takes the relativistic Hamiltonian (1) and now makes the operator replacements

E

--+

;

ili t'

P -+

-

[equations (6.1), (6.2)] in this equation.

iIi grad

(2)

One obt:tins (3)

This equation is known :ts the Klein-Gordon equation. \Ve can write it in a way which brings out its relativistic invariancc. by using a four-vector notation. \Ve shall define the four·vcctor x" to have components (Xl> x2, xJ, X4=ixO - ict). \Ve shall also write this (x, x4) and generally will usc Greek subscripts, )., I', ) , . . . running from I to 4, to denote four-vectors, whereas Latin subscripts, running n·om I to 3, will denote spatial coordinates. i,i, k, I, III, The scalar product of two four-vectors .�I' and)'" is defined by '

•

•

•

(4,) •

By rd:uiviSlie i,wari:U1te we: alwa�.. mean im·ariancc: und.,.. Lorc:"t� Iransform:uiDIU.

203

46.

THE DIRAC EQ.UATION FOR A FREE PARTICLE

We shall use the convention usual in relativity theory of writing this simply as (4b)

i.e. summation over repeated Greek suffixes in a product is implied unless stated otherwise. The constant (Ii/me) in equation (3) , characteristic of the mass of the particle, has the dimensions of a length and is called the Compton wave­ length of the particle. For simplicity we shall denote it by 11M. The Klein-Gordon equation then takes the simple form where

(5)

(02 _ M2) V' (X.) = 0,

(6) Equation (5) is relativistically invariant, provided the wave function V'(x,) is invariant under Lorentz transformations, i.e. provided it is a scalar, borrowing this tcrm from three-dimensional vector analysis. The use of the Klein-Gordon equation as the relativistic theory for electrons meets several difficulties. We shall not discuss these. It suffices for our purposes to see that equation (5) cannot be the correct relativistic equation for electrons. For equation (5) is a wave equation for a one-component wave function·, whereas we have seen that even non-relativistically a particle with spin t requires a two-component wave function, corresponding to the two degrees of freedom of the spin. Indeed one can show that in the non-relativistic limit of small kinetic energies, the Klein-Gordon equation reduces to the Schrodinger equation without spin and not to the non-relativistic Pauli theory, developed in section 26. There we introduced the spin and the magnetic moment in an ad hoc manner. Going back to scction 26, we would like to stress that, by analogy with equation (26.2), one would expect a particle of intrinsic spin 8 to possess an intrinsic magnetic moment (we take e>O so that the electron has the charge e) -

-e

and not

-

B

2mc -e

-8

me

as postulated in equation (26.4) and as required to agree with a vast amount of spectroscopic and other experimental data. It is one of the • One could of course consider '" as consisting of two or more scalar wave functions with­ out violating relativistic invariance. These would then be completely independent of each other. Under three-dimensional rotations, for example, they would not transform correctly to represent a particle with spin (if. Exercise VI.6 and section 42) and thus cannot represent an dectron.

204

46.

THE DIRAC EQ.UATION FOR A FREE PARTICLE

many virtues of the Dirac equation of the electron that it automatically gives the correct spin and magnetic moment. In going from the Schrodinger to the Klein-Gordon equation we obtained symmetry between space and time coordinates by having second-order derivatives throughout. Dirac had the ingenious idea of working with a relativistic equation linear in space and time coordinates. We shall write this equation, which is known as the Dirac equation, (7)

where the y" are so far undetermined. [In equation (7), remember that summation over ,.,. from 1 to 4 is understood.] The Dirac equation looks very different from the Klein-Gordon equation. We do of course still want the classical Hamiltonian, equation ( 1 ), to hold and for many purposes of interpretation one wants an equation resembling the Klein-Gordon equation. The characteristics of the Klein-Gordon equation are that it is of the second order in all derivatives and does not contain any first-order derivatives. We obtain such an equation from the Dirac equation by operating on it with the operator

which gives (8) This equation becomes identical with the Klein-Gordon equation if the ".a satisfy (9) If the yS are ordinary numbers, these relations cannot make sense. Ordinary numbers do not anticommute. However, if one interprets the ys as matrices, it may be possible to satisfY equations (9) since matrices do not commute in general, as we have seen in Chapter V. (The rules for operating with matrices, discussed there, will be used continually in this chapter.) The quantity "" % x" in equation (7) is now both a differential and a matrix operator. It is thus not un­ reasonable to take V' to be a column vector, the number of components being equal to the number of rows in ",,' "" operating on V' is then the operation of matrix multiplication. Using Latin letters at the begin­ ning of the alphabet, a, h, c, 0 • 0 , to label rows and columns of these matrices and vectors (in distinction to i,j, k, 0 • 0 , used to label spatial components offour-vectors) we write the Dirac equation (7) in the form

.f(",,'" a�" + Ma"')'P(6) (

x.) =0,

205

(7a)

46.

THE DUlAC P.Q.UATION FOR A "Rt:E PARTICLE

or in still more detail as

a¥l(ll

Y,.

ax

a.,i1) ax,

"

0. . . o

I...

- 0.

(7b)

(In both these equations, summation over /1 is of course implied.) It is lIsually most convenient to suppress the indices a, b, . . . , labelling clements of matrices and vectors, as in equations (7) 10 (9). One must then remember that one is dealing with matrices which are non­ commuting. \Vhcn in doubt and in the course of detailed calculations, it may be advisable to usc the more explicit notation employcd in cquation (h). Wc sec that the Dirac equation has one of the properties we desired: the wave function has several components and we shall sec that these arc connected with spin. For this reason such wave functions are called spillors* . We have not so far stated lhe number of components of these spinor wave functions or the number of rows ilnd columns or the y-malrices. We shall presently give all explicit representation of the ys, satisfying the allticommutation relations (9), in terms of·... x 4 matrices. l�urther­ more, it can be shown that nothing is gilined by considering larger matrices. The corresponding wave functiong are four-component spinors. For many purposes one docs not need an explicit matrix representa­ tion of the )'S but merely Ihe fact that they arc quantities satisrying the anticommutalion relations (9). One can see that it is possible 10 get a set offour'� x '� malriccs which satisfy the amicomnllllation relations (9). To show this we note that in virtue or equations (9) any product of iln arbitrary number of ys is, apart from sign, equal 10 one of 16 distinCI basic quamilics:

}

[I l [4) ( 10) [G] YlY,. [41 YlY,. Y. YIY2Y3Y" [ 11 where )., II, I' = I, . . . 4 and are subject to the restrictions staled. The numbers in square bracken give lhe numbers of distinct quantities of each lype. Similarly a sum of products of ys can be written as il sum of the above 16 basic quantities with suitable numerical coefficients .

I y,

( ""identity)

• A .pinor i5 rigorously defined by illl transronnalion Ilropcrlie$ under Lorentz lrans· fonnalions. These properties will be: considered in K<:tion

·19.

206

46.

THE DIRAC EQ..UATION FOR A FREE PARTICLE

(These may of course be complex numbers.) Now a 4- x 4- matrix has 16 elements. It can hence be written as the sum, with suitable co­ efficients, of 16 linearly independent 4 x 4 matrices. (For these we could, for example, take matrices with zeros everywhere except for one element which is unity, but one can of course equally well take more complicated linearly independent matrices.) It follows that the 1 6 basic quantities, and i n particular the 1'.. themselves, can b e represented by 4 x 4 matrices. It is clear from the above that this representation is not unique. We can write the 1 6 basic quantities in a different form if we intro­ duce the quantity ( 1 1) which is seen to satisfy the same anticommutation relations as the I'A, ( ). = I , . . . 4) : (12) y; = 1, 1'.. 1'5 + 1'51'.. =0.

}

We can then take for our 1 6 basic quantities : 1 �rI'.. ..

%I'sl'I'p 1'5'

().
..

[I] [4] [6] [ 4] [ 1]

}

( 13)

On account of the anticommutation relations (9) and ( 1 2), the quantities ( 1 3) agree with those in (1 0), except for factors ± i. These have been introduced so that the square of each of the basic quantities equals the identity I. Furthermore, in the explicit matrix representa­ tion which we shall choose, the 1'.. will be Hermitian matrices so that with the above choice offactors all the 1 6 basic matrices are Hermitian. The particular 4 x 4 matrix representation which we shall use can be expressed in terms of 2 x 2 submatrices as follows :

I'*= [i�* -�a*l' I'4 = [� - ?l

( 14)

and from equation ( 1 1 ) one finds

ak

I's = [ _ � - �].

( 1 5)

Here are the 2 x 2 Pauli spin matrices [ equations (25.3 1 ) ] and 1 and 0 denote the 2 x 2 unit and null matrices respectively ·. These I'-matrices are Hermitian and satisfy the anticommutation relations (9) and ( 12). 4

•

x4

No confusion with the numbers 1 and 0 ahould arise if one remembers that the rs matrices.

207

are

16.

THE DIRAC EQUATION VOR A VREE PARTICLE

In the literature, one orten finds, instead orthe }'I" the related matrices ( i6) whose representation In terms or those or the ys is easily found (Exercise X.2). We also introduce the generalization or the Pauli spin matrices, which will play the analogous part in the relativist.ic theory, by qJ; =

[a. 0 ] o

a,

= iY"YsrA,

(k - i , 2, 3)

(i7)

where we have lIsed a 'hat' ( A ) to distinguish the 4 x 4 matrices. Alternatively we can write these as where k, l, TTl arc in cyclic order. Having treated these m:ltI'ices in detail, we consider the rorm the particle density and current take in the relativistic theory. In analogy to the non·rebtivistic theory (if sections 7 and 26), Dirac defined the particle density at the space-time point x. by ( i9) where ",* is the adjoint to rp, i.e. it is the compic....: conjugate transpose or 'P' ,.vc shall modify aliI' notation somewhat. We shall denote the com· plex conjugate or Ii no longer by a bar (II) but by an asterisk, thus A*. We shall also lise, instead or the adjoint spinor tp*, the related spinor iji defined by (20) The reason for this is that the quantities whieh have simple trans­ formation properties under torentz transformations arc naturally written in terms of iji rather than V'*' Doth VI* and iji arc or course row vectors and always occur as the Icft·hand factor in a matrix element, as required by the laws or matrix algebra. One can easily show that ir V,(.I:.} satisfies the Dirae equation (46.7), then iji(x.) satisfies (2 i ) (Exercise X.4). In terms ofiji, the density, equation (19), is (22) 20B

47.

THE SPIN AND MAGNETIC MOMENT OF THE ELECTRON

Correspondingly, we define a current-density vector

jA:(x.) = icip(xp) Yk'P(Xp). Let consider the four-vectort sp(xp) , defined by SA: = c!j4' S4 =ie i.e. by

jA:(xp) by

us

(23)

(24a) (24b)

It satisfies the continuity equation

�!e

AX =0' I'

jk

(25)

as the (Exercise X.5). This is the justification for interpreting current-density. The same problem in the non-relativistic theory was treated in Chapter II (if particularly Exercise II.4) . 47.

The spin and m.agnetic m.om.ent of the electron We shall now show that a particle which satisfies the Dirac equation has spin t. For this purpose we shall show that it is not the orbital angular momentum 1 which commutes with the Hamiltonian but the operator· (1 ) where a is the vector with components Cl4' From equation (46.7) we can at once obtain the Hamiltonian of the Dirac theory. For it can be written

(2 ) with

=icy4yk/lk + me2y4 [using equation (46.2) and M =mc/li] . H

(3)

Consider the commutator [/3' H) . The second term in equation (3) commutes with 13• From the first term we obtain, using the results of Exercise IV. 1 3, [/3 , H] (4) and generally one obtains (5) [I, H) where y is the vector Thus we see that the orbital angular momentum is not a constant of the motion.

Y4'

= - 1icY4( YtP2 - Y2Pl) , = - fteY4(Y AP)

t We shall see in section 49 that I,. is a four-vector. Throughout this section j will denote the operator defined by equation (I). This should not lead to any confusion with the thrce-current vector which, in the last section, was also denoted by j ; equations (46.23) and (46.24). •

209

47. TliE SPIN AND MAGNETIC MOMENT OF TilE ELECTRON Consider next [03' H] which according to equations (46.18) is the same as [ - iY 1 Yl' HJ. Since Y.. commutes with YIYl, again only the first term in equation (3) need be considered. Now from the anli­ comml1tation relations (46.9) one casily proves that

}

so that

[ YIYl, YoIY 1 ] = -2YoI Y2 [ YIY1, Y4Yl] "" + 2Y4YI [ Y1Y2, Y4YJJ -O

(6)

(7) and generally

[a, H] - 2cY, (YAP) .

(8)

From equations { l L (5) anel (8) it follows that [j, H] - [1 +l/'a, /1] -0.

(9)

This justific.'> our interpretation of j as the total angular momentum of the particle with the resulting conclusions about its spin. In section 48, dealing with solutions of the Dirac equation of a frce particle we shall come back to the qucstion of spin. The proof which we have here given also applies in the more general case of a Dirac particle in a spherically symmetric external field (for e:-:ample in the hydrogen alom). Such a potential leads to an additional term in the Hamiltonian which commutes with all rOlations and hence with the IJ (cfSeclions ·1-\ and 42). Since it depends only on the spatial variables and not on the spin, it also commutes with a, and hence with the total angular momentum j, as required from general considcrations. In section 49 it will be shown that lhe operator a is a pseuclo-vector as required for an angular momentum [if equations (41.8) and (H .9»). We next investigate the magnetic moment, associated wilh this spin of the electron. For that purpose we consider the effect of a magnetic field H, constant in time. Such a field is described by a time-inde­ pendent vector potential A (i.e. by a four-vector A" with 114 -0), and is given by (cf Abraham-Becker6) H -curl A.

(10)

It can be shown quile generally that the effect of an electromagnetic field specified by a four-vector 1I,,(.'t.) on a particle of charge ( - e) is correctly described by replacing the four-momentum /',. in the equations of Illolion by the four-vector· ( 1 1) • 11,;, result iJ oblained rrom .he rel:lI;Vislie equation of motion or a chnrgcd particle in nn electromagnetic fidd, :l$ gin:n by Lorentz. It \J shown thaI Ihis equ:lIion can be derived from a L.."lgr.mgi:m by means of the usual J.�'grnngi:Ln or Hamiltonian ronm,lisms provided that the momentum p,. is replaced by the four,vCClor (PI' +lAp/c) throughout. For demils, (/ MOil and Sneddonlo, §:'2.

21 0

47.

THE SPIN AND MAGNETIC MOMENT OF THE ELECTRON

Thus, the Dirac equation of an electron in the field Ap becomes, on account of (46.2), yp

(� � ) &p

+ Ap tp + Mtp =0.

(12)

To interpret this, we again convert this to a second-order equation by operating on equation ( 1 2) with

["AYP( O�" + �AA) ( O�p + �Ap) - M2]

giving

( 1 3)

The terms with ), = I' give (since y,,2 = 1 )

( 14a) The terms with ),=1=1' give [on account of equations (46.9)] " !Y Yp

{( �

OA

) ( � � ) ( � � ) ( � k;A )} (�) (��: ��;)

+�AA

&p

+ Ap

- op

+

p A

o J.

A tp

+

-

tp. ( 1 4b)

) ]

( 15)

= t YAYP

Combining these results, equation ( 13) becomes

[2:" (.!..OXA + lie!!AA) 2 + 2lieie y"yp ( oAax"p

_

& A A &xp

_

M2 'I' = 0.

From the definition of curl (Appendix A), equations ( 1 0) and (46. 1 8), one obtains a ie 2 e L.. + A" - -aH - M2 '1' =0, ( 16) lie " OXA lie

[""(

-

)

A

]

remembering that we are considering a static magnetic field. We write this equation as 2 (� - M) ( � + M) tp = [ - 2: (� + �AA:) + � aH] tp A: &XA: lie OX4 &"4 Ii.:

( 1 7)

and take the non-relativistic limit, by considering a particle whose velocity is small compared to the velocity of light. Its energy will then be very nearly equal to its rest energy me2• Since for an eigenstate of energy E,

211

48. PLANE WAVE SOLUTIONS OF THE DIRAC EQUATION

it follows that in thefirst factor on the left-hand side of equation ( 1 7) ." a '.e. we can put '''at =

me2

•

a 8X4

-

=

-

M

(18)

•

Equation ( 1 7) then gives the 'Schrodinger equation'

iii 81p al

=

[me2 - 2mli2 i (�oXI; Tic!!AI;) 2 +

1;_1

+

�

aH]

2me

'P'

( 1 9)

Since the potential energy of a magnet of magnetic moment M in a magnetic field H is ( - MH), the last term in equation ( 1 9) shows that associated with the electron, as described by the Dirac equation, is. a magnetic moment � (!Ii&), me

(20)

as required by experiment. The correct prediction of this magnetic moment was one of the first and most remarkable successes of the Dirac theory of the electron. Since then it has had overwhelming detailed confirmation in many branches of physics ; for example in quantum electrodynamics where it leads to an excellent description of the interaction of electrons with the electromagnetic field. 48. Plane wave solutions of the Dirac equation Let us obtain solutions of the Dirac equation of a free particle, equa­ tion (46.7). We expect such solutions to exist, representing plane waves corresponding to states of definite momentum p. From the relativistic Hamiltonian (46. 1 ) it follows that for a particle of momentum p the energy E can assume two values (1)

where (2) The existence of these negative energy states leads to difficulties in a naive interpretation of the theory. We shall return to this point later and, to begin with, find the explicit solutions. For a given value of p and E,l the Dirac equation admits four linearly independent solutions, which we can take to be

'1', =

{

U,(P,)d,.../II

u,(P.)e-iltl'.11I

(r = 1, 2) (r = 3, 4)

}

(3)

where P. = (P, iE,lc) . The first two of these solutions (r = I, 2) corre­ spond to electron states with momentum p and energy E" the last two 212

48.

PLANE WAVE SOLUTIONS OF THE DIRAC EQ.UATION

(, = 3, 4) to momentum - p and energy - E,. That there exist exacdy four solutions follows from the fact that the Dirac equation really represents a system of four simultaneous linear equations for the four components of the wave function. In equation (3) , the u, arc constant four-component spinor wave-functions ( i.e. for each fixed " the u, has four components u,cG), a = 1 , . . . 4) . To find these constant spinors, we substitute equations (3) into the Dirac equation and solve the resulting set of linear equations for the u,. For this purpose we must of course use a particular matrix representation for the ,,-matrices, for example that defined in section 46 which we shall use throughout. One then finds : (4a)

(4b)

(4c)

(4d) The multiplicative factor [(E, + me2) /2me2] i is of course not determined by the Dirac equation which is homogeneous in the unknowns. We have chosen it to give a convenient normalization. We shall not here deal with the orthogonality and normalization properties of the plane wave solutions (3) and (4). The most important properties, demon­ strating the expediency of our choice, are left as exercises to the reader (Exercises X.6-7). The Dirac equation and its solutions become particularly simple in the rest frame of the electron : p = 0. Substituting equations (3) into the Dirac equation for this case, one obtains for the positive energy solutions ( "4 - 1 )u, = 0, (, = 1, 2), and for the negative energy solutions (1'4 + l) u, = O, 213

(, = 3, 4) .

48.

PLANE WAVE SOLUTIONS OP THE DIRAC BQUATION

1n our particular matrix representation, Y4 is given by eq\l4\tion (46.1'�). It follows at once that the eigenvectors fI, are given by

This agrces with equ ations (4a-d). We next considcr the spin of thc electron in its rest system. We sec from the definition of the 4 x 4 spin matrix a, equation (46. 17), tlmt

.

.

}

�3111 -1I 1 �311J =11) UJU4 = - II� UJIIZ - -112

(6)

i.e. in the rest system wc can interprct 11\ an d ".\ as sl:l.les of the electron with spin pointing in the positive z-direction (one of posilivc, thc other of negative cnergy) ; similarly 112 and 114 corrcspond to states with thc spin pointing in the neg:l.tivc z-direetion. This interpl"c l'atioll can still be approximately maintained in the non-relativistic case of small velocities. In general it breaks down: the II, arc not eigenfunctions of a�. This is not surprising, since we have seen (section 'H) that the spin is not a constant of thc motion of the Dirac J-bmiltonian. How­ ever, the component of the spin vector a in the direction of the momentum p is a const:l.nt of thc motion (Exercise X.B). We consider thc non-relativistic case of particles whose velocity v is small compared to the velocity of light: v«c. Then E� ':::!. mc2 and Pk ':::!. IIWh 50 tha t terms of the form cp.I(E, + /1/C2) are of order vIc and hence small in the non-rebtivistic limit. Hence in the positive energy solutions, fll :l.nd U2, the third and fourth components arc small, and in thc negative energy solutions, IIJ and fl4, it is the first and second com­ ponents that are small in the non-relativistic limit. These con­ clusions are orcourse consistent with the interpretation or equations (5) as representing the eigenvectors in the rest system of the electron_ In practice one often requires the non-relativistic limit of such matrix elements as (u" Y.fl,). To deal wi th this question, we rcmember that the positive energy solutions III and fl2 (we shall brieRy write either of them 11+) decompose into twO large components ( I and 2) and two small components (3 and 4), so that we can write (7a) Here 11; i tselr st:l.nds for the two-componcnt vector or the large com­ ponents, tic. Similarly we write for either of the negative cnergy solutions III or 114,

"- - [j].

(7b)

48.

PLANE WAVE SOLUTIONS OF THE DIRAC EQ.UATION

Remembering that the zrr are of order (vlc)uL, and using the explicit matrix representations of the I'-matrices (whose elements are 0, ± I , ±i) one can easily give the order of magnitude of matrix elements. As an example, we calculate (u+ , I'4U+) = [u�*, u!*]

[1 ] [Uf] 0

o

- I u!

= u;*u; - u!*u! � u;*u;

(8) = (uf, uf ), neglecting a term of order (vlc) 2. We have thus reduced this matrix element to its non-relativistic form in terms of two-component wave­ functions. An essential difference in such matrix elements arises between diagonal and off-diagonal operators, i.e. matrices of the types

[: �]

and

[ ] �:

respectively. (Examples of diagonal matrices are 1'4 and (J.l i of off­ diagonal matrices : I'.l and 1' 5') For they mix up large and small com­ ponents differendy. As an example of an off-diagonal matrix, consider (u+, I'su+) :

= - u;*u! u! *uf. (9) Here large and small components are mixed up and so this matriz element is of order vIc compared to (u+, u+) . But of course 1'.5 gives large matrix elements between positive and negative energy states ; e.g. -

(u+, I'su_) = [u;*, u!*] =

-

[ _ � �] [:!] -

u;*u� - u!*u!

� - u f *u�, ( 10) neglecting terms of order (v/c) 2 . These examples should suffice to enable the reader to deal wi th any case. I t should again be stressed that the explicit solutions and detailed deduction from them, made in this section, depend on the particular matrix representation which we have chosen for the 1'#'

8 + Q..M.

215

49.

RELATIVISTIC INVARlANCE

We come back to the physical interpretation of the negative energy states. One is here faced with the difficulty that quantum jumps from the positive to the negative energy states are possible ; so we cannot ignore the negative energy solutions, as is done in classical theory. To begin with, we must admit that these difficulties cannot be properly resolved within the framework of a one-particle theory such as developed here. One requires a many-particle theory in which electrons can be created and absorbed, i.e. a field theory, analogous to Maxwell's theory of the electromagnetic field which is a field theory for photons. As we shall not treat field theory in this book we can only give a very qualitative discussion of how the difficulties associated with the negative energy states are resolved. For this purpose, Dirac postulated that the negative energy states are ordinarily all occupied. (Remember, we are dealing with electrons so that, according to the Pauli principle, there can at most be one particle in each state.) This state with all negative energy levels filled is then to be thought of as the vacuum and the negative energy sea of electrons are not to produce any physical effects. An unoccupied negative energy state, i.e. a hole in the sea of negative energy electrons, with energy E, and momentum p can then be interpreted as a positron of energy E, and momentum p. The descrip­ tion of electrons and positrons which we have here given appears very unsymmetrical in positively and negatively charged particles. In fact one can show that the field theory is completely symmetrical in electrons and positrons, the apparent asymmetry being due to the particular way we introduced positrons starting from the single particle Dirac equation for electrons. The theory developed along these lines is remarkably successful. The existence of the positron, predicted by Dirac from the above theory, has been confirmed and the theory affords a quantita­ tively correct description of such phenomena as Bremsstrahlung and Compton scattering, as well as pair creation and pair annihilation. (Pair creation is the process when an electron from the negative energy sea is excited into a positive energy state leaving an empty hole, a positron, the excitation energy being provided by a high-energy photon, i.e. by a gamma-ray. Pair annihilation is the inverse process.) -

49.

-

Relativistic invariance

We now wish to consider certain aspects of the relativistic invariance of the Dirac theory which are especially important in practice. We begin by stating briefly the basic properties of Lorentz trans­ formations. Consider a linear homogeneous transformation from a coordinate system x. to a coordinate system xv', defined by

X,: = a"r,' 216

(1)

49.

RELATIVISTIC INVARlANCE

This transformation is a homogeneous Lorentz transformation if it leaves the lengths of vectors invariant, i.e. (2)

This is the condition that the velocity of light should have the same value c in all Lorentz frames. Substitution of equation ( 1 ) into (2) gives a condition on the transformation matrix a".. : a".ap4 = 8..,. (3) Using this equation one finds at once the inverse of equation ( 1 ) : ( l a) x. = a,,�,,', and substituting equation ( l a) into (2), one finds, analogously to (3) : a).pa".. = 8Jpo

(3a)

Equations (3a) are however not additional new conditions on the matrix a". but follow from equations (3). The determinant of a Lorentz transformation has the value ± 1 . For, from equation (3), (l ap.al ) 2 = l a"a l l a.,,1 = l a�" l l a"fJl = la�.arjl l = 1 8aj11 = 1, where the index T denotes the transpose. Hence

(4) l a",\ = ± 1 . We now restrict the transformations to the proper Lorentz trans­ formations by confining ourselves to transformations with determinant +1: \aJp\ = 1, (5)

and in addition we require that

a44>0. (6) This means that we do not consider transformations which reverse the sense of time, interchanging future and past light cones. Of the improper Lorentz transformations (which are defined by having a determinant equal to - 1 ) we shall consider spatial inversion, defined by the diagonal matrix [a".] = -1 (7) -1 -1

[

agreeing with our earlier definition (p. 39 and sections 39 and 4 1 ) : (8) 217

49.

RELATIVISTIC INVARIANCE

Thus inversion is the same as going from a right-handed to a left-handed spatial coordinate system. The proper Lorentz transformations include as a particular case the identity all" = allP (9)

which simply leaves the coordinate system unchanged. One easily shows that the proper Lorentz transformations form a group, in the sense in which we used this term in Chapter IX (Exercise X.9) . Hence one need for many purposes only consider the infinitesimal Lorentz transformations. Lastly, we must mention the reality character of the matrix elements al'''' From our original definition (Section 46) the XI are real and X4 is pure imaginary; we require this of course in all coordinate systems. One easily sees that this is ensured if we have

}

ak/, a44 real au, a4! pure imaginary •

( 10)

This property is also preserved by two successive Lorentz transforma­ tions. We now wish to study the properties of the Dirac equation which follow from the fact that we are trying to formulate a relativistically invariant theory. We would like to remind the reader of what we mean by relativistic invariance, using Maxwell's wave equations for the four-potential A,. (x,,) as an example. An observer 0, using a co­ ordinate system (x.), will write these 81 A ,,(x.) = es,. (x,,) , --;r8X..IIX..

where s,. is the classical charge-current density. Consider a second observer 0' using a coordinate system (x:) related to the system (xp) by a Lorentz transformation. The principle of special relativity then requires that there should be nothing to choose between the two co­ ordinate systems : the observer 0 should in no way have a preferred position with respect to 0' or vice versa. This means that 0' must write Maxwell's equations as 82 A ' ' ' ' ' I' (x. ) = es" (x. ) , 8-'Xl 8Xli.e. the equations must be invariant in form under a Lorentz transfor­ mation, just as in classical mechanics Newton's equations of motion are invariant in form under a Galilean transformation. This invariance in form follows at once. From equations ( I ) and (3) 8 2/8x.. 8xA is invariant under Lorentz transformations and A,. and s,. are four-vectors and thus transform in the same way under Lorentz transformations.

218

49.

RELATIVISTIC INVARIANCE

After these preliminaries we discuss the relativistic i nvariance of the Dirac equation

(yp a�p + M) 'P(X.) =0.

(11)

We shall take the y-matrices as constant, i.e the same in any Lorentz equation ( 1 ) . frame. Consider a particular Lorentz transformation This will induce a linear transformation i n the four-dimensional spinor space Uust as in section 42 we could think of three-dimensional rotations inducing a transformation in spin-space) so that the same state of the The Dirac equation system will be represented by a new spinor is invariant under Lorentz transformations if equation ( 1 1 ) transforms into

ap.,

'P' (x:).

( 1 2) This is the case provided that, for a given Lorentz transformation, can be written in the form

'P' (x/)

'P' (X: ) = S'P (x.) S

( 1 3)

where is a constant 4 x 4 matrix, determined by the Lorentz trans­ formation 011.' which possesses an inverse Using equations ( 1 ) and ( 1 3), the Dirac equation ( 1 1 ) becomes

$- 1.

a pS- I'P' (X.' ) + MS-I 'P' (x: ) =0, yl,;:;-;O}. i.e. premultiplying by S: oApSYpS-1 / ,'P' (x.') + M'P' (x.' ) =0. "XA rJX}.

( 1 4)

This equation is identical with equation ( 1 2) if or, rewriting this, if

oApSYpS-l = YA

( 1 5a) ( 1 5b)

sp'

In section 46 above we defined the charge-current density We require Sll to be a four-vector, so that the continuity equation (46.25) permits the usual invariant interpretation. This means that it must transform according to i.e.

sp(xA) s:(x,,') =a,psp(xA), iji' y,tp' = o.pijiyp'P' -+

On account of equations ( 1 3) and I 5b) we can write this equation as

a.piji'Syptp a.uijiyp'P· =

219

49.

RELATIVISTIC INVARIANCE

Hence .p(X.l) must transform according to .p' (x.') = .p (X.t) S-l. t But from equation ( 13) itself one finds directly that

( 16 )

.p ' (x.t') = tp'* (X.t ') "4 =tp* (X.t)S*"4 =.p(X.th '4S*"4.

Comparison with equation (16) shows that, for consistency, we must require S to satisfy the condition ( 1 7) To complete the proof of the Lorentz invariance of the Dirac equation, it must be shown that for every Lorentz transformation a", a matrix S with the required properties (15) and ( 1 7) exists. Since the Lorentz transformations form a group, it suffices to construct this matrix for the infinitesimal Lorentz transformations from which any finite Lorentz transformation can then be generated. We shall omit the proof of this existence theorem (if, for example, reference 1 7 ; or in English : S. S. Schweber, H. A. Bethe and F. de Hoffmann, Mesons and Fields, volume 1). For our purposes it suffices to know that a matrix S with these properties, equations ( 1 5) and ( 1 7), exists for every Lorentz transformation. From the tp and .p and the ,,-matrices one can construct five quantities which have definite transformation properties under Lorentz trans­ formations ; they are scalars, vectors, etc. These five couariants-as quantities with definite transformation properties are called-play a very important part in relativistic quantum mechanics, for e."{ample in the theory of p-decay (Bethe and Morrison 1 9, Blatt and Weisskop(20 ). These five covariants are closely related to the five combinations of ,,-matrices given in equations (46.10) or (46. 13). We define them as : (i) scalar or invariant (1 component) : I = .ptp.

( l Sa)

(ii) four-vector (4 components) :

( lSb) s" = iip""tp. We have denoted this by sp as it is identical with the charge-current density vector considered in section 46, equations (46.22-25). (iii) antisymm.etric tensor of the second rank or six-vector (6 com­ ponents) : Jll". = i.p",.".tp, p=!=v, (lSc) = 0 , p =v,

}

and one sees at once that Mil' = - M,p. 220

49.

(iv)

RELATIVISTIC INVARIANCE

pseudo-vector or axial vector (4 components) : Kp = iipys Yp'l"

( l Bd)

(v) pseudo-scalar ( 1 component) :

N = iipystp.

( 1 8e)

The factors i have been included in these definitions so that the different components should have the reality character required of them. We see, for example, from equation ( I Sb) that sp has real spatial com­ ponents, whereas the time-like component S4 is purely imaginary [compare equation (46.24a)] . Similarly in equation ( I Sc) the Mil are real, the MH = - J\14i purely imaginary, as required for a six-vector. It is now easy to demonstrate the transformation properties of the above quantities, using equations ( 1 3), ( 1 6) and ( 1 5) . We have already seen that these equations imply that sJ" equation ( I Sb), trans­ forms like a four-vector : Sp' = op.,$.. ( 19b) We consider two of the other cases as typical examples : (i) scalar : from equation ( I Ba) I' = ip'tp' = ipS-IStp = V''I'

��

(ii) pseudo-scalar : from equation ( 1 8e) ,

I.

( 1 9a)

N' = iip'ystp' = iipS-l ysStp = irS- I YlY2 Y3 t4Stp = iipS-l y1SS- l Y2 '

• .

Stp

= aUa2pa3.04aiip YA'Y/,Y.Ya'l"

(20)

In this four-fold sum over )., Il, v, C1 only terms with all four indices different from each other contribute. [This follows, since for any fixed v and C1 say, the sum over the four terms i. = Il vanishes on account of the anticommutation relations (46.9) and the orthogonality relation (3a) .] We define EApra to be zero if the indices )., Il, v, C1 are not all different, to be equal to + 1 if they are in the order 1 234 or an even permutation of this order and to be equal to - 1 if they are an odd permutation of this order. Equation (20) can then be written in the form (2 1 ) N' = a U.a2p03.04aE}.p",,iipt 1 Y2 1'3 Y4tp. The E-symbol in this equation originates from the rearrangement of YAYp Y.Ya in equation (20) in the order 1'1 1'2 )'3 ,'4 using the anticommuta­ tion relations (46.9) . Now one can easily see that the determinant lap. 1 can be expanded in the form (22) 221

EXERCISES X SO

that equation (2 1 ) finally gives

N' - Jal'.JN.

( 1 9c)

Hence it follows that nnder proper Lorentz transformations (Ja, 1 + I ) JV is invariant, whereas under improper Lorentz transformations (Ja, 1 - - I), in particular under inversion, it changes sign. These arc exactly the transfonnation properties which define a pseudo-scalar. The same method is easily applied to thc remaining two covariants (18c) and ( l 8d). One finds ••

""

••

( 1 9c)

and

( 19d) \\'hich again arc the characteristic transformation properties of tensors of the second rank and of pseudo-vectors respectively. Jn particular, under inversion, equation (7), K, _ K,' = K" whereas a vector changes sign, l.g. r _ r' .,. - r. \Ve note in particular, as stated earlier (section H), that the spin ft, transforms like a pseudo-vector. On account ofeQuatiol1 (46. 17) we can write IjI*U,V' as (23)

which is the spatial part of the pseudo-vector /(1" equation ( H ld). The importance of these eovariauts is due to the fact that from them one builds up relativistically invariant expressions (analogous to the scalar product x,J',.) which represent the interactions between particles or fields. The requirement of invariance pl:lces a very severe restric­ tion on the possible forms of imeraction. The intercsted reader is referred to the theory of p-decay which affords a particularly good example of these matteI'S. EXERCISES X

Show that the y-matrices dcfined in equations (46. 14-15) satisfy nmutation rclatiolU (46.9) and (-16.12). Derive the matrix representations of (1.1' equation (46. 16), from Derive equations ("6.1S) from (46.17). Prove equation (46.21). X.5. Prove the continuity cquation (·16.25). X.6. For the spinors II" defined in equations (4S.1), prove that: (i) (1111

1/2 ) =

(1/.11

but all other inner products (u" 11,):;i:0. (ii) U,II,

=

II�

0, if r:;i:s. 222

) =0

X.7.

( ha t

E..XERCISES x

For the complete plane wave

(i) (V'

••

(ii)

the

V',) = O if r#.s;

V'. are nonnalized to

E,/mcl

(iii) this normalization s i invariant

solutions V'"

equations (-18.3). show

particles pcr unit volume;

under Lorcntz

transformations.

� Pro\'e that the component of the spin vector a in the direction of t�lentum p, i.t. (erp), is a constant of the;:: motion for thc Dirac equation.

X.9.

8-

Prove the group property of the proper Lorentz transformations.

223

HINTS FOR SOLVING THE EXERCISES CHAPTER I

1.1. THE fundamental domain being - L <x
(g(x),j(x» =

f�l(x)J(x) dx.

To prove that the set is normed and orthogonal one need only show that (11'"" 11',.) = 8""" (1J'"" 1J'n) = 8""" (1J'.., tpn) = ° (H 1) for all m and n. 1.2. To obtain the an and h", we successively form the inner product of equation (1.2) with 1J'", and 11'.. using equations (H I ) . For example (tp",(x),f(x» = h."

i.e

JLf- f(x) sin 7xdx L

h.,, = and similarly

f

(m = l , 2,

L

1

00 = 1/2L

r 1/L 1

L

.

.

•

)

,

L

f(X) dx _L mit

f(x) cos xdx (m = 1 , 2, . . ) . T " -L ) and the Xn (n = 0, ± I , ± 2, To show that the tpn' 1J'. (n = 0, I , 2, . . . ) span the same space, if L = it, we have that a." =

.

.

.

.

(i) '1'0 = Xo· (ii) For any positive integer n, the manifold spanned by tpn' 1J',. is the same as that spanned by XII' X-II) in fact cos nx = ·Hem- + e-iu) i.e.

sin nx = ·!;(elu - e-iAr) 1 'P. = 1/2 ( Xn + X-n) 1

tpn = i1/2 ( X' - X-n) ' Hence any vector of the form 'ntpn + dn1J'.. can be written in the form C,.X. + DnX-n' 224

HINTS FOR SOLVING THE EXERCISES

1.3. If f{x) is odd, i.e. f( - x) = -f(x) , then since sin ( - Ax) = -sin (Ax) and cos ( - lx) =cos (lx) , f( - x) cos [ - m;x] = -f(x) cos [7x] f( -x) sin [ -7x] = f(x) sin [7x] and therefore = m2=fO'L . mn dx (m = 1 , 2, 3, ) , m - yL f(x) sm L X , o in agreement with the analysis in section 2. If f(x} is even, r( -x) =f(x) , the general Fourier series similarly reduces to a cosine series. We see that we can think of the normed orthogonal sets ['Po, 'Ph 'P2, ] and [lPh lP2' ] defining mutually orthogonal subspaces of the total space spanned by the 'P" and the fP" together. 1.4. From Exercise 1.2 we have a bo

}

a

.

.

•

I.e.

• • •

as

I nx cos nxdx=� r:1 (cos nn - I) = .r:I [( - 1 )n - l ], :fn2 nn 2

f - cos njl = -- ( - l)". =-I fnx sin nxdx= -n n

an = � / vn

b "

• • •

o

v

v

yn

Yjl

o Hence from equation (1.2) I e - 1 )" - I ] cos nx - -I ( - 1 )" sin nx}. f(x) n4 + 2:"' {nn-ir n 1.5. We have generally for m and n zero or positive integers, that (x", x") = f�+nd.\" = + 1n + 1 [X"'+n+lr-Il = {m +� + I ' if (m + n) = cven 0, if (m + n) =odd. Hence (fPo, lPo) = 2, i.e. 'Po = �2lPo= �2' We next let XI = lPl - 'Po('Po, lPl) = lPl =x, hence (Xh Xl) =�, i.e. 'PI = J�lPI = J�x. We next let yn

=-

_I

II - I

m

225

HINTS FOR SOLVING THE EXERCISES

X2 = '1'2 - 'PI ('Ph '1'2) - (tpO, '1'2) = '1'2 - v'2 Ttpo = x2 - 31 1 8 ( X2, X2) = (x2, x2) -32 (x2, 1 ) +9(1, 1) = 'Po

hence and therefore

45

'1'2 = J�(�X2 - �). By considering further functions '1'3 = X3, '1'4 =X4, • • • we can generate · a set of normed orthogonal polynomials, tpo, tph 'P2, • • • which form a complete set of functions for the interval - 1 <x < 1 . These poly­ nomials are called Legendre polynomials. They are of great im­ portance in quantum mechanics. 1.6. From (1.6c)

f�.,lf(x)12dx = f�LI�gXI 2dx =2L,

hence the Fourier integral theorem is applicable.

(1.6b) then gives

sin (ko - k) L (H 2) e(k) = 2n.!.fL L n(ko k) • Equation (1.6a) can be interpreted as follows. eikx is a plane wave of wave-number k/2n, i.e. wavelength =2n/k. f(x) represents a superposition of plane waves with amplitudes e (k). Given the resulting amplitude f(x), equation (1.6b) gives the amplitude of any one monochromatic component ew.. Equation (1.6c) represents a sinusoidal wave-train of length 2L and wavelength 2n/ko• In practice we have of course always only wave­ trains of finite length, never infinitely long wave-trains. Equation (H 2), or better the intensity distribution 2 (ko -k) L (H 3) le(k) 12 =sinn2(k o -k)2 , '(� -1)...1.. e' 0 �

ffk"

_

.it

shows that such finite wave-trains are never strictly monochromatic. The intensity distribution (H 3) is indicated in Figure 15. The distribution has a maximum at

k =ko, le(ko) 1 2 = L2x2 • The first minima occur at k = ko ± 1. ' 226

HINTS FOR SOLVINO THE EXERCISES

On account of the factor (ko - k) 2 in the denominator of (H 3), con­ tributions to the amplitudef(x) by wave-numbers k with Ik - ko l >1f/L are very small. Practically all the contributions come from the band of wave-numbers ko - (1f/L) < k< ko + (1f/L), i.e. we have an intensity distribution of approximate spread 21f/L *. Increasing L, the width

Figure 15. Spectral intensity distribution tif sinusoidal U'(lVl­ train tif finite length 2L and wavelength 2n/ko

,

k

of this waveband decreases ; for L»),o = 21f/ko, i.e. a wave-train many wavelengths long, we obtain a very sharp, narrow distribution. The above analysis and its interpretation is well illustrated in optics. Spectral lines due to finite wave-trains are not infinitely sharp, but have a finite, if small, width. • In fact, writing x= (k - ko)L, one finds by numerical integration that

f!O+tlIL ko-tllL f'"

n I �J =

Ic(k)l2d� L

whereas

- co

= -

n

/c(k) l2Jk

=

n

-n

-

227

sin2x -

n x1

dx = 0·90

� sin2xd x= .

nxl

n

L -

n

•

HINTS FOR SOLVING THE EXERCISES

1.7. Fromf= ('1', '1')'1' - ('1', '1')'1', it follows that (J,f) > 0 becomes (J,f) = (('1', '1') '1' ('1', '1')'1', ('1', '1')'1' - ('1', '1')'1') = ('1', '1') ('1', '1') ('1', '1') - ('1', '1') ('1', '1') ('1', '1') - ('1', '1') ('1', '1') ('1', '1') + ('1', '1') ('1', '1') ('1', '1') = ('1', rp) { (rp, cp) (tp, '1') - (cp, tp) (cp, tpH = (cp, cp) {IIcpll 2/1tpll2 - I (cp, '1') 1 2} >0 -

and since (cp, cp) >0, Schwarz's inequality follows : I (cp, '1')
I

I.e.

where a is a complex number. From equation ( 1 .4*) one has at once, that for an ordinary vector space, I (u, v) I = lIull llvll l cos
01

1 (}lu (}2U C2 al2 = ax2

+ (}ib'2U2

•

We obtain equation ( 1.8a) and the boundary conditions (L8b) by writing u(x, y, t) = w (x, y) T(t) and separating the variables. In ( I.8a) we again separate the vari­ ables, w (x, y) = X(x) T(y) , and obtain two eigenvalue problems for X(x) and T{y) identical with (2.5). We then get for the eigensolutions and corresponding eigen­ values ( ) . 17m . nn Wm• x, y = S1O X SID rY L n

).",. n -=

where

m and

n

(iY(m2 + n2),

are restricted to the values m, n = 1 , 2, 3, 228

.

•

.

HINTS FOR SOLVING THE EXERCISES

The eigenvalues A..., ,, are two-fold degenerate for to the two different solutions Wn, 11I and WI1I, D' combination aw"" D bw", 1II

m =l=n corresponding

Hence any linear

+

is also an eigenfunction of the problem belonging to the same eigen­ values ). , '" i."., n'

n

=

CHAPTER

II

11. 1 . The classical angular momentum vector L is

Replacing the classical quantities by the corresponding operators [equations (6. 1 ) and (6,2)] Lop

r,A ( -iii grad,) or for the components 8 8 ) , etc, (L,,)op = - t1i"'J. ( Y'a - Z' =

2.

I- I

•

n

,�I

Z,

a ry,

11.2. The classical conservation of energy equation is

1 ( 1 1 e2 1 2m(P1 2 + P22) + 2MP2 - 2e2 r; + ;:;) + '12 = E, where rh r2 are the position vectors of the two electrons relative to the nucleus, '1 = Irl l, etc, '1 2 I rl -r2 1; m, M are the masses of electron and nucleus respectively, P h P are the momenta of the electrons =

P2,

and nucleus respectively. Using equations (6. 1 )-(6.3) we obtain the time-dependent Schro­ dinger equation for R, I), R being the position coordinate of the nucleus,

u(rh r2, [ - :�(VI 2 + V22) - :�V2 - 2e2(� + �) + :IJ u(rh r2, R, t) =ili;'u(rh r2, R, t), 2 where V1 = 82/8x1 2 + 8 1/ 8y1 2 + 02/8z1 2 etc, and V 2 = 81/oX2 + 82/or2 + 8 2/0.<,2. If the nucleus were infinitely heavy (M/m co) the kinetic energy term ( _1i2V 2/2kf) would not occur in the SchrOdinger equation. Since M/m-::: 7400, one can to very good approximation neglect this -

term.

229

HINTS FOR SOLVING THE EXERCISES

II.3. With E = E(p) , Eo = E(Po) , we put px - Et = (PoX - Eot) + {(P - Po) x - (E - Eo) t} and can then write equation (II.3)

where

u(x, t) =

A exp,,i (PoX - Eot),

as

j'O+-'Po/1 /10--'10/1

[i {(P -Po) x - (E - Eo) t}] dp. By analogy with a plane wave we call A the amplitude of the wave packet. Surfaces of constant amplitude are defined by A=

a (p) exp

(p - Po) x - (E - Eo) t = const., and they are propagated with the velocity dx E - Eo c, = dt = P o -P For a sufficiently sharp wave-packet, (E - Eo) /(P - Po) approximates the differential coefficient at P =Po arbitrarily closely, •

c, =

(��'-/Io'

Using equations (5. 1 ) and (5.3), equation (6.8) follows. IIA. Equation (II.4) follows at once from equation (6. 13), by applying Gauss's theorem to the surface integral, and using the fact that (6. 13) holds for any domain of integration V. II.5. For a wave-train of length 2L, equation (I.6c), the uncer­ tainty in position of the particle is Ll x�2L. (H 4) From the discussion following equation (H 3), and Figure 15, the uncertainty in the wave-number k is of the order of magnitude of 2n/L, i.e. since p =1ik, (H 5) LJp =1iLlk�1i2n/L =h/L and from (H 4) and (H 5) Ll x Llp�2h. •

III

CHAPTER

(r, p) for (rh ra, Ph H(r, p)tp(r) Etp(r).

IlI . 1 . We shall write the SchrOdinger equation

•

•

=

230

•

•

.

.

Pal .

tp(r) satisfies (H 6)

HINTS FOR SOLVING THE EXERCISES

Under inversion, r _ - r, and (remembering that p = - iii grad, so that p - - p under inversion) , equation (H 6) transforms into H( - r, - p)",( - r) = Etp( - r) . On account of equation (111. 1 ), this equation is the same as H(r, p) V'( - r) = E,,, ( - r) (H 7) i.e. '1'( - r) is also an eigenfunction of H belonging to the same energy eigenvalue E. But since we assume that E is non-degenerate, "' ( - r) must he essentially the same function as tp(r), i.e. it must be a multiple of 'P(r) : (H 8) '1' ( - r) = ).'P(r), applying (H 8) twice gives 'P(r) = ).2'P(r),

). = ± I . If we allow E to be a degenerate eigenvalue then we cannot deduce equation (H 8), but equation (H 7) still holds : tp(r) and '1'( - r), and consequently any linear combination of them, are eigenfunctions of H belonging to the same eigenvalue E. We can then choose two linear combinations, which span the same manifold as tp(r) and '1'( - r) , and which have a definite parity. These two linear combinations are tp(r) ± tp( - r) . 111.2. The probability current vector j is given by equation (6. 1 2) : j=

1i 2 ml. ( ip grad 'I' - 'P grad ip) .

For 'P = ei�'Ir, the only non-vanishing component of j JS the radial component . lik 1 v J, = m ;:2 = ;:2 where v is the magnitude of the particle velocity. Thus we are dealing with an isotropic wave of outgoing particles ; the number of particles crossing a sphere r = const., per unit time, is 4Jlv. If we had considered tp = e-ib/r we would have found v . J, = - ;:2 ' i.e. an isotropic wave of incoming particles. 111.3. Let = m J + m2, then mJ m = r + 2r , r = rl - r2' R Al J Al 2

M

From these equations one easily proves that

1 1 1 1 -V'1 2 + -V'22 = - V'R2 + -V' 2, m2 ml f.l

M

231

HINTS FOR SOLVING THE EXERCISES

where R and r have components (X, r, Z) and (x,y, <;), and VR'1. = aX'1. + ar'1. + aZ2 V 2 = ox2 + cry2 + 0<;2 ' and is the reduced of the two masses ml and m2, defined by 0'1.

I'

0'1.

0'1.

0'1.

'

0'1.

0'1.

mass

mlm2 1' = -ml + m2 ' t) = � (R, t)


Writing 'P(rIJ r , (III.3) is separable,2 giving . a

'1i8t�(R, t) =

-

the SchrOdinger equation 112 VR2�(R, t) (H 9)

2M

o 1i'1. iIiFt
(H 10)

Equation (H 9) is the Schrodingcr equation of a free particle of mass M fi re�� � �� � � �� � ��� �� � � (arbitrary) coordinate system. Equation (H 1 0) describes the relative motion of the two particles. It represents the motion of a particle of mass in the potential field VCr) This separation into centre of mass and relative motion is the same as occurs in classical mechanics. IlIA·. The problem is treated in essentially the same manner as the problem in section 8. The constants of inte�ation in the three regions x Vo, this leads to [ 1 + 4E(E Vo) ] -I , T [ 1 + Vo2 sin2 (k2a) ] - 1, V02 sin2 (k2a) - 4E(E Vo) where k2 is defined in equation (8.3). This problem has been con­ sidered in detail by Schiff12 (section 1 7). 111.5. The solutions must be of the form 'P=AelCX, 1(= {i� (Vo - E) y, when x< - a, 'P=BeWr+Ce-i!Jt, k = {��Er when Ix l a. These solutions and their first derivatives must be equated at x ± a to determine the constants of integration, apart from a common factor determined from the normalization of the wave function. (i) Using the fact that the eigensolutions must be even or odd, one can straight away say that there are two types of solutions; both behave like when x >a. .

p.

R

=

-

_

-

=

tp = De-""

232

HINTS FOR SOLVING THE EXERCISES

For O<x< a, they behave like 'P = B cos kx and 'P C sin kx respectively. (ii) Matching logarithmic derivatives for the even eigenfunctions gives at once k tan ka = IC. This is a transcendental equation in which all quantities except E are known. Its solutions give the allowed discrete set of energy eigen­ values. For the odd eigenfunctions the situation is similar, the equation for the energy eigenvalues being k cot ka = IC. This problem has been discussed in detail in Schiff12 (pp. 36-40) ; in particular, a simple method for solving the above transcendental equations is given. III.6. The wave equation of the hydrogen atom is given by ( 1 2 . 1 ) with V(r) = e2jr. For m one should strictly take the reduced mass =

-

-

m=

melectron melectron

•

mprOlon

+ mp.oton

,

but since mprotonjmelectron� 1 840, m�meleclron in all atomic problems. Spherically symmetric solutions, i.e. solutions depending on r only, are obtained from ( 1 2.7) with 1. = 0. For the ground state, the electron is bound, i.e. E< O. Hence we put a2 = 2mEJli2• The approximate asymptotic solution ( 1 2. 1 0) now becomes e-ar. Substituting this in ( 1 2.7), one finds that it represents an exact solution, provided that me2 a = li2 ; -

the normed wave-function, derived from

is

r

• • 11

l 'Pol 2dT = 1 ,

'Po =

(�re-�.

It represents the ground state since it contains no nodes. The corre­ sponding energy level, i.e. the ionization energy of the hydrogen atom is

Eo =

- 2fl 2 = - 1 3 ·5 me4

eV

(the electron volt (eV) being the energy acquired by an electron in falling through a potential difference of l volt. 1 eV = 1 ·59 X 1 0 1 2 erg) . l /a is called the Bohr radius of the hydrogen atom. It is the most -

23 3

HINTS FOR SOLVING THE E!(ERCISfS

probable distance between proton and electron as one can easily verify finding lhe maxirillun of r1Iy'oP. 1/a-0·528 x 10-' em, 10-' em being typical for the order of magnitude of :l.Iomic dimensions. The corresponding wave function of an electron moving i n the Coulomb field of a nucleus of charge Ze, neglecting the effect of any other electrons present, is obtained by replacing e2 by Ze2 everyby

where, i.e.

'� ) ' a ' 'Po '" « --;:;- e "'.

the correct wave fu nction for singly-ionized helium (Z-2), or dOllbly-ioni1.cd lithium (,(' ... 3). If wc ncglcct the Coulomb rcpulsion bctween the two clectrons in helium, the wave equation for helium is separable and the ground state wave function is simply a product of two wave func tions of the abovc kind with ,(' .... 2 : 8 l lPO(rh r:) _ e_:<1(" +'j).

This is, for example,

( :)

IV.!.

CII"PTER

IV

From equations (IV.I) nnd ( 1 5.5) 8 J! I 1J;-;nP;, P;- - 8q;'

(i - 1 , 2, 3).

Equation (IV.I) is the Hamiltonian function for n panicle of mass m, with cartesian coordinates (11) 12, ql), m oving i n a potential ficlel V(1h Q2. ql)' Hamilton's equations merely stale the conventional definition of momentum and Newton's law of motion. From (12.2), (15.1) and ( 15.2), T =t m (.i 2 +j2 + ,t2) -tm(i"2 + rliJ2 + r2 sin2 Orp 2) )1- )I(r). I·fence from (15.3) aT P· - a; - mr . aT p, _ _ _ mr20 ao aT ' ' 2 0p. P. _ -;-; _ mr-� Slll ,� The Hamiltonian function is IV.2.

1

•

j

234

HINTS FOR SOLVING THE EXERCISES

and Hamilton's equations follow at once by elementary differentiation . 1 r = m P,

· _ P, -

tiC.

_

]

[

oV _ I p _ I_ p + 2+ 2 or mr3 6 sin 2 0 9'

[Note that the generalized momenta do not necessarily have the dimensions of momentum, i.e. of (mass x length)/time ; from (1 5.3) it follows that the dimensions of Pi are such that Pi qi is an energy. Thus P, has the dimensions of momentum, Po and P9' those of angular momentum.] IV.3. Let the operator A have the complete normed orthogonal sequence of eigenfunctions tph tp2, • • •, tp", • • • and let a1
'1' = 2.c,.tp", n-I

hence

co

X = 2.d,.tp,,; n- I

i.t. (AX,

'1') = (X, A'P), i.e. A is Hermitian. IVA. From equation ( 1 6. 1 9) (L1I�) 2 = <1�2>,

since = 0. And from equations ( 1 6.20) , ( 1 4.5), ( 1 3.2 1 ) and it follows that

{"IR(r) I 2r2dr = 1 1

= (/�'P, /�'P) = 21 + li l

and since 2. m 2 =!/(l + 1 ) (2 1 + I ), 111 - 1

•

2 1i 2

:��2 = (21 + 1 )",�:n2

=1h2/ (1 + I ) .

IV.5. (i) Since ( U*) * = U, equations ( 1 8. 1 4) are also the condi­ tions for U* to be unitary : they are symmetrical in U and U* ; 235

HINTS FOR SOLVING THE EXERCISES

( ii) ( UV) ( UV) * = UVV*U* = UU* =1, etc; and similarly, since ( V- 1 ) * = V, (iii) (UV- 1 ) ( UV- 1 ) * = UV- 1 VU* =I, etc. IV.6. If X is any inverse operator of A, then XA = I and AX = I. Multiplying the first of these equations by A- 1 on the right-hand side and using ( 1 8. 1 8), X =A-1 follows, i.e. A - I is unique. Similarly, if X = (AB) - 1 is the inverse operator of AB, then ABX = 1 and XAB = 1. Multiplying the first of these equations successively by A-1 and B- 1 on the left-hand side one obtains BX = A- 1 , and X =B- 1A- 1 , and this X obviously also satisfies the second equation XAB = I. Hence (AB) - 1 =B- IA -1. This relation was to be expected, since a similar relation, equation (18.9), holds for the adjoint operators, and for unitary operators, for which U* = U- 1, the two equations must agree. IV.7. From equations ( 18. 1 6) and ( I B. 1 7) it follows that the vectors lJ'h 11'2, • • • 'Pn' • • • are normalized and orthogonal. They are also a complete set. For if they were not complete there would exist a vector f orthogonal to every member of the set, (1, 11',,) = 0, for all n ; i.e. (U-lj, rp,,) = 0 for all n ; i n other words, there would exist a vector, U-1f, orthogonal to all 'i'", in contradiction to the assumption that the rpn are complete. IV.B.

.

.

[P" P,] = 0, smce for any wave function '1',

Equation (5b) follows from a ." a ." a , ." � :" - U�' = 11'0,,'1" CJX (IX CJX

x

_

,

,

�

�

� 'I' = � 'I"

ux,uX,

tlX,uX,

,

IV.9. From equations ( 1 4.2) and ( 1 4.3), the required result follows at once. For the operator a/arp commutes with both position variables , and 0, but not with rp. But rp does not appear in ( 1 4.3) .

lJ'(x) has a definite x) (x) = f�=xl V' (x) 1 2dx and the integrand is an odd function of x whereas the range of inte­ gration is oo<x< so that (x) =0. I I . To find the momentum distribution we represent 'P( IV. 1 0. From Exercise II!.1 we know that parity, so that 1 1J'(X) 1 2 = IV'( - 1 2• Now -

00,

IV. r) as a superposition of eigenfunctions of definite momentum, i.e. of plane waves. The correctly normalized plane waves are

�) teipr/"

rp(p, r) = (2

2 36

(H 1 1)

HINTS FOR SOLVING THE EXERCISES

so that, according to (4. 1 7) and (4. 1 8),

Itp(p', r)'P(p, r)dr = 8(p' - p) ,

the triple integral being over all values of r, i.e. thell write

V'(r)

=

-

(H 12) co< X< co, etc.

j.4(p) 9'(p, r)dp,

We

(H 13)

the triple integral being over all values of p, i.e. - co
A (p) =

�)tf

(2

V'(r)e-iPr/"dr.

(H 1 4)

all

space

This equation could have been derived directly, without introducing the 8-function, from the Fourier integral theorem (if. Exercise 1.6) ; in fact, the momentum distribution is the Fourier transform of the wave function. Using the above equations, we can easily check that we used correctly normed momentum wave functions, i.e. that the momentum distribution is correctly normed, i.e.

For

JI A (P) 12dp = (2�) 3ffI�(r)V'(s)eiP(r-.)/Adrdsdp f'P(r) V'(s) 8(r - s)drds J 1'P(r) 1 2dr = 1

since 'P(r) was assumed normalized. IV. 1 2.

[A, B + C] =A(B + C) - (B +C)A = [A, B] + [A, C] [A, BC] = ABC - BCA = (AB - BA)C + B(AC - CA) = [A, B]C + B[A, C].

The third relation (IV. 1 2c) follows from (IV. 1 2b) by inverting the order of the factors, since [A, B] = - [B, A] . IV. I 3. Using the commutation relations ( 1 9.5) and Exercise IV. 12, equations (IV. l 3a-d) are easily proved. Some typical proofs are : 237

lII!\'TS "OR SOLVING

TIn:

EXERCISES

[/.. x) _ [;p, -
[I.. 1') - [I.. II) + [I.. I.') - [/" 1,)/, +1,[/" I,) + [I.. 1,)1, + 1,[/" I,) -ild.[, + l,iM, + ( - ild,) t. + (( - iIiI,) .... O. CIIAPnR V N

N

i_I

'_I

(AB)... -2.A.u.B,. -2.0..h�O..JOM = o..b"o..... ; Ihis is a diagonal matrix and is symmetric in A and B, hence AB-BA. V.2 From (22.19) one gets ,\'

.v

"" .- 1

._1

2. A..c..co-= o2.lc.P,

and taking complex cOluugatcs or this equation, since A"", = IT... : ".

N

N

.."

._1

",0_1

"" . _ 1

rl2.lc.ll= � 1I...c,.c.- 2. A...c,.c. = 2. A.OI'.c.., interchanging the summation suffixes V.S.

Suppose

m

.., . _ 1

and

rI.

Hence a - ii.

x

x

211...c.-=0Ic., 2A_d.. -ald

.- ,

Hence

.-

x

,

•.

x

2:A_c.,:a. - OIl:c.d.,

and I.t.

"" .-1 ".

)(

",.- 1

.-1

.-1

(1-1 15)

2.A"",It.,c. -a22.d.c., (since Dl =a2). x

x

2: A...c..d. -01.2.1;.c.

... . - 1

._1

238

(1-1 16)

HINTS FOR SOLVING THE EXERCISES

and subtracting equations (H 15) and (H 1 6)

O = (a.

hence if

N

"2.

- a,.) d,.e., ,. - 1

a. =l=a2, [en] is orthogonal to [dn].

VA. From the law of matrix multiplication (22 . 2)

S.2=�[� �] [� �] =� [� �] =!l2I and similarly s,2 =S�2 =!lZ Ij hence 82 =il2 I. Thus the operators s� and 82 are diagonal and hence simultaneously observable. s� has the eigenvalues ± lj2, 82 has the degenerate eigenvalue il2• One easily verifies that the operators ., s", s� do not commute, s.!,, =�" [O1 o1 ] [� -0 i] = �i [0l -0 1] s"s.= � [� -io ] [01 01 ] =�i [ - 01 0l ] '. hence [S., s,,] =ihs., s

e.g.

,

and two similar relations obtained by cyclic interchange of x, y and .t. These commutation relations, like the above properties of these operators which we have already met, are generally true for angular momentum operators not only for the particular 2 x 2 representation we used above. V.S. Let n 1 , 2) be the 2 x 2 matrix which transforms into the eigenfunctions f1''' f1'z of A, i.e. from equation the vectors (22 .23), f1'" + (n I, 2). The eigenvalue problem (22.2 1 ) is in our case + Jl =0 (J. - 0,.) + (J. - an) U211 = 0 . Each of these equations gives the ratio They are consistent if these two ratios are the same, i.e. if

Um,,( , �" �2 m

=

= UIn Et U2nE2, = UI" U2" Jl Ut" Uln/U211• -;;- = ).-Jl a,,· ,t

-

a,.

[This equation is just the consistency condition (22.22),

(H 1 7)

I ,t A -Jl a" I = 0.] From the quadratic equation in (H 1 7), we obtain two solutions al = p, a2 =;. + p -

Jl

9 + Q..M.

a,.

,t -

a,.,

239

HINTS FOR SOLVING THE EXERCISES

to we find UIII U21 = - 1 , Aa. weCorresponding find Ul21 U22 = + 1 . For U to be unitary 2 Uul 2 + IU12 1 2 = 1 and I U2d 2 + I U22 1 2 = 1 , so that we can I UII = UI2 = Hence the orthonormal eigensolutions are 911 = y1 2 (EI - E2), 912 = y1 2 ( EI + E2) .

the eigenvalues of and corresponding to we must have take 1 1 Y2.

as

al

V.6. One can prove (23.4) by the same method which is used in section 23. In the Schrodinger picture, the wave functions V'I and X, are time-dependent and so that

d AV'I) = (axl' A'I'I) + ( aAV'I) + ( at at = (�, ��V'I) + (X" �� ) - ( X"

di (X"

so that

iA�(x" AV'I) = (x,,{ili��

'1'1

+

Xu

a'l'I) ATt �:'1'1)'

X"

[A, H]�,).

(H 18

)

In the Heisenberg picture, the wave functions are always time­ independent, so that the whole time variation must be taken into account through

('1'1 ='1'0, aV'I/at=O, etc)

A(t}:

i1i�(�, AV'I) =ili(XU �V'I) '

(H 1 9)

Comparing equations (H 1 8) and (H 19) we see that the two modes of description are consistent if the Heisenberg equation of motion of the operator is

A(t)

iii

� iii �� + [A, H] . =

V.7. The required matrix element is

= f:a)V'n(x)xV'",(x)dx = 1[a)91D (E) E91..( E)dE = (2D+ 11 m.n)iJa) e-:tEH"( E) H,,,a)dE a) 1 I ) iJ-a)e-:tHm( E) {lHn+l ( E) +nH,,_ I ( E)}dE (2"+"'n1. m .n (H 20) a

"'n

I

_ a)

a

where we used ( 1 1 .23) and the recurrence relation ( 1 1 . 1 9). From the orthogonality relation ( 1 1 .22) it follows that the only non-vanishing 240

HINTS FOR SOLVINO THE EXERCISES

terms in (H 20) occur for m =n + l or m = n - l . Using ( 1 1 .23) to obtain the correct normalization, one easily obtains

(

I n + 1 ) i = (i 2 J

(m = n + l ),

1 (n)i = (i 2 '

(m = n - l ),

=0,

(n

- I :y!:m :y!:n + 1).

CHAPTER VI VI. I.

[A+ J M�] = [AfM) Af�] + i[M.7J M(] = - mM, - IiMJt = - IiA+, [A+, M2] = 0, from (25.7), A+A_ = (MJt + iM,) ( MJt - iM,) = M,,2 + Ml - i(MJtM, - M,MJt) = M2 - M�2 + 1iM�. VI.2. From (25.23) , if lJ'(q, n, m) and lJ'(q, n, m ± l ) are to be normed :

IN:I: (q, n, m) 1 2 = (q, n, m IA ;:A :l: l q, n, m> = =1i2n(n + I) - 1i2m2 "T 1i2m = 1i2 {(n"T m) (n ± m + I)}, where we used the fact that (A:I:) * =A", and equations (25. 14) for A ;:A:I:' Taking the square root, one finds that N:I: (q, n, m) = eiali {(n"T m) (n ± m + I )P and in the arbitrary phase factor e!a we have taken a = O. VI.3. We have, from equation (25.31 ), that

[

ncr

�

= L..n,(J, ,_1

] [;

- [nl n3+ ln,. 2, nl -- inn32]

_

•

VI.S. The general 2 x 2 Hermitian matrix can be written in terms of four real numbers a, h, c and d as

a

h - ic

h + iC d

=

;

a d a d +

h - ic

241

];

h + ic a d_a d

;

HINTS FOR SOLVINO THE EXERCISES

VI.6. (i) Since the direction of the z-axis is in no way preferred, it follows that the component of 0' in any direction has the eigenvalues ± I . We must find two eigenfunctions Ul and U2 such that (H 2 1 ) u"ul = u" a,P2 = - U2' Since a and {J fonn a basis in S , U l and U2 must b e linear combina­ tions of a and {J, (H 22) and (H 23) IA I 2 + I B I 2 = 1 , ICl z + IDI 2 = 1 so that Ul and U2 are nonned. From (25.3 1 ) and (26.7) one finds that a,.a ={J a:/a = i{J a,a = a a,,{J = a a� = - ia a� = - {J and since a,. = em = a,. sin 8 cos qJ + a:/ sin 8 sin qJ + u, cos 8 the eigenvalue equation for u" (H 2 1 ), becomes (A{J + Ba) sin 0 cos qJ + i(A{J - Ba) sin 0 sin qJ + (Aa - B{J) cos 8 =Aa + B{J, and taking the inner products with a and (J in turn we get B sin 8 e-ito + A cos 8 =A A sin 0 e}'r - B cos 8 =B. One can easily check that these two equations are consistent, i.e. AlB has the same value calculated from either of them. From the first equation, one finds that B sin 8 e-i" =A( 1 - cos 0) B =A tan (812)ei", and from (H 23) 8 IA I = cos 2 ' We choose the phase factor in A as unity, i.e. o A = cos 2 so that and Ul = a cos

; + {J sin ;ei".

(H 24)

We could similarly find U2t but we determine it more easily from the fact that it must be orthogonal to u" i.e. (Uh uz) = 0, 242

HINTS FOR SOLVING THE EXERCISES

which, with suitable choice of phase factor in "2, easily gives "2 = a sin

� - P cos �e;9'.

(H 25)

The above calculation admits a simple interpretation. Starting with a coordinate system (x,y, z) we have rotated it about the origin to form a new system (x', )", z') so that the new z-axis, i.e. the z'-axis, points in the direction of the unit vector D. The component of the spin vector (J in the direction of the z'-axis is (Jc' = (J" and "1 and U2 are its eigenfunctions. (ii) To find the probability of observing ±1i/2 when measuring s" =1i(J1I/2 on a system in a state = 'PI (r) a + 'P2(r) p we must expand this state in eigenfunctions of (JnI i.e. since . O o . 0 a = "1 cos + "2 sm ' p = "1 sm - "2 cos 2 e-'9', 2 2 2 0 . 0 0 o . ITI r = " 1 'PI COS 2 + 'P2 sm 2 e-'9' + "2 'PI sm 2 - 'P2 cos 2 e-'9' The probability of measuring + 1i/2 for SIlO irrespective of the position of the electron, is then (} (} . O . (} -'.9' 'PI cos 2 + 'P2 sm r-'9'. 'PI cos 2 + 'P2 sm 2e

'P

.} {

{

(

0) .

(

}

•

)

•

�

�

.

= cos 2 ( 'Ph 'PI) + sin 2 ('P2' 'P2) + sin (} Re [ ('Ph 'P2)e- '9I] ,

where Re(z) denotes the real part of the complex number z = x + V', i.e. Re(z) =x. One similarly finds the probability of measuring - li/2 to be sin2

g('Ph 'PI) + cos2 g(V'2. 'P2) - sin

(} Re [ ( 'Ph 'P2) e-i9'] .

These two probabilities add up to unity as required. (iii) We do not know the wave function of the system completely but only that it is of the form tP = '1 a + '2ei3P, where ei3 represents an undetermined phase factor. Though multi­ plying the wave function as a whole by a phase factor does not lead to a physically new situation, multiplying part of a wave function by a phase factor does lead to observable differences. Carrying out the same analysis for tP which we did for in (ii) , we find for the prob­ ability of s" having the value ±1i/2,

IClli��:}; IC2I i::}� +

243

'P

±sin ORe['l c2ei(9'-�)] .

HINTS FOR SOLVING THE EXERCISES

We must now average over the unknown phase 8 which can assume all values 0 <8 <2". Since

d"Llne-i3d8 = 0,

the probabilities, after averaging over 8, become Jsin2 ).!! JCOS2� IC1 1 1.sin 2 J2 + IC2 1 1.cos2J2 ° Cases (ii) and (iii) show an interesting difference. In case (iii) we are dealing with a problem which is axially symmetric about the z-axis. Hence the azimuthal angle rp cannot occur in any physically observable quantities, such as the above probabilities. The wave-function lJI in case (ii), on the other hand, is not axially symmetric on account of the phase relationship between the two parts '1'1 a and 'I'zfJ of the wave function lJI, which we assumed known. In a symmetric case we would have (Sit) =0; for 'IJ' one easily finds that Ii (Sit) = 2 (['PI a + 'P2ft] , (tlt['PI a + 'P2ft]) Ii

= 2 ('P1 a + tpzfJ, 'PIP +'P2 a) = 1iRe[ ('Ph 'P2) ] :/=0, showing at once that there is no symmetry about the z-axis. Conse­ quently physically observable results, such as the probabilities we calculated, will in case (ii) depend on both 0 and rp. Thus, a know­ ledge of the wave functions '1'1 and 'P2 gives one much more informa­ tion about the system than a knowledge of ICl l and IC2 1 only. V1.7. From equations (2 7.3) and (2 7.4), it follows that

Since

[Mit' M.1] = [MIt( I ) + M.. (2) , M.1(I) + M,(2)] = [M.( l ), M,( l ) ] + [M.(2), M.1(2)] = i1i {M�(I) + M� (2)} = iliM" etc. M2 =M2 (1) +M2 (2) + 2M(I)M(2),

and from (27.3) and (25.7)

etc, it follows that

[M2 (k), MIt(l)] =0, (k, 1 = 1, 2)

[M2, Mit] = 2[M(I)M(2), M ( l ) + MIt(2)] ; ..

and

[M (2)M(I), MIt(I) ] = M,(2) [M,(I), MIt(I)] + M�(2)[M�(I), Mil ) ] = i1i { - M,(2)M�(1) + M( 2)M,(I)} ,

and similarly

[M(1)M(2), MIt(2)] = i" { - M,(1) M� (2) + M( 1 )M,(2)}, 244

HINTS FOR SOLVING THE EXERCISES

so that

[M2, M..] =0,

etc, proving relations (25.6-7) . We need not derive equations (25. 12-14) anew, as in obtaining these, equations (25.6-7) only were used. VI.8. We are combining the three orbital angular momentum eigenfunctions of a p-state (I ml ) ' (m, with the The resultant 6-dimen­ and spin states decomposes into a 4-dimen­ sional product manifold x sional manifold corresponding to j and a 2-dimensional manifold corresponding to j Using the methods of section 27, we obtain for the eigenfunctions I , j, i.e) :

= I), Y'1 ( 1, = - 1 , 0, I) Y'2(!, -!) =p IDl1 Y'2(!,6!) = a. (I) 2 =3/2 IDlt IDli = !.

Y'{a IDlt:t; () (H 26a) Y'(I , ! ; i, i) = Y'1 ( 1, I) Y'2{!, t) = Y'l(l, I)a, Y'(I, ! ; i, t) = y1 3 {V2Y'I(I , 0) Y'2(!, !) + Y'1 ( 1, 1 ) Y'2 {!, -!)} (H 26b) = y1 3 {'V'2Y'I(I , O) a+ Y'1 ( 1, I)P} ; Y'{I , ! ; i, -!) = y1 3 {'V'2Y'I(I , 0) Y'2{!, -!) + Y'I(I , - I)Y'2{!, !)} (H 26c)

t; ) 1Y' (I, P(I, t, ! Y'( I , ! ; t), A + {a Y'I(I , 0)Y'2{!, t) + hY'I(I, I) Y'2 (!, -!)} =0

(H 26d)

by choosing a linear combination of (b) IDli : We obtain 1, equation (H 2Gb), and I )P orthogonal to or so that A+ operating on it vanishes :

Y'1 ( l, O) a

i.e.

hence

Ifl(l, !; t, !) = y1 3 {Y'1 { 1, 0)Y'2 (t, t) - y 2 P1 (l, 1) Y'2 (t, -t)} (H 27a) = y1 3 { Y'1 ( 1, O) a - V2Y'1 ( 1, I)P} 1 1JI( I , t; t, -t) = 3 { - Y'l (l, 0)Y'2 {!' -t) + y 2 P1(I, - 1 )Y'2(!, tn y .r

. /-

(H 27b) 245

HINTS FOR SOLVING THE EXERCISES

VI.9. To see how the 8-dimensional spin space 61 x 6z X 63 decom­ poses into manifolds corresponding to given eigenvalues of SZ, we firSt combine 61 and 62 according to the scheme of Example 2, section 2 7 :

61

x

6z =IDll ( I 2) +IDlo( 12)

(where we have written IDll(12) and IDlo(1 2) to indicate that these are product spaces of particles I and 2), and then we combine each of these with 63, according to the schemes

IDll( l2) IDlo( 12)

X

x

63 =IDlt +IDli'

(H 28a)

63 = IDli""

(H 28b)

where we have used primes to distinguish the two doublet manifolds. We span the manifold IDll( 1 2) by the eigenfunctions (27 . 1 6a--c) and IDlo ( 1 2) by (2 7 . 1 6d) . In the decomposition (H 28a) we then combine an angular momentum I with an angular momentum t. This is exactly the problem we treated in Exercise VI.8, and the results we obtained in that exercise hold here. For in considering the addition of angular momenta in section 27 we were not interested in what the actual wave functions are, but only that they should satisfY certain eigenvalue equations. Hence we obtain the eigenfunctions !P( I , !- ; !, m) and !P( l , 1; !-, m) spanning the manifolds IDl. and IDli' in (H 28a) by com­ bining equations (H 26a-d), (H 27a-b) and (27. 1 6a--c) :

(a)

IDlt :

!PH, !) = al aZa3

(H 29a)

1

!P(t, t) = v3 {at azP3 + aZaJPl + a3alPZ}

(H 29b)

- = � {alPzP3 + aJJ3Pl + a3PJP2} !P(t, - i) = PIPzPJ ; !P(!,

t)

(b) IDlt' :

!P(t,

t)

(H 29c)

3

=

!PH, - !-) =

(H 29d)

� {�2 ( atPZ + azPt ) aJ - V2at azPJ} )3{ - )2 ( a1Pz + azPl) P3 + V2PlPz a3}' 3

We easily obtain the other 2-dimensional manifold combining the singlet state (27. 1 6d)

I v2 ( alPZ - a2P1 ) 246

IDlt

(H 30a) (H 30b) arising from

with aJ and PJ : (e) IDlt" :

HINTS FOR SOLVING THE EXERCISES

'P(!, !) = v21 ( aJP2 - a2PJ) a3 'P(!, -!) = v1 2 ( alP2 - a2PJ)P3'

(H 3 I a) (H 3 I b)

If we again consider the symmetry under the permutation of particles, it follows from the fact that the triplet states of particles I and 2 are symmetric and that IDl: and IDlt' are derived from these, that °IDlt and IDlt' contain only wave functions symmetric in 1 and 2. Furthermore, IDl, contains only wave functions symmetric in all three particles, as one sees from equations (H 29a-d) . Similarly IDlt· con­ tains only wave functions antisymmetric in particles I and 2 as it is derived from the singlet state

�2 { a1P2 - a2P1} '

We could of course equally well have combined the spin states of any other pair of particles to begin with. This corresponds to per­ muting the suffixes 1 , 2 and 3, labelling the particles, in the above equations. This does not affect IDl: which is completely symmetric, but the resultant 2-dimensional manifolds IDlt' and IDlt" are different. Together they, of course, span the same 4-dimensional manifold

61 x 62 x 63 -IDlI•

In practice, this ambiguity ofstates does not usually arise. On account of the Pauli principle (section 28) , the symmetry properties of the wave functions are usually prescribed. VI. 10. By simply interchanging the labelling 1 and 2 of the particles we have H(2, 1 ) '1'(2, 1 ) = EV'(2, 1), and since H( l , 2) = H(2, 1 ) the result follows. VI. I l .

Substituting (28.3) into (28.4) gives

(l a I 2 - lbI 2) I V' ( I, 2 ) 1 2 + ( lbI 2 - l a I 2) I V' (2, 1 ) 1 2 + (ab - ba ) V' ( I , 2) '1' (2 , I ) + (b a - ab) V' (2, I ) V' { I , 2) = 0. Since V'{ I , 2) is an arbitrary wave function, this equation is only satisfied if la l 2 = lbl 2 and ab = bii.

b

H ence, if a =Aeia and = Bei(J (A, B, a, P real and A >0, B >0), we require ei(a-(J) = eiCJl-a), A =B

i.e. (a - P) = nn, (n = 0, 1 , 2, . . . ) ,

so

247

that a = ± b, and (28.5) follows.

HINTS FOR SOLVING THE EXERCISES

2

VI. 1 2. Let I and denote the protons, and 3 the neutron. There are two doublet spin states which we obtained in Exercise VI.9, equations (H 30) and (H 3 1� ; we write them

= J2 {a1fJ2 - alfJl) a3 1 E"-0'2; 3) = ...j2 ( alfJ2 - alfJl) fJ3

E"+ ( � ; 3)

and

= J3{J2( alfJ2 + a2fJl) a3 -v'2alalfJ3} E,- ( f2 ; 3) = J3{ -�2 ( alfJ2+alfJl ) fJ3+V2fJlfJ2a3}.

E,+ ( rn ; 3)

The total wave function must be antisymmetric in the proton co­ ordinates, but there is no restriction as far as the single neutron is concerned. If q>,( � ; 3) and q>,,( i2 ; 3) denote spatial wave functions of the indicated symmetry properties then the total wave function is of the form

4}:I: ( � ; 3) =aq>,(rn ; 3) E,,:!: ( i1! ; 3)

q>

+ h ..( f2 ; 3) E,:!:(T2 ; 3), = ±1i/2. 28.)

As for the

where the ± signs indicate the spin states with SJ; a-particle, one finds that tendency to be in a singlet state. (Gj end of section

Ihl
CHAPTER VII

VII.2. The perturbation of any energy level En, equation given by equation

(29.13). VM = af:..,X3 1V'n(X) \ 2dx. V(x) =ax3 is V',,(x), l 'fn( -X)l 2 = I ",,,(x) 1 2, V",, = O.

Since the perturbation and the oscillator wave function definite parity, so that

(11.15),

is

V( - x) = - V(x),

an odd function, equation ( 1 1 . 23), has always a it follows that

This result is generally true for states of definite parity and perturba­ tions of odd parity. Second-order perturbation theory will, in general, give a displacement of levels.

11, 1 .1 ) (29.13) E.=(n + �)1iQ).+ V..

VII.3. Using the notation of section levels are, according to ( 1 5 and

,

248

the perturbed energy

HINTS FOR SOLVING THE EXERCISES

f.

where v_ - .

{X''P,(X)) 2tfx -

� f. {E'to, (m 'dE.

where we have used the fact that the oscillator eigenfunctions, equation ( 1 1 .23), are real functions. Applying the recurrence relation ( 1 1 . 19) twice, one obtains E2Hn( E) = 1 Hn+- 2 (E) + ! (2n + I)H,, ( �) + n (n l } H,, 2(E) , -

-

and using the orthogonality relation ( 1 1 .22), one obtains

V

""

=3

(

)

a1i2 4 m 2cu02 ( 2n 2 + 2n + l ) .

VIlA. To carry out the second-order perturbation calculation, we must solve equation (29.8c), as well as (29.8a-b). Taking equations (29.5) to (29. 14) over unchanged and putting CD

f/'x(2) � b,f/',(O), 1 =

,-

equation (29.8c) gives on substitution co

co

co

co

� b,E,(O}f/',(O) + V� a,f/',(O) = Ex(O)� b,f/',(O) + Ex(l)� a,rp,(O) + EX(2)f/'x (0) ; , -1 ,-1 ,- 1 ,-1 and taking inner products with rp,(O) : CD

b,E,(O) + � a, V.. = b,Ex(O) + a,Ex(1) + EX(2)8x,. ,-1

Separating the cases s = N and equations (29. 1 3- 1 4), and

s :pN,

the last equation gives, using

� ' Vx, V,x Ex(2) = L. , Ex(OJ _ E,(0)'

(s = N) ,

axVIN Vxxv'x V"V,x �, b, = f (Ex(O) _ E,(O» (Ex(O) - E,(O» - (Ex(O) _ E,(O» 2 + Ex(O) _ E,�O)'

(�:pN)

where, as before, the term r =N is excluded from the summation

2:'.

aN and bx from the normalization condition IIrpNCO) + ef/'x( l) + elrpX(2) II = I which must now be satisfied up to terms in 62• In this way, we obtain

We again determine

the two conditions

aN+ aX= O bN + hN = - 1:/0,.1 2•

,

These two equations again fix the real parts of aN and bx• As in section 29, we can again put the imaginary parts of ax and bx equal to 249

HINTS FOR SOLVING THE EXERCISES

zero, corresponding that

to

a particular choice of phase factor for

tpN, so

aN =O bN= - t2:l a,1 2. ,

Using these results, we finally obtain the following expression for the energy levels and the energy eigenfunctions according to second-order perturbation theory:

12 ," ' I(OVN, EN = EN(0) + 8 VNN + 82 £.. (H 32) ) EN - E,(O) ' V,N ,"' (0) , tpN = tpN(0) + 8£.. , tp EN(0) _ E,(O) V" V,N V,NVNN ," ' tp,(0) ,"' + 82� � (EN(O) E,(O») (EN(O) _ E,(O») - (EN(O) E,(O») 2 2 (H 33) - �"'ItpN(0) (E (O)I V-,NIE (O») 2 ' , N

{

VIi.5.

e/e.

[

The matrix H -

Under the

}

_

r'

1>,

J

El + eEl '·

u]'

where

[g]

,

A and p

m m [�] m:] m [gl } (H + e V)

-+-

-

+e

The eigenvalue problem

(E, + eE, ')

i.e. to first order in e:

Hence

]

has the eigenvectors �, -

E perturhation, H � H H V, ¥, �

are of the first order in e, and El becomes

_

-E'

+ sE' '

El ' = Vl 1 E E A( I - 2) = 8 V21 ' P(EI - E3) = 8 V31

[s]

which is just what we should expect from equations (29. 1 3-14). If El = E2, the above method breaks down: A is no longer small. This is b
250

[�],

eigenval ue EI • eigenvector

HINTS FOR SOLVING THE EXERCISES

Hence we now write the corresponding perturbed where only . i• •mall, of order e. To first order in _,

}.

the eigenvalue problem now gives (denoting the perturbed eigenvalue by El + eE')

e ( Vll " + V1 2l') = eE'). e ( V21" + V22I') = eE'1' E3v + e ( V3 1 " + V321') = E1v

The last equation gives

e V3 ). + V321') V= ( 1 E. - E3

•

The first two equations are homogenous in ). and I' and only possess a nontrivial solution if

V1 2 1 = 0. I Vll - E' V22 - E' V21

This is the secular equation which is quadratic in E' and thus gives two solutions EI ' and E2'. Corresponding to each of these, we obtain a value of "II' and hence, apart from normalization, an eigenvector (if sections 22 and 30). CHAPTER VIII

VIII. I .

For a solution depending on the radial coordinate

V 2 reduces to

only,

V2 = .!.2 �(r2 !!..) . r dr

dr

Hence

so that

r

( V2 + k2) ( -

::) = eib( -;nV2�) = eib8(r)

,

(H 34)

from equation (33.2) . We can omit the exponential factor eUr from the right-hand term of equation (H 34), since 8 (r) vanishes, except at r = O, and in that case the exponential becomes unity, proving the required result. The result for the second solution ( - e-lkr/4nr) follows at once by taking complex conjugates throughout the above proof. 251

HINTS FOR SOLVING THE EXERCISES

VIII.2. From the discussion leading to equation (33.16), we expect the required transition probability per unit time to be

�1(eiklr9',{q), VCr, q)V'(r, q»1 2(;!)3 8(E' - E) ,

( H 35)

where E is the initial energy of the system, equation (36.3) , and E' the final energy :

E' = ' + 2}'Ti2k 2 E

I •

35)

One easily verifies this by integrating (H over tIk, to eliminate the 8-function, and dividing by the flux of incident particles, Tiko/p, to give the same expression for the cross-section as before, equation (36.1 4). VIII.3. To obtain the first Born approximation, we replace the true wave function by the zero-order approximation

'P(r, q)

tfkor9'o(q)

[if equation (36.4)] on the right-hand side of equation (36.6). gives for the function F, defined in equation (36.8),

W(r)

{r), F,(r) = eikor!dq9',(q) U(r, q)9'o(q) =eikor� W(r) .

This

represents a mean potential, averaged over the internal coordi­ nates of the scatterer, which the incident particle experiences. The cross-section then becomes, from equations (36. 1 2) and (36.1 4) ,

= (hl(2�2rl!W(r) eiCk.-kllrdr r, [if equation (34. 1 )]. Ti (ko - k,) again represents the momentum which the incident particle has transferred to the target during the a(O, 9')

collision. VIII.4. If and are the position vectors of the two particles, then the wave function of the two-particle system can be written (if Exercise III.3)

rl

r2

rl + r2) x(r) (r = rl -r2), 'P(r" r2) =IP(2-

the wave function tP which describes the motion of the centre of mass of the system being symmetric in the coordinates of the two particles (since they have the same mass) , and describing the relative motion. If we treat the two particles 1 and as different, and hence dis­ tinguishable, to begin with, then the asymptotic form of the wave function in the C.o.M. system

2

is

x(r)

ei" + !d4" f(O, 9'), r

252

HINTS FOR SOLVING THE EXERCISES

( r, 0, rp) being the polar coordinates of the vector r giving the position of particle relative to particle Since the polar coordinates of the vector r (corresponding to interchanging the two particles) are (r, n 0, n + 9'), [if equation the asymptotic forms of the wave functions, symmetric and antisymmetric in the spatial coordi­ nates respectively, are

-

-

2. (13.24) ]

1

-

( ei" ± e -i.t�) + db[J( O, rp) ± /(n 0, n + 9') ]. r Hence we obtain for the cross-section, by arguments analogous to those in section a(O, tp) 11(0, 9') ±/(n - 0, n + 9') We can also see this result in another way. If particle is scattered through angles n 0, n + 9', particle flies off in the opposite direction, 0, 9" Hence the scattering amplitude corresponding to this process must also be included in the cross-section a(O, 9'} if the two particles are identical. VIII.5. A qualitative justification for neglecting exchange effects in the case of fast electrons was given at the end of section Since we are dealing with fast electrons, the validity criterion for the Born approximation is satisfied, so that we shall describe incident and scattered electrons by plane waves. To calculate the cross-section, we use the method of section Equations and then give for the case of elastic scattering in the Born approximation

32,

' 2.

=

-

2

I

36.

(36.12)

/(0) =

36.

(36.8)

- 4�Ie-i"rdrIds9'(s) ��( -� + Ir: s ,) rp(s) eikor,

for an electron to be scattered from a state with momentum liko to one with momentum lik (both states having the same kinetic energy the wave function of the atomic electron in the ground state of the hydrogen atom being

E=n2ko2/2m =1i2k2/2m),

I

rp(s) = vn ate-
(r, s are position vectors of incident and atomic electrons relative to the nucleus.) We can write this expression for the scattering amplitude

= - 2:n2IV(r}ei(ko-klrdr where e2 } 1 VCr} = - - + e2 Irp (s} 1 2 _r 1r - s I ds . Comparison with equation (34.1) shows that VCr) represents the mean potential which the hydrogen atom presents to the incident electron. This is physically reasonable. The first term ( -e 2/r) represents the 253 /(0)

HINTS FOR SOLVING TilE EXERCISES

Coulomb field of the nucleus, the second term (the integral) represents the Coulomb field of the atomic electron whose charge density dis­ tribution is _ eltp(s) 12. Evaluating V(r), we obtain V(r) = -e2 (� + a) e-2"', showing that we are dealing with a spherically symmetric potential as we indicated by writing V(r) rather than VCr). We can now use equations (3'1,.3) to (34.5) directly, rather than repeat the angular integration, and obtain /(0) -

;.':;f U + aje-'. sin(K,),d,

2me2(Saz+ K2) = fi2(4a 2 + K2) 2 . Hence we obtain for the differelllial cross-section (H 36) where K=2k sin(OJ2), equation (34.5). Integrating equation (H 36) over all scattering angles we obtain Ihe total cross-section,

L"a(O) sin 0 dO

u,c, ,,," 2n = 271: so that

I

I

KdK a(O)"""'I2 '

K_.

_� (me2) 2(12a4 + ISa2k2 + 7k4) a2(a2+kz)J 3 li2

Vrn.6. From equations (38.29) and (32.12) atOI

2l + 1 ) (2l + l )ej{�J--d,') sin 8/ sin 81' - 1.�i2.2.( / /' •

'

x

•

f

271: +l p/(x)Pr(x)dx. -I

From ( 1 3 . 1 6-17) tlus integral equals 2811'/(2l+ I ) , giving at once (38 .30) .

VIlLS. Consider the potential ).V(r), with ). a continuously varying parameter: 0,).<1. The phase shirts SI will then be functions of )..

254

HINTS FOR SOLVING THE EXERCISES

Correspondingly, we write instead of (38.38) :

[

d2 v,(r, i.) + k2 - i.U(r) dr2

-

1)]

1(1 + � v,(r, i.) =0

and a similar equation with ()' + .1 i.) instead of ).. section 38, one finds

d sin 8,().) di.

=

-

Proceeding as in

1 r = U( r) [v,(r, i.) ] 2dr kJo

which is always <0 or �o, according as U�o or U <0 respectively.

VIII.9. We can solve the wave equation (38.38) explicitly for the case under consideration (if. section 8 and Exercises 111.4. and III.5 where similar problems are treated) : if r
vo (r) = A sin kir,

vo(r) = sin (kr + 80), if r>a,

where k and k.. are related to the bombarding energy E and the potential depth Vo by

!i2k·2 E = 1i2k2 2", ' (E + Vol = 2� .

The two constants A and 80 are to be chosen so that Vo and dvo/dr are continuous at r = a. This gives tan

k (ka + 80) =f. tan (kjil), ,

yielding 80• As E, i.e. k tends to zero, 80 _ o. The corresponding cross-section is given by 0'0 = (4n sin2 80) Ik2• The last equation gives

0'0 = 4na2 since

[tan koa - 1 ] 2, koa

CHAPTER IX

IX. I . For n = 2, 62 consists of only two elements, the identity E and the permutation Pu, and hence is Abelian. (The same argument proves that the inversion group is Abelian.) For n >2 , we need only consider 63• For the permutations of 6) all occur in 6n and hence if these are not commutative, 6n cannot be Abelian.

255

HINTS FOR SOLVING THE EXERCISES

A, B, C PJ2P13(ABC) =P12(CBA) = (BCA) P13P12(ABC) =Pu (BAC) = (CAB),

PiJ

arranged in definite order, let be the Given three objects operator which exchanges the objects in the ith and jth places. Then

so that63, and hence

6" with n >3

as

well, are not Abelian.

IX.2. Carrying out the permutation is simply a relabelling of the variables of integration in the integral defining the scalar product, and this is hence invariant. We illustrate it for n = 2 :

(f(r., r2), g(r., r2» = (f(r2' rl), g{r2, rl» = (P12f{r., r2), Pug (r., rz».

IX.3. The proofS are closely analogous to that o f section 4 1 for spinless particles. We shall indicate the proof of one of the three results only, namely for the operators T, of the total function space. From the first of equations (42.6) and (42.8), it follows that for any wave function

{S, • • • }

{9'I(r)a+9'2{r)p} Tr(r, a) {9'I(r)a + 9'2(r)P} =r(t-Ir, 1-1 0) {9'1 (t-Ir) [ci a +C2P] + 9'2(1-lr}[Cl a +C4P]} = r(r, 0) {9'I(t-Ir)[Cl a+ czP] +9'2(t-Ir)[C3a+C4P]} =r(r, 0) T{9'I (r)a + 9'2(r)P} T-t T-I[9'l(r)a + 9'2(r)P] = 9'1 (tr)[Ct 'a +C2'P] + 9'2(tr)[Cl'a +C4'P] c/ a' {J': a' =cI' a+ c2'p, {J' =c3'a+ c/{J. S S •

The converse follows exactly as equation (41 .4) if we define the inverse operator appropriately. From equation (42Aa) this definition is

where the

are the coefficients of the solution of (42.2) for

and

IXA. For two particles is a good quantum number. So we still have a pure triplet state. L is no longer a good quantum number, = 1 to though J = 1 is. The L values which can combine with give J = 1 are L = 0, 1, 2. However, parity is a good quantum number, and for two particles with relative orbital angular momentum L the parity is - ) ) L ; i.e. the and states have opposite parity to the P state. Hence the ground state becomes a mixture of and states when tensor forces are present.

(

S

D

3S1

T-1V'1

V'I (V'., Ttpz) TV' 2

3Dl

is any vector in 9Rh then is in 9RI (since 9n 1 is in­ IX.5. If = variant under &). Hence, iftpz is any vector in9R2J then 0, i.e. =0. Hence Ttpz is orthogonal to and since is any vector of 9n., is orthogonal to the whole space But 9R is invariant and '1'2 a vector ofIDl, hence lies in 9R and therefore in 9R2•

T'P2

256

T-IV'h 'P V'h ( V'I z) IDlI•

HINTS FOR SOLVING THE EXERCISES

IX.G. The theorem is proved, if we can prove that any invariant manifold £In of the group & = {S, T, } is completely reducible into one-dimensional invariant manifolds. Let £In be such an n-dimensional manifold. Since the operators of OJ all commute with each other, they possess a set of orthonormal mutual eigenvectors tp h 11'2, • • • spanning the whole function space � : •

Stpi = S;9'b Ttpi = litp;,

•

•

(i = I , 2 ,

,

). We can decompose each vector tpi uniquely into two parts •

•

•

•

.

.

tpi = Ei + 7]i

such that Ei and 7]i lie in the invariant manifolds £In and � -£In (if Exercise IX.5) respectively. Hence the equation

TEi = tiEi + (Ii - T)7]i

implies that

TEi = liEi' T7]i = 'i7]i (since TEi must lie in £In, and ( Ii - T)7]i is orthogonal to £In), i.e. the

manifolds £In and � -£In are separately spanned by orthonormal sets of mutual eigenvectors of the group &, and each such vector defines a one-dimensional invariant subspace.

IX. 7. 1 and (J are pseudo-vectors under total rotations and inversion. Hence we at once find the selection rules :

(i) no change of parity; (ii) IJJ J' J = 0, ± 1 (not 0

-

-

0) .

Let us next again assume that total spin and orbital angular momenta by themselves are good quantum numbers. The orbital and spin parts lead to different selection rules for L and S. (J and 1 are of course invariant as far as rotations in orbital and spin space respec­ tively are concerned. Hence we obtain these selection rules : (a) orbital part of interaction :

.1S= O IJ L = O,

(not 0 _ 0) ;

± 1,

(b) spin part of interaction :

.1S =O, ± 1, (not 0 _ 0) . IJL = O.

IX.B.

We must show that

f-

- ilie

11"

ax

is proportional to (45.19) . I f E and and 11" then

II'

257

a

tpdr

E'

(H 37)

are the energies of the states

HINTS FOR SaLVINO THE EXERCISES

V2rp 2m(E - V)rp V21rp 2m{E' - V) rp , 2 fdr(rp'- xV2rp -rpxV2rp') - m(E' - E)frp'xrp - dr. ""

""

and hence

- 1i 2 -

"2

-;

=

,,2

(H 38)

The right-hand term is proportional to the electric dipole term (45. 19). We need only show that the left-hand term i s proportional t o equation (H 37). We can replace the left-hand term in (H 38) by

fdr{qi'V2(Xrp) - (Xrp)V2q?} - 2f"rpI!:dr.

(H 39)

(This

follows, since one easily proves from equations (A4), (A5) and (A6) in Appendix A, that

V2(Xrp) = xVlrp + 2:.)

The first term in (H 39) can be transformed into a surface integral, by Green's theorem, equation (Al l ) , over a surface at infinity, which vanishes. The second term is proportional to (H 37), proving our result. IX.9.

The operator

� P.(k) 4 c(k» 2.(�4 4-1 + x p

occurring in the matrix element of the quadrupole transitions has parity 1), and it was so chosen as to transform according to under rotations. Since

(+

il2

.1L12 X;&Ij = { ilt.i)22'+j + . . . + ilI2-jl

tr'\

tl'>

if ] =0

1. if] >"l"'

we at once have these selection rules (with the same notation as in section 45) :

(i) (ii)

L1] = O, ± l , ± 2 ;

)

(

nat o -+ 0 not 0 -+ 1 not 1 -+ 0

no change in parity,

If L and 8 are also, separately, good quantum numbers, one obtains the additional selection rules :

(iii)

L18 = 0 ;

(iv)

L1 L = L1]

[as under

(i)]

258

HINTS FOR SOLVING THE EXERCISES

CHAPTER

X.2. 0 a.t = [ (I.t

X.3. (46.9) . X.4.

c,.t =

X

(I.t] 0

- i"1 "2"3 ".t, from which (46. 1 8) at once follows, using

From the Dirac equation : 0", 0", ".t- + "4 � + M", = O. oX.t I tlXo

Taking complex conjugates, transposing and postmultiplying by

"4 :

0",* 01J'* -Y.tY 4 - . - Y42 + MV' *Y 4 = 0 18xO OXk

and since

and Vi = "'*"4, equation (46.21 ) follows.

".t Y4 = - " 4Yk

X.5. Differentiating (46.24b) and using the two Dirac equations (46.7) and (46.2 1 ) , (46.25) follows at once. X.6. (u" u,)

4

= L U,(4) ·U,(4) ; a- I

l1,u,

=

•

L

at 6-1

u,(0) · " 4Q6 U,<").

The desired results follow directly, using (46 . 1 4) and (48.4) . X.7. (i) For half the cases this follows from the orthogonality of the spinor parts (if Exercise X.6.) ; for the remaining cases it follows since the spatial parts of "'1 and 1J'2 are orthogonal to those of "1'3 and ( ii) = (u" u,) = Ep/mc2, from equations (48.4), i.e. tp, is normalized to a density of Ep/mc2 particles per unit volume. In the rest system of the particles this reduces to the usual density of 1 particle per unit volume. (iii) The reason for the normalization to a density (! = Ep/mc2 particles per unit volume is that this is Lorentz invariant. To see this, consider a volume V containing N particles having energy Ep. Under a Lorentz transformation to a new coordinate system having a relative velocity v = {Jc, V - V' = V( I - {J 2)�, Ep _ Ep' =Ep(I - {J2)4. Hence

tp4 '

(tp" tp,)

(!

,

_

N N _

- V' - V

V

_

N EP'

V' - V

Ep EP' EP' E-p - mc2 Ep - mc2' _

_

X.8. We must prove [ap, H] = 0 . This follows at once from equa­ tion (47.8), which gives [ aP, H] = 2CY 4P (Y " P) = 0. X•9 • If XA - xI'I = apAxA and xI'i - Xpll = bpp.xp.I, then x" _ X." = cpAx,., where C.A = bppa/
HINTS FOR SOLVING THE EXERCISES

Also IC,JI = Ib,,,l laIlJI = 1, showing that the composite transformation is also a proper Lorentz transformation. From C•.t = b."a".t and (49. 1 0) one sees that CII and C44 are real, Cf4 and C4� are pure imaginary. For in the former case the factors a and b are both real or both imaginary. In the latter case, one factor is always real but the other imaginary.

260

APPENDIX A :

VECTOR ANALYSIS

IN this appendix we give a brief summary of the definitions and theorems of vector analysis required in this book. The reader not familiar with these will find clear and simple expositions in the books by Abraham-Becker6 or Hague 7. (1) If A is a vector with components (Ax, A" A.c) we define its magnitude A = IA I = Y {A..2 + A.,2 + A.c2} . (A 1 ) (2) If A and B are vectors, components (A.., A" A,) and (B.., B" B,) respectively, we define (a) the scalar product AD =A..Bx +A,B, + A)1.c = AB cos 8, (A 2) where 8 is the angle between the vectors A and B ; in particular AD = 0, if A is orthogonal to B; (b) the vector product A A B as the vector C with components C.. = A,B, - A)1, C, = A.cB.. - A..B, C.c =A..B, - A,B... If i, j, k are unit vectors in the directions of the coordinate axes, so that A =iA.. +jA, + kA.c, and i2 =j2 = k2 = 1 , ij =jk =ki = O, we can write the vector product as a determinant j k C = AAB = i (A 3) A.. A, A, B" B, B, . C is a vector of magnitude AB sin 8, whose direction is normal to the plane containing A and B, such that A, B and C form a right-handed coordinate system. (3) Let tp(r), tp(r) , x(r), . . . be scalar fields (e.g . temperature, electrostatic potential, etc), and let F(r), G(r), be vector fields (e.g. fields of force), we then define (a) the gradient of tp(r) as the vector .8tp .8tp . 8tp grad tp = 1 x + +k (A 4) v t[ . 8 J(!y 8;; ' (b) the divergence of F(r) as the scalar 8F.. 8F, 8F.c " � t- : d'IV F = - + - + (A 5) " 8x 8y 8t '

}

•

2 61

•

•

APPENDIX A: VEOTOR ANALYSIS

(e)

the curl of F(r) the vector curl F (r) = i as

j a ay F,

a ax F.

If, in

k a aZ. F�

particular, F=grad rp, we obtain dlV grad rp=V2rp = aPxrp", + Piryrp2 + PaZ.rp" ; (A 6) ifF=rpG, then (A 7) div (rpG) = rp div G + G grad rp ; ifG=grad V' (A 8) div (rp grad V') =rpV"tp+ grad rp grad V'. (4) We state Gauss's theorem for a vector field F(r) : (A 9) fy(div F)dT= fSFAdS. Here is the surface bounding the volume V. The left-band integral is a volume integral over V, the right-hand integral is a surface integral over S, being a surface element of S and Fn the component of the vector F in the direction of the outward normal to dS. Ifwe take F= rp grad V', then FA = rpatp/an, where a/an denotes differ­ entiation with respect to the outward normal, and Gauss's theorem assumes the form (A 10) grad V + : 'vdr �grad 9' Ipd.. Interchanging rp and V' in this equation and subtracting the resulting equation from (A 10), we obtain (A 11) -,,!!}ISEquations (A 10) and (A 1 1 ) are known Green's theorem. •

S

-

dS

L'P� - f fJ'P!:

f

L('Pvz,p-"V"P)d" as

262

BIBLIOGRAPHY following bibliography, which is clearly in no way exhaustive, is merely a list of books helpful to the understanding of this book or suggestions for further reading.

THE

A.

Mathematics (eigenvalue problems, spedalfuTl&tions, etc) Fourier Series and Boundary Value Problems McGraw­

1 CHURCHILL, R. V.

Hill, 194 1 . (Eigenvalue problems, Fourier series and integrals, Legendre and Bessel functions.) 2 COURANT, R. Differential and Integral Calculus Blackie, 1942. (Vol 1 , chap. IX : Fourier series ; vol. 2, chap. IV : Fourier integrals.) .

3

MatheT1UJticalIy advQTI&ed comprehensive treatments are the following : COURANT, R. and HILBERT, D. Methoden der mathematischen Physik vol. 1 ,

Springer, 193 1 (and Interscience, 1 943) . There exists now an English translation, with additions, of this work : Methods of Mathematical Physics vol. 1 , Interscience, 1953. 4 SOMMERFELD, A. Partial Differential Equations in Physics Academic Press, 1949. 5 VON NEUMANN, ]. Mathematische Grundlagen der Qjlantenmechanik Springer, 1932 (and Dover, 1943) . English translation by R. T. BEYER, Prince­ ton University Press, 1 955. (A rigorous abstract treatment of the mathematical scheme of quantum mechanics and its interpretation.)

B. 6

7

Vector analysis

M. and BECKER, R. Classical Theory of Electricity and Magne­ tism Blackie, 1 946 (chapters I and 2). HAGUE, B. An Introduction to Vector Analysis Methuen, 1 950.

ABRAHAM,

C.

Q,uantum mechanics 8 BORN, M. Atomic Physics 3rd ed., Blackie, 1944. II HEITLER, W. Elementary Wave Mechanics Oxford University Press, 1 945.

(An elementary introduction to the physical interpretation of wave mechanics. ) 1 0 MOTT, N. F. and SNEDDON, I. N. Wave Mechanics and its Applications Oxford University Press, 1940. 1 1 PAULING, L. and WILSON, E. B. Introduction to Qpantum Mechanics McGraw-Hill, 1 935. 1 2 S CHIFF, L. I. Qpantum Mechanics McGraw-Hill, 1 949. (This book, which contains many applications, approaches quantum mechanics from a viewpoint rather similar to our own.) 1 3 SOMMERFELD, A. Atombau und Spektrallinien vol. 2, 2nd ed., F. Vieweg, Brunswick, 1 944.

263

BmLIOORAPHY

1 4 SOMMERFELD, A.

Wave Mechamcs Methuen, 1930. (This is the English translation of the first edition ( 1 929) of Ref. 1 3. Hence 1 3 is much more up-to-date and gives a better and more detailed treatment of many topics.) TIllfollowing are more advanced treatises on quantum mechanics :

1 5 DIRAC, P. A.

M. TIll Principles oj Qpantum Mechanics 3rd ed., Oxford University Press, 1 947. 1 6 HEISENBERG, W. Physical Prindp/es oj the Q;uzntum Theory University of Chicago Press, 1 930. 1 7 PAUU, W. Die allgemtinen Pri�ipien tier Q;uzntenmechanik, lIandbuch dtr Physik vol. 24, part 1 , Springer, 1 933. 18 VON NEUMANN, J. See reference 5.

Applications oj quantum mechanics to particularfields 1 9 BETHE , H. A. and MORRISON, P. Elementary Nuclear Theory, 2nd ed., Wiley, 1 956.

D.

J. M., and WEISSKOPF, V. F. Tlltoretical Nuclear Physics Wiley, 1 952. 11 CONDON, E. U. and SHORTLEY, G. H. Theory oj A tomic Spectra Cam­ bridge University Press, 1 95 1 . 1 1 MOTT, N. F . and MASSEY, H. S. W. Theory oj Atomic Collisions Oxford University Press, 1 949. 2 3 VAN DER WAERDEN, B. L. Die gruppentheoretische Methode in der Qpanten­ mechanik Springer, 1 932 (and Edwards, ( 944) . 1 4 WENTZEL, G. Wellenmechanik dtr Stoss- und StrahlungsIJorgiinge, Handbuch der Physik vol. 24, part 1 , Springer, 1 933 (and Edwards, (943). 25 MASSEY, H. S. W. Theory oj Atomic Collisiolls, Handbuch der Physik vol. xxxvi (Atoms II), Springer, 1 956. 10 BLATT,

264

INDEX· Adjoint, 72-73, 95 Allowed transitions, 201 Angular momentum, 14, 25, se, also Spin addition of, 27, 128-130, 245-247 classically, 54 conservation of, 8 1 , 1 82 operators, 34, 54, 1 04, lOS, 229 commutation relations for, 79, 84, 89, 104, lOS, 130, 237-238, 239, 244-245 eigenfunctions and eigenvalues of, 5455, 67-ti8, 104, 105-108, 130 .total, 1 16, 1 2 1 Associated Legendre functions, 50-53 se, Matrix representation, Group representation Bessel functions, 166-168 Bohr magneton, 141 Born approximation, 34, 160, 1 75, 252 Boson, 127 Bra vector, 102

Basis,

Centre-of-mnss coordinate system, 35 Clebsch-Gordan coefficient, 120 Commutation relations, 79, 84, 86, 89, 104 Commutator, 72, 77 Commuting observables (or operators), 7778 complete set of, 80, 8 1 , 1 15-1 16 Compatibility of several observables, 7578, 82 Complete orthonormal set, 4, 1 0 Compton wave-length, 204 Constant of the motion, 69, 8 1 , 1 0 1 , 1 151 16, 126, 182, 185, 186 Coulomb scattering, 1 51-153 Cross-section, 144, 145, St, also Scattering, Born approximation Current-density vector, 3 1 , 35, 209 Delta function, 1 7-19, 1 45 Density of states, 42-43, 148 Dirac equation for free particle, 46 for particle in electro-magnetic field, 2 10-21 2 negative energy states, 2 1 2, 2 16 non-relativistic limit, 2 1 1-212, 2 1 3-215 plane-wave solutions, 48 relativistic invariance, 49 Eigenfunctions, 6, 12, 27, 32, 62-ti3 complete set ofsimultancous, 75-78, 1 00 completeness of, 12-13, 34, 62-ti3 degenerate, 13-14, 63 expansion in, 13, 1 7, 34, 62-ti3 interpretation of, 27, 55, 66-68 linearly independent, 13 orthogonality of, 9, 33-34, 40, 63, 100, 102, 238-239 •

Eigenvalue, 6, 12, 27, 32, 62-63 degenerate, 1 3-14, 100 interpretation of, 27, 55, 63, 66-68 problem, 6, 12, 27, 32 in matrix form, 97-100, 103, 239-240 reality of, 33, 63, 100, 102, 238 spectrum continuous, 4, 17, 62 discrete, 14, 32-33, 62 mixed, I6 Electro-magnetic transitions, 197-200 Electron magnetic moment, 26, 47 spin, 26, 47, 204 Exchange integral, 139 operators, 126, 184 Exclusion principle, 127 Expectation value, 63, 66-68, 7 1 , 8 1 , 1 15 Fermion, 127 Forbidden traruitions, 201 Four-vector, 220 Fourier integral, IS, 19, 20, 87 series, 7, 20 Function even and odd, 8, 39 length of, 9 magnitude of, 9 space, 10, 1 1 , 60, 76-77 complex, I I dimensionality of, 10 manifold of, 13 product, 1 1 3-1 15, 1 16-1 1 7, 183 subspace of, 1 3 Gauss's theorem, 262 Green's theorem, 262 Ground state of alpha particle, 129-130 deuteron, 201, 256 lHe, 130 helium atom, 57, 136, 234 hydrogen atom, 57, 233 Group, 1 78 Abelian, 1 79, 189, 201 , 255-256 axial rotation, 1 78-1 79, 189-190 inversion (or reflection), 1 79, 1 88 Lorentz, 2 1 8 permutation, 1 79, 1 89, 201 , 255 representation, 43, 1 87 basis for, 187 decomposition of, 187 equivalent, 187 irTeducible, 187, 202 product, 189 reducible, 187-188 reduction of, 1 87

The DWDbe.. ill heavy type refer 10 the IeCIlcma

265

INDEX Group-eonld. rotation, 40, 1 79, 189 transfonnations (or operators), of, 39, 178 invariants of, 41, 186 unitary, 180, 201 256 velocity, 30, 35, 23 0

Measurement, 65-70 complete (maximal), 80 one observable, of, 65-68, 70-71 several commuting observables, of, 19 several non-commuting observables, of, 20 state of system, of, 79-80 state of system after, 68-70

Hamiltonian function, 27, 59, 64 invariancc under in�ion, 56, 182, 185, 186, 1 93 pennutations, 125, 137, 1 77, 194 rotations, 105, 1 25, 1 77, 181, 182, 185, 186, 194 spherically symmetric, 47, 54, 68, 104, 105, 1 15-1 16, 125, 140, l ol l , 181, 182 Hamilton's equations, 59, 88, 234 Hankel funCtiOns, 166-168 Heisenberg, Ste also Uncertainty principle equations of motion, 101, 103, 240 picture or representation, 100-101 Hermite polynomials, 45-46 HenocUtian, 62-63, 72-73, 95 Hilbert apacc, 1 1, 60, 76-77

Non-commuting o£Cl'll tors, let! Operators Nonn, �, 41 5, 9, 12 Nonnalized, 2, 9 Nucleon, 129 Observable, 25-28, 61-63 Operator, 26-28, 6 1--63 commuting with the Hamiltonian, 80-81, 101, 1 15-1 16, 126 equations of motion of, set Heisenberg inverse, 74 unit, 73 Operators commutability of, 77-78, 100 complete set of commuting, 80 functions of, 18, 78 non-commuting, 82 Optical theorem, 1 72 Orthogonal, 3, 4, 9 Orthonormal, 3, 9

Identical particles, 28, 130, 137-140, 160161 , 247-248 Inner product, 2, 4, 9, 1 1 , 9 1 , 1 14 invariance under unitary transformations, 73 Intercombination lines, 198 Invariancc, Stl Hamiltonian function Invariants, Stl Group of transformations Inversion, 39, 56, 177, 182, 185, 2 1 7, 23 1 Ket vector, 102 Klein-Gordon equation, 203 Kronecker 8 symbol, 3

Laboratory coordinate system, 35 Laporte's rule, 198 Legendre polynomials, 50-51 Life-time, 86 Linear independence, 4, I I Lorentz transformation, 49

Magnetic moment, 47, 1 1 1-1 12 Manifold, 4, 13, 76-77, 79-80, 98, 100 decomposition of, 187 invariant, 186 irreducible, 187 reducible, 187, 188 reduction of, 1 87 Matrix, 92 diagonal, 94, 102 eigenvectors and eigenvalues of, 97-100, 102 multiplication, 9 1 , 93-94representation, 21, see also Group representation basis of, 90 change of basis of, 95-97 eigenvalue problem in, 97-100 of an operator, 92 of a vector, 9 1 transposed, 95 unit, 94Mean square deviation, 67-68, 82

Parity, 39-40, 45, 5 1 , 53, 56, 88, 182, 185, 193, 230-231 . Partial wave analysis, 38 Pauli matrices, su Spin matrices principle, set! Exclusion principle Permutation operator,ste Exchange operators Perturbation theory time-dependent, 37 time-independent, Chapter VII ( 1 3 1 -141) applications, 31 first-order degenerate, 30 first-order non-degenerate, 29 second-order non-degenerate, l ol l , 249250 Phase factor, 28, 60, 61, 108, 109, 120, 133, 243-244, 250 vclocity, 29 Phase-mift, 1 7 1 Planck's constant, 22, 23 Plane wave, 10, 23, 28, 29 Positron, 2 16 Probability current, 3 1 , 35 interpretation of quantum mechanics, 28-29, 65-68, 70-7 1 , 1 15 Pseudo-scalar, 221 Pseudo-vector, 182-183, 221, 222 Quantum number angular momentum, 55 good, 182, 198 magnetic, 55, 141 Radiative transitions, 197-200 Reduced mass, 1 43, 157, 232 Relativistic invariancc, 49

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