Quantum mechanics for mathematicians

o

. .

stand for the gradient of a function f at a point q E �n with Cartesian coordinates ( q1, ... , qn). We denote by A• (M)

E9 Ak (M) n

=

k=O

the graded algebra of smooth differential forms on M with respect to the wedge product, and by d the de Rham differential- a graded derivation of A• (M) of degree 1 such that df is a differential of a function f E A0 (M) = C00 (M). Let Vect(M) be the Lie algebra of smooth vector fields on M with the Lie bracket [, ], given by a commutator of vector fields. For X E Vect (M) we denote by £x and ix, respectively, the Lie derivative along X and the inner product with X. The Lie derivative is a degree 0 derivation of A• (M) which commutes with d and satisfies £x(f) = X(f) for f E A0(M) , and the inner product is a degree -1 derivation of A• (M) satisfying ix(f) = 0 -

3

4

1 . Classical Mechanics

and ix(df) = X(f) for f E A0 (M). They satisfy Cartan formulas

£x =ixod+doix = (d+ix)2, i[x ,Y] = £x oiy- iy £x. For a smooth mapping of manifolds f : M ---; N we denote by f* : TM ---; TN and f* : T * N ---; T* M, respectively, the induced mappings on tan­ o

gent and cotangent bundles. Other notations, including those traditional for classical mechanics, will be introduced in the main text. 1.

Lagrangian Mechanics

1 . 1 . Generalized coordinates. Classical mechanics describes systems of finitely many interacting particles 1 . A system is called closed if its particles do not interact with the outside material bodies. The position of a system in space is specified by positions of its particles and defines a point in a smooth, finite-dimensional manifold M, the configuration space of a system. Coordinates on M are called generalized coordinates of a system, and the dimension n dim M is called the number ofdegrees of freedom 2 . The state of a system at any instant of time is described by a point q E M and by a tangent vector v E Tq M at this point. The basic principle of classical mechanics is the Newton-Laplace determinacy principle which asserts that a state of a system at a given instant completely determines its motion at all times t (in the future and in the past). The motion is described by the classical trajectory- a path "Y(t) in the configuration space M . In generalized coordinates ')'(t) = (q1 ( t ) , . . . , q n (t) ) , and corresponding =

dqi

derivatives qi = di are called generalized velocities. The Newton-Laplace principle is a fundamental experimental fact confirmed by our perception 2 i of everyday experiences. It implies that generalized accelerations iji are uniquely defined by generalized coordinates qi and generalized velocities qi, so that classical trajectories satisfy a system of second order ordinary differential equations, called equations of motion. In the next section we formulate the most general principle governing the motion of mechanical systems. =

�t;

1 . 2 . The principle of the least action. A Lagrangian system on a con­ figuration space M is defined by a smooth, real-valued function L on T M x lR - the direct product of a tangent bundle TM of M and the time axis3 called the Lagrangian function (or simply, Lagrangian). The motion of a

1 A particle is a material body whose dimensions may be neglected in describing i ts motion. 2 Systems with infinitely many degrees of freedom are described by classical field theory. 3It follows from the Newton-Laplace principle that L could depend only on generalized co­ ordinates and velocities, and on time.

1.

Lagrangian Mechanics

5

Lagrangian system ( M, L) is described by the principle of the least action in the configuration space ( or Hamilton's principle) , formulated as follows. Let P(M)��:�� b: [to, tt] M ; r(to) = qo, 1(t1) = qt} be the space of smooth parametrized paths in M connecting points qo and q 1 . The path space P(M) = P(M)��:�� is an infinite-dimensional real Frechet manifold, and the tangent space TyP(M) to P(M) at 1 E P(M) consists of all smooth vector fields along the path 1 in M which vanish at the endpoints q0 and q 1 . A smooth path r in P(M), passing through 1 E P(M), is called a variation with fixed ends of the path 1(t ) in M. A variation r is a family lc(t) = f(t, E ) of paths in M given by a smooth map r: [to, tl] X [-Eo, Eo] M such that r(t, 0) r(t) for to :::; t :::; tl and r(to, c ) = Qo, r(tl, c) = Ql for -Eo ::; E ::; Eo. The tangent vector �

=

�

=

Or=

�

ar

uE

J c=O

E

T'YP(M)

corresponding to a variation lc(t) is traditionally called an infinitesimal variation. Explicitly, o1(t) = r * ( gc ) (t, 0) E T'Y(t)M , to :S t :S t1,

where £ is a tangent vector to the interval [-Eo, Eo] at 0. Finally, a tangential lift of a path 1 : [to , t 1 ] � M is the path 1' : [to, t1] T M defined by r'(t) = '* (-ft) E T'Y(t)M, to ::; t::; t1, where %t is a tangent vector to [to, t1] at t. In other words, r'(t) is the velocity vector of a path 1(t) at time t. �

Definition. The action functional S :

(M, L) is defined by

S(r)

P(M) � lR of a Lagrangian system

ltl L( '(t ) , t) dt . r

=

to Principle of the Least Action ( Hamilton's principle ) . A path 1 E PM describes the motion of a Lagrangian system ( M, L) between the position qo E M at time to and the position Ql E M at time t1 if and only if it is a critical point of the action functional S , d S(lc) 0 dE c=O for all variations lc(t) of r(t) with fixed ends.

I

=

The critical points of the action functional are called extremals and the principle of the least action states that a Lagrangian system (M, L)

1.

6

Classical Mechanics

moves along the extremals4. The extremals are characterized by equations of motion - a system of second order differential equations in local coordinates on TM. The equations of motion have the most elegant form for the following choice of local coordinates on TM. Definition. Let ( U, <.p) be a coordinate chart on M with local coordinates

q = (q1,... , qn). Coordinates (q, v) = (q1,...,qn,v1, ... ,vn) on a chart TU on TM, where v = (v1, ... ,vn) are coordinates in the fiber

corresponding to the basis dinates.

0�1, � . . ., {)

n

for TqM, are called standard coor­

Standard coordinates are Cartesian coordinates on <.p* (TU) C Tll�n and have the property that for (q, v) E TU and f E C00(U),

�n x �n

v( f )

n iof of =L = va. V {ii q q i=1

Let (U, c.p) and (U', c.p' ) be coordinate charts on M with the transition func­ tions F = (F1 , . , pn) = c.p' c.p-1 : c.p(U n U') c.p'(U n U'), and let (q, v ) and ( q' , v') , respectively, be the standard coordinates on TU and TU'. We .

o

.

----t

{

()pi

have q' = F(q) and v' = F*(q)v, where F* (q) = � (q) J uq

}

n

is a matrix-

.. 1 valued function on c.p(U n U'). Thus "vertical" coordinates v = (v1, ... , vn) t,J=

----t

in the fibers of T M M transform like components of a tangent vector on M under the change of coordinates on M. The tangential lift 1' ( t) of a path 1(t) in M in standard coordinates on TU is (q(t) , q(t ) ) = (q1 (t) , . . . ,qn(t), t/(t), . . . , qn(t) ) , where the dot stands for the time derivative, so that L (T'(t) , t) = L ( q ( t), q (t), t).

Following a centuries long tradition5, we will usually denote standard coor­ dinates by

(q, q) = (q1, ...,qn,q1, ... ,qn),

where the dot does not stand for the time derivative. Since we only con­ sider paths in TM that are tangential lifts of paths in M, there will be no confusion6 . 4 The principle of the least action does not state that an extremal connecting points qo and a min i mum of S, nor that such an extremal is unique. It also does not state that any two points can be connected by an extremal. 5 Used in all texts on classical mechanics and theoretical physics. 6 We reserve the notation ( q( t), v( t)) for general paths in T M.

q1

is

1. Lagrangian Mechanics

7

L)

Theorem 1 . 1 . The equations of motion of a Lagrangian system (M, in standard coordinates on TM are given by the Euler-Lagrange equations

��

(q(t), q( t) , t )-

! (�� (q(t), q(t), t))

=

0.

Proof. Suppose first that an extremal r(t) lies in a coordinate chart U of

M. Then a simple computation in standard coordinates, using integration

by parts, gives 0

=

= =

=

I S(re) : I e=O irtotlL ) ) t irtotl (�� oqi + �q� oil) n t 8L 8L) oqtdt. +I: 8L · 1 t1 . I: 1 1 (fii 8 q to o q - 'dt 8 ·i i= l i= l t d

d c e=O

(q(t, c , q(t, c , t) dt

c

i=l

dt

q

n

d

·ioqt

q

The second sum in the last line vanishes due to the property oqi(to) = oqi(t1) = 0, i = 1, ... , n. The first sum is zero for arbitrary smooth functions oqi on the interval [t0, t1) which vanish at the endpoints. This implies that for each term in the sum the integrand is identically zero,

��

(q(t), q(t), t)-

:t (�� (q(t), q(t), t) )

=

0,

i = 1, .

. . , n.

Since the restriction of an extremal of the action functional S to a coordi­ nate chart on M is again an extremal, each extremal in standard coordinates 0 on T M satisfies Euler-Lagrange equations. Remark. In calculus of variations, the directional derivative of a functional S with respect to a tangent vector V E T1P (M)- the Gato derivative­

is defined by

8v S =

d

I

d c e=O

S(re) ,

where re is a path in P(M) with a tangent vector V at 'Yo = r· The result of the above computation ( when 1 lies in a coordinate chart U C M) can be written as d f t1 t ( �� - d �� ) ( q (t) , q(t), t)vi(t)dt ito i=l uq uq = iftot1 (88Lq - ddt88Lq ) (q(t),q(. t),t)v(t)dt.

8vS = (1. 1)

t

1. Classical Mechanics

8 Here V ( t) =

t vi (t) 88

qi

i=l

is a vector field along the path

1

in M. Formula

(1 . 1 ) is called the formula for the first v ari ation of the action with fixed ends. The principle of the least action is a statement that <5vS(r) = 0 for

all V

E

TyP(M).

, t1] ---+ P(M) = {r : [to -M parametrized paths in M. The tangent space TyP(M) to } of all smooth --P(M) at 1 E P(M) is the space of all smooth vector fields along the path 1 in M ( no condition at the endpoints ) . The computation in the proof of Theorem 1.1 yields the following formula for the first v ari ation of the action Remark. It is also convenient to consider a space

with free ends :

<5v8 =

(1.2)

£1- -d -;::;--:-

Ldt on TM

x

JR.

where i'

=

Problem

d

t+

over the 1-chain i' on TM x JR.,

( )=

Sr

JR.

v

8L lt1 -;::;-;- V

uq

to

Show that the action functional is given by the evaluation of the

P roblem 1 . 1 . 1-form

d

ltl (8L t 8L) to uq uq

-

j Ldt, ;y

{(r (t) , t); to :::; t:::; tt} and Ldt (w, c gt ) '

1.2. Let

f E C00(M).

= cL(q, v ) ,

wE

T(q,v)TM,

Show that Lagrangian systems

(M, L)

c E

and

( M, L+df) (where df is a fibre-wise linear function on TM) have the same equations

of motion.

Give examples of Lagrangian systems such that an extremal con­

Problem 1.3.

()

necting two given points i is not a local minimum;

exist.

of

(ii ) is not unique; (iii) does not

For 'Y an extremal of the action functional S, the

Problem 1.4.

S is defined by

2 S= 8v1V2

{)2 I

� Cl C2

Cl =<2=0

second variation

S(r<1.<2),

w here r<1,e2 is a smooth two-parameter family of paths in

'Y<1,o and 'Yo,e2

M such that the paths = 'Y E P(M) have tangent vectors V1 and For a Lagrangian system (M, L) find the second variation of Sand

in

V2, respectively.

P(M)

at the point 'Yo,o

verify that for given V1 and V2 it does not depend on the choice of 'Y<1 ,<2•

1 . 3 . Examples of Lagrangian systems. To describe a mechanical phe­

nomena it is necessary to choose a frame of reference. The properties of the sp ace-time where the motion takes place depend on this choice. The space-time is characterized by the following postulates 7 . 7 Strictly speaking, these postulates are valid only in the non-relativistic limit of special relativity, when the speed of light in the vacuum is assumed to be infinite.

1. Lagrangian Mechanics

9

Newtonian Space-Time. The space is a three-dimensional affine Eucli­ dean space E3. A choice of the origin 0 E E3 - a reference point establishes the isomorphism E3 � JR3 , where the vector space JR3 carries the Euclidean inner product and has a fixed orientation. The time is one­ dimensional - a time axis lR - and the space-time is a direct product E 3 x R An inertial reference frame is a coordinate system with respect to the origin 0 E E3 , initial time to, and an orthonormal basis in JR3 . In an inertial frame the space is homogeneous and isotropic and the time is homogeneous . The laws of motion are invariant with respect to the transformations r

� g

· r + ro,

t

�t

+

to,

where r, ro E R3 and g is an orthogonal linear transformation in R3 . The time in classical mechanics is absolute.

The Galilean group is the group of all affine transformations of E3 x lR which preserve time intervals and which for every t E lR are isometries in E3 . Every Galilean transformation is a composition of rotation, space-time translation, and a transformation r

� r + vt, t � t,

where v E R3 . Any two inertial frames are related by a Galilean transfor­ mation. Galileo's Relativity Principle. The laws of motion are invariant with

respect to the Galilean group.

These postulates impose restrictions on Lagrangians of mechanical sys­ tems. Thus it follows from the first postulate that the Lagrangian L of a closed system does not explicitly depend on time. Physical systems are de­ scribed by special Lagrangians, in agreement with the experimental facts about the motion of material bodies. Example 1 . 1 ( Free particle ) . The configuration space for a free particle is M = JR3 , and it can be deduced from Galileo's relativity principle that the

Lagrangian for a free particle is

L

=

!mr2•

Here m > 08 is the mass of a particle and i-2 = li-1 2 is the length square of the velocity vector r E TrR3 � JR 3 . Euler-Lagrange equations give Newton's law of inertia,

r = 0. 80therwise the action functional is not bounded from below.

1.

10

Classical Mechanics

Example 1 . 2 ( Interacting particles ) . A closed system of N interacting par­ ticles in JR3 with masses m1, . . . , m N is described by a configuration space

M = JR3N = JR3 X

... X

JR3

with a position vector r = ( r1, ... , rN) , where ra E JR3 is the position vector of the a-th particle, a = 1, ... , N. It is found that the Lagrangian is given by

L

where

=

N

N

L � mar� - V( r) T - V, =

a=1

N

·2 T = ""1 L..J 2ma ra a=1

is called kinetic energy of a system and V( r) is potential energy. The Euler­ Lagrange equations give Newton's equations F.

where

a

__ 8V -

Ora

is the force on the a-th particle, a = 1 , .. . , N. Forces of this form are called cons ervative. It follows from homogeneity of space that potential en­ ergy V( r) of a closed system of N interacting particles with conservative forces depends only on relative positions of the particles, which leads to the equation a=1

In particular, for a closed system of two particles F1 + F2 0, which is the equality of action and reaction forces, also called Newton's third law. The potential energy of a closed system with only pair-wise interaction between the particles has the form V( r)

L

=

1�a
=

Vab( ra- rb)·

It follows from the isotropy of space that V( r) depends only on relative distances between the particles, so that the Lagrangian of a closed system of N particles with pair-wise interaction has the form N L

= L..J

""1

a=1

·2

2mara

1. Lagrangian Mechanics

11

T

If the potential energy V(r) is a homogeneous function of degree p , = VV(r) , then the average values and V of kinetic energy and potential energy over a closed trajectory are related by the virial theorem

V(.Ar)

2T

( 1 .3)

= pV.

Indeed, let be a periodic trajectory with period 0, i.e. , ) r(O) r . Using integration by parts, Newton's equations, and Euler's homogeneous function theorem, we get r(t) = ( T)

T >

r(O)

= r (T ,

Example 1 . 3 (Universal gravitation). According to Newton's law of gravi­ tation, the potential energy of the gravitational force between two particles

with masses

ma and mb is

V(ra-rb)=-G I

mamb Ta-

rb I'

where G is the gravitational constant. The configuration space of N particles with gravitational interaction is M {(r1, . .. ,rN) E lR3N : =I rb for a =I b, a, b = 1 , ... ,N}. Example 1 .4 (Particle in an external potential field) . Here M = JR3 and

=

L

=

Ta !mr2-

V(r, t),

where potential energy can explicitly depend on time. Equations of motion are Newton ' s equations = F =-

mr

��·

If V = V(lrl) is a function only of the distance lrl, the potential field is called central. Example 1 . 5 (Charged particle in electromagnetic field9 ) . Consider a par­ ticle of chargee and mass JR3 moving in a time-independent electromag­ netic field with scalar and vector potentials
min

L

= -mr22 -

+

e

(rA ) -c-

-
A

'

9This is a non-relativistic limit of an example in classical electrodynamics.

12

1. Classical Mechanics

where c is the speed of light. The Euler-Lagrange equations give Newton's equations with the Lorentz force,

( � B) ,

mr =e E +

where

x

x

is the cross-product of vectors in JR3 , and E = - Br.p

and B =curl A 8r are electric and magnetic 1 0 fields, respectively.

Example 1 . 6 (Small oscillations). Consider a particle of mass m with n de­ grees of freedom moving in a potential field V(q), and suppose that potential energy U has a minimum at q 0. Expanding V(q) in Taylor series around 0 and keeping only quadratic terms, one obtains a Lagrangian system which describes small oscillations from equilibrium. Explicitly, = !mq2- Vo(q), where Vo is a positive-definite quadratic form on JR.n given by 1 � 82V .. Vo(q) = 2 L..J 8 i8 j (O)q�qJ . �. J.=1 q q . Since every quadratic form can be diagonalized by an orthogonal trans­ formation, we can assume from the very beginning that coordinates q = (q1, . , qn ) are chosen so that Vo (q) is diagonal and =

L

..

( 1 .4)

n

L = !m(q2- Lwf(qi)2),

i=1 where w1, . . . , Wn > 0. Such coordinates q are called normal coordinates. In normal coordinates Euler-Lagrange equations take the form iji + wfqi = 0, i = 1, . , n, and describe n decoupled (i.e. , non-interacting) harmonic oscillators with fre quencies w1, ... , Wn.

..

Example 1 . 7 (Free particle on a Riemannian manifold) . Let (M, ds2) be a Riemannian manifold with the Riemannian metric ds2. In local coordinates x 1 , ... ,x n on M , ds2 = gJ.Lv(x)dxi-Ldxv,

where following tradition we assume the summation over repeated indices. The Lagrangian of a free particle on M is L(v) = �(v,v) �llvll2, v E TM, lDNotation B = rot A

=

is also used.

1.

Lagrangian Mechanics

13

where ( ) stands for the inner product in fibers of T M given by the Rie­ mannian metric. The corresponding functional = 1 {t1 1 { t1 ito is called the action functional in Riemannian geometry. The Euler-Lagrange equations are ,

9p,v(x)iPxvdt S(r) = i o l r' (t) l 2 dt t gp,.>, X·p,·X>- ' 9p,vX"11- + aax>-gp,v X·p,·X>- - 2�aaxv and after multiplying by the inverse metric tensor guv and summation over _

v

they take the form

rY = 1,

. . .

, n,

gl/A - agP,>-l/ ) ruP,l/ = �2 gUA ( aaxl/gp,A aaxilax

where

+

are Christoffel's symbols. The Euler-Lagrange equations of a free particle moving on a Riemannian manifold are geodesic equations. Let V' be the Levi-Civita connection - the metric connection in the tangent bundle TM and let V'� be a covariant derivative with respect to the vector field� E Vect ( M ) . Explicitly,

-

(\7�17)11- = (�77uxv11- + r�.>-17>- ) ev, where e = e(x) uxllf:la '17 1711-(x) uxllf:la . For a path 1( t) = ( xll- ( t)) denote by V' a covariant derivative along d17 11- ( t) r�.>,(l(t))xl/ (t)17A (t ) , where 17 1711-(t) f:la (V'..y77)11-( t) = dt uxll=

1,

..y

+

=

is a vector field along I· Formula ( 1 . 1 ) can now be written in an invariant form { t 1 (V'..y"y,

5S - i o 61)dt, t which is known as the formula for the first variation of the action in Rie­ =

mannian geometry.

Example 1 . 8 ( The rigid body ) . The configuration space of a rigid body in �3 with a fixed point is a Lie group G = S0(3) of orientation preserving orthogonal linear transformations in �3. Every left-invariant Riemannian metric ( ) on G defines a Lagrangian L : TG � by L(v) =�(v,v), vETG. According to the previous example, equations of motion of a rigid body are geodesic equations on G with respect to the Riemannian metric ( ) . Let g = .so(3) be the Lie algebra of G. A velocity vector g E T9G defines the by n (Lg ) g E g, where L9: G---+ G are angular ,

velocity of the body

---+

=

-

,

l

*

1.

14

Classical Mechanics

left translations on G. In terms of angular velocity, the Lagrangian takes the form L = !(O, O)e, where ( , )e is an inner product on g TeG given by the Riemannian metric ( , ) . The Lie algebra g - the Lie algebra of 3 x 3 skew-symmetric matrices­ has the invariant inner product (u, v)o = -! Truv ( the Killing form ) , so that (0, O)e (A· 0, O)o for some symmetric linear operator A : g g which is positive-definite with respect to the Killing form. Such a linear operator A is called the inertia tensor of the body. The principal axes of inertia of the body are orthonormal eigenvectors e1, e2, e3 of A; corresponding eigenvalues h, h, h are called the principal moments oj inertia. Setting 0 01 e 1 + 02e2+ 03e3n, we get !(hOi+ 120� + hO�). In this parametrization, the Euler-Lagrange equations become Euler's equa­ tions h01 = (12- h)0203, h02 (h- h)0103, h03 (h- h)0102. Euler's equations describe the rotation of a free rigid body around a fixed point. In the system of coordinates with axes which are the principal axes of inertia, principal moments of inertia of the body are h, h, h. =

=

----+

=

L

=

=

=

1 . 5 . Determine the motion of a charged particle in a constant uniform magnetic field. Show that if the initial velocity v3 0 in the z-axis (taken in the cmvt r direction of the field, B = (0, 0, B)), the trajectories are circles of radii = eB in a plane perpendicular to the field (the xy-plane ) , where Vt = Jvi + v� is the initial velocity in the xy-plane. The centers ( x0 , y0 ) of circles are given by

Problem

=

xo =

cmv1 x + , eB

cmv2

Yo = - --;]3 + y,

where ( x, y) are points on a circle of radius

r.

Problem 1 . 6 . Show that the Euler-Lagrange equations for the Lagr angian L(v)

llvll, v E TM ,

=

coincide with the geode sic equations written with respect to a con­ stant multiple of the natural parameter.

1. 7. Prove that for a particle in a potential field, discussed in Example 1.4, the second variation of the action functional, defined in Problem 1.4 , is given by

Problem

1 1 This establishes the Lie algebra isomorphism by the cross-product .

g

�

IR3, where the Lie bracket in IR3 is given

1. Lagrangian Mechanics where 6 1 r, 62 r E T-y PIR 3 , r I is the

15

=

r (t) i s the classical trajectory, .:1

�2� (t)

3 x 3 identity matrix, and

ur

=

{ ; 8°

2

(r(t))

linear differential operator .:1, acting on vector fields along operator.

Ta Tb

=

}3

!2 �:� (t) , 2 A second-order

-m .

a , b= l /, is called

I -

the Jacobi

Problem 1 . 8. Find normal coordinates and frequencies for the Lagrangian sys­ tem considered in Example 1 .6 with V0 ( q ) � a2 L: 7= 1 (qi + 1 - qi ) 2 , where qn + l = q 1 . =

Problem 1 .9 . Prove that the second variation of the action functional in Rie­ mannian geometry is given by

t� (.:J(6n) , 621) dt. ito - \7 � - R('Y, · h is the Jacobi

62 8

=

Here 6n, 6zr E T-yPM, .:1 = operator, and R is a curvature operator - a fibre-wise linear mapping R : TM 0 TM ---+ End ( TM) of vector bundles, defined by R(� , ry) \7'7 \?e - 'Ve\7'7 + Y' [e,7JJ : TM ---+ TM, where �' ry E Vect ( M ) . Problem 1 . 10. Choosing the principal axes of inertia as a basis in JR3 , show =

that the Lie algebra isomorphism

�

g

JR3 is given by

JR3 .

g 3

(

�3 -�3 _g: 0 01

- 02

)

f---'>

Problem 1 . 1 1 . Show that for every symmetric A E End g there exists a sym­ metric 3 x 3 matrix A such that A f2 A n + f2A, and find A for diagonal A. Problem 1 . 12. Derive Euler's equations for a rigid body. ( Hint: Use that L = - � Tr Af22 , where n = g - 1 9 and 6n = -g-16g f2 + g- 1 6g, and obtain the Euler­ ( nl , fh , fh)

E

·

=

Lagrange equations in the matrix form Afl + flA = Af22 - f22 A.)

1 .4. Symmetries and Noether's theorem. To describe the motion of a mechanical system one needs to solve the corresponding Euler-Lagrange equations - a system of second order ordinary differential equations for the generalized coordinates. This could be a very difficult problem. There­ fore of particular interest are those functions of generalized coordinates and velocities which remain constant during the motion. Definition. A smooth function T M lR is called the integral of motion (first integral, or conservation law) for a Lagrangian system ( M, L ) if

I

:

--+

:/(r'(t )) = 0

for all extremals 1 of the action functional. Definition. The energy of a Lagrangian system ( M, L) is a function E on T M x lR defined in standard coordinates on T M by E( q , q,

qi �� ( q , q, t ) - L ( q , q , t ). t) = t =

i l

q

1. Classical Mechanics

16

Lemma 1 . 1 . The energy

R

E q �� - L is a well- defined function on TM =

x

Proof. Let (U, r.p) and (U', r.p' ) be coordinate charts on M with the transition functions = = r.p' o r.p- 1 : r.p U n U') --+ r.p' (U n U') . Corre­ sponding standard coordinates are related by = and and and (see Section 1 . 2 ) . We have = =

( F (F1, . . . , Fn) q (q, ) (q' , q' ) q' F(q) q' F* (q)q dq1 F*(q)dq dq' G(q , q)dq + F.(q)dq (for some matrix-valued function G(q, q)), so that 8L dt dL = o8Lq' dq + o8Lq' dq + Ft ( o8L F*(q) + o8Lq G(q , q). ) dq + o8L F*(q)dq. + at8L dt q' ' q' 8L dq + 8L dq. + 8td 8L t . 8q 8q Thus under a change of coordinates 8L 8L ., 8L . 8L 8q' F* ( q ) 8q and q 8q' = q 8q , 0 so that E is a well-defined function on TM. Corollary 1 . 2 . Under a change of local coordinates on M, components of 8L 8L , . . . , aq_n 8L ) transform lzke. components of a 1 -form on M. oq (q, q,. t) = ( {)(jl •/

I

=

=

=

Proposition 1 . 1 (Conservation of energy) . The energy of a closed system

is an integral of motion.

Proof. For an extremal 1 set Euler-Lagrange equations,

E(t ) E(t(t )) . We have, according to the dE _ d ( 8L ) q. 8L q. - 8L q. - 8L q. - 8L dt - dt 8q 8q 8q 8q 8t ( !!_ ( 8L ) 8L ) q. 8L 8L . dt 8q 8q 8t 8t =

+

=

_

Since for a closed system

��

=

_

=

_

0, the energy is conserved.

0

Conservation of energy for a closed mechanical system is a fundamental law of physics which follows from the homogeneity of time. For a general closed system of N interacting particles considered in Example 1.2,

E

=

N

N

l: ma r� - L = L ! ma r� + V (r ). a=l a=l

1. Lagrangian Mechanics

17

In other words, the total energy E = T + V is a sum of the kinetic energy and the potential energy. Definition. A Lagrangian L

:

TM -+ lR is invariant with respect to the diffeomorphism g : M -+ M if L(g* (v ) ) = L (v) for all v E TM. The diffeo­ morphism g is called the symmetry of a closed Lagrangian system ( M, L) . A Lie group G is the symmetry group of (M, L) ( group of continuous symme­ tries) if there is a left G-action on M such that for every g E G the mapping M 3 x g · x E M is a symmetry. �

Continuous symmetries give rise to conservation laws.

Theorem 1 .3 ( Noether ) . Suppose that a Lagrangian L : TM -+ lR is invari­ ant under a one-parameter group {gs } sElR of diffeomorphisms of M. Then the Lagrangian system (M, L ) admits an integral of motion I, given in stan­

dard coordinates on TM by

8L I (q , q) = � � a · i ( q , q) ( '

where X =

i= l

'

q

dg� ( q ) d8

I ) () . a , s=O

=

8L q

i= ai ( q ) 0°qi is the vector field on M associated with the flow

i=l

g8 •

The integral of motion I is called the Noether integral.

Proof. It follows from Corollary 1 .2 that I is a well-defined function on TM. Now differentiating L ( (gs) * (l' (t) ) ) L (1' (t) ) with respect to s at s 0 and using the Euler-Lagrange equations we get =

0

=

= ��a + �� a = : ( ��) a + �� �: = : ( �� a) , t

t

where ( t ) = ( a 1 ( T ( t ) ) , . . . , an ( T (t)) ) . D Remark. A vector field X on M is called an infinitesimal symmetry if the corresponding local flow g8 of X ( defined for each s E lR on some Us � M) is a symmetry: L (g8 ) * = L on U8 • Every vector field X on M lifts to a vector field X' on TM, defined by a local flow on TM induced from the corresponding local flow on M. In standard coordinates on TM,

a

o

( 1 . 5)

X

_

� � ai (

i=l

)0

q fji

an d

X1

_

· j ai 0 )0 +� � q 0O ( ) 0 · i · i,j =l q q i=l

� � ai ( q

fji

q

j q q It is easy to verify that X is an infinitesimal symmetry if and only if dL (X') = 0 on TM, which in standard coordinates has the form

( 1 . 6)

-

8L L1 at ( q ) fji q t' = n

.

+

n

-

L q_J

' ' =l t,J

·

0a i

8L ) 8qj ( q 8q·i

=

0.

1. Classical Mechanics

18

T M x lR JR. Namely, on the extended configuration space M1 define a time-independent Lagrangian by

x

Remark. Noether's theorem generalizes to time-dependent Lagrangians L : ---+

L1

L l (q, q, i) T,

=

L

( *' ) q,

T

=

M

lR

i,

where T ) are local coordinates on M1 and T , i) are standard coordi­ nates on TM1 . The Noether integral h for a closed system ( M1 , L 1 ) defines an integral of motion for a system (M, L) by the formula

(q,

(q, q,

I

I(q, q, t ) = h (q, t , q, 1 ) .

L1 is invariant with

When the Lagrangian L does not depend on time, respect to the one-parameter group of translations T Noether integral h gives I = -E. =

�1

t-t T

+

s,

and the

Noether ' s theorem can be generalized as follows.

Proposition 1 . 2 . Suppose that for the Lagrangian L : T M ---+ lR there exist a vector field X on M and a function on T M such that for every path 1

K

in M,

dL(X' ) (T( t)) = :t K(T'( t)) .

Then

I=

ai(q ) ��q (q, q) - K(q, q) t l i=

is an integral of motion for the Lagrangian system (M, L) . Proof. Using Euler-Lagrange equations, we have along the extremal /,

0

Example 1 . 9 ( Conservation of momentum ) . Let M = V be a vector space,

and suppose that a Lagrangian L is invariant with respect to a one-parameter group + sv , v E V. According to Noether ' s theorem,

g5 ( q) q =

i a8L·i i= l q

I=� L...J

v

is an integral of motion. Now let (M, L) be a closed Lagrangian system of N interacting particles considered in Example 1 .2. We have M = V JR3N , and the Lagrangian L is invariant under simultaneous translation of =

1.

=

coordinates ra v

19

Lagrangian Mechanics

= ( c,

. . . , c) E

I

=

( r� , r� , r� ) of all particles by the same vector c E JR3 . Thus JR3N and for every c ( c 1 , c2 , c3 ) E JR3 ,

� �

N

a= 1

(

=

c

1

8L 2 8L 3 8L {)r. a1 + c {)r. a2 + c {)r. a3

)

=

1 2 c P1 + c P2

+ c3 P3

is an integral of motion. The integrals of motion H , P2 , P3 define the vector p

N �

{)L

=� · a=l aTa

( or

E

JR3

rather a vector in the dual space to JR3 ) , called the momentum of the system. Explicitly, N so that the total momentum of a closed system is the sum of momenta of individual particles. Conservation of momentum is a fundamental physical law which reflects the homogeneity of space. 8 Traditionally, Pi = �1 are called generalized momenta corresponding to 8q 8L generalized coordinates qi , and Fi = 1 are called gene ralized fo rces. In 8q these notations, the Euler-Lagrange equations have the same form

p=F as Newton's equations in Cartesian coordinates. Conservation of momentum implies Newton's third law. Example 1 . 10 ( Conservation of angular momentum ) . Let M = V be a vector space with Euclidean inner product. Let G = SO(V) be the connected

Lie group of automorphisms of V preserving the inner product, and let g = .so (V) be the Lie algebra of G. Suppose that a Lagrangian L is invariant with respect to the action of a one-parameter subgroup g5 (q ) = esx q of G on V, where E g and ex is the exponential map. According to Noether's theorem, n {)L ·

x

I = L (x i=l

·

.

qy [j7i q

is an integral of motion. Now let ( M, L) be a closed Lagrangian system of N interacting particles considered in Example 1.2. We have M V = JR3N , and the Lagrangian L is invariant under a simultaneous rotation of coordinates ra of all particles by the same orthogonal transformation in JR3 . =

1. Classical Mechanics

20 Thus x = ( u, . . . , u) E .so (3) EB

·

·

·

EB .so(3) ,

and for every u E .so (3) ,

N N

8L

8L

fJL

I = � ( u · ra) !::1 · 1 + ( u · ra )2 8 . 2 + (u · ra )3 8 . 3 u

(

)

� � � a"""' =l is an integral of motion. Let u = u1 X1 + u2 X2 + u 3 X3 , where X1 = 8 81 ' x2 = - 81 80 0 ' x3 = � is the basis in .so (3) ':::::: JR3 cor0 0 0 0 0 responding to the rotations about the vectors e1 , e2, e 3 of the standard orthonormal basis in JR3 ( see Problem 1 . 10) . We get 3 I = u1M1 + u2Mz + u M3 , where M = ( M1, M2 , M3) E JR3 ( or rather a vector in the dual space to .so (3)) is given by N [)L M = l: ra x or .

( J)

1

( 6)

( -5 g )

a

a=l

The vector M is called the angular momentum of the system. Explicitly, N

M = l: ra

a=1

X

ma ra ,

so that the total angular momentum of a closed system is the sum of angular momenta of individual particles. Conservation of angular momentum is a fundamental physical law which reflects the isotropy of space. Problem 1 . 13. Find how total momentum and total angular momentum trans­ form under the Galilean transformations. 1 . 5 . One-dimensional motion. The motion of systems with one degree

of freedom is called one-dimensional. In terms of a Cartesian coordinate x on M = lR the Lagrangian takes the form L = � mx2 - V(x) . The conservation of energy 1 . E 2 m 2 + V(x)

x

=

allows us to solve the equation of motion in a closed form by separation of variables. We have

dx dt

so that

t

=

=

vm2

/mj

V2

(E -

V (x)), dx .

y'E - V(x)

21

1 . Lagrangian Mechanics

The inverse function x(t) is a general solution of Newton's equation dV mx.. = dx ' with two arbitrary constants, the energy E and the constant of integration. Since kinetic energy is non-negative, for a given value of E the actual motion takes place in the region of JR. where V(x) � E. The points where V(x) = E are called turning points. The motion which is confined between two turning points is called finite. The finite motion is periodic - the particle oscillates between the turning points x1 and x 2 with the period dx T ( E ) = v'2m rX2 lx 1 JE - V(x) If the region V(x) � E is unbounded, then the motion is called infinite and the particle eventually goes to infinity. The regions where V (x) > E are forbidden. On the phase plane with coordinates (x, y) Newton's equation reduces to the first order system dV mx = y, y. = - dx · 'frajectories correspond to the phase curves (x(t) , y(t) ) , which lie on the level sets 2 + V(x) = E m of the energy function. The points (xo , 0) , where xo is a critical point of the potential energy V(x) , correspond to the equilibrium solutions. The local minima correspond to the stable solutions and local maxima correspond to the unstable solutions. For the values of E which do not correspond to the equilibrium solutions the level sets are smooth curves. These curves are closed if the motion is finite. The simplest non-trivial one-dimensional system, besides the free par­ ticle, is the harmonic oscillator with V(x) !kx2 (k > 0) , considered in Example 1 . 6 . The general solution of the equation of motion is x (t) = A cos (wt + a) ,

;

=

where A is the amplitude, w =

{f;, is the frequency, and

2 7r .

a is the

phase

of a simple harmonic motion with the period T = The energy is E = w �mw 2 A 2 and the motion is finite with the same period T for E > 0.

Problem 1 . 14. Show that for V(x) = -x4 there are phase curves which do not exist for all times. Prove that if V (x) ;::: 0 for all x, then all phase curves exist for all times.

22

1. Classical Mechanics

Problem 1 . 1 5 . The simple pendulum is a Lagrangian system with M = 81 �/2 7r Z and L = � 0 2 + cos O. Find the period T of the pendulum as a function of =

the amplitude of the oscillations.

P roblem 1 . 16 . Suppose that the potential energy V(x) is even, V(O) 0, and V (x) is a one-to-one monotonically increasing function for x � 0. Prove that the inverse function x (V) and the period T(E) are related by the Abel transform v T(E)dE E dx dV { { 1 and x (V) = T(E) 2V2ffi . Jo dV vE - V 21r../2m Jo vV - E =

=

1 . 6 . The motion in a central field and the Kepler problem. The motion of a system of two interacting particles - the - can also be solved completely. Namely, in this case ( see Example 1 .2) M .IR6 and m1 r· 21 + -m2 r· 22 - V ( jr1 - r 1 ) . L -2 2 2 Introducing on .IR 6 new coordinates r r1 and R m1 + m2 2 + m1 m2 ' we get L ! m R2 + ! J.L r 2 - V ( j r i ) , where m m1 + m2 IS. the and J.L = m1m2 IS. the ml + m2 of a two-body system. The Lagrangian L depends only on the velocity R of the center of mass and not on its position R. A generalized coordinate with this property is called It follows from the Euler-Lagrange equa­ tions that generalized momentum corresponding to the cyclic coordinate is conserved. In our case it is a total momentum of the system, . aL P = -. = mR ' an so that the center of mass R moves uniformly. Thus in the frame of reference where R = 0, the two-body problem reduces to the problem of a single particle of mass J.L in the external central field V ( I r I ) . In spherical coordinates in .IR3 ' x = sin '!? cos cp, y = r sin '!? sin cp , z = cos '!?, where 0 :S '!? < 1r , 0 :S cp < 27r, its Lagrangian takes the form L ! J.L ( r2 + r2J2 + r2 sin2 '!? cp 2 ) - V( r ) .

two-body problem

=

=

=

=

total mass

=

reduced mass

cyclic.

r

r

=

It follows from the conservation of angular momentum M J.LT x r that during motion the position vector r lies in the plane P orthogonal to M in .IR3 . Introducing polar coordinates ( r , x ) in the plane P we get : e J2 sin2 '!? cp 2 , so that + L !J.L(r 2 + r 2 x 2 ) - V ( r ) . =

=

23

1 . Lagrangian Mechanics

The coordinate x is cyclic and its generalized momentum J-Lr 2 x coincides with I M I if X > 0 and with - IM I if X < 0. Denoting this quantity by M , we get the equation ( 1 7) which is equivalent to Kepler 's second law 12 . Using ( 1 . 7) we get for the total energy .

( 1 .8) Thus the radial motion reduces to a one-dimensional motion on the half-line > 0 with the effective potential energy

r

Veff ( r)

=

V ( r) +

M2 2 J-Lr 2 ,

-

where the second term is called the centrifugal energy. As in the previous section, the solution is given by

t

(1 .9)

=

�!

V2

dr JE - Vef f ( r ) .

It follows from (1 .7) that the angle x is a monotonic function of t, given by another quadrature M dr ( 1 . 10) X= .J2ii r2 JE - Vejj ( r) ' yielding an equation of the trajectory in polar coordinates. The set Vejj ( r ) :-:=; E is a union of annuli 0 :-:=; rmin :-:=; r :-:=; rmax :-:=; oo , and the motion is finite if 0 < rmin :-:=; r ::; rmax < oo . Though for a finite motion r(t) oscillates between rmin and rmax , corresponding trajectories are not necessarily closed. The necessary and sufficient condition for a finite motion to have a closed trajectory is that the angle dr �x = M VXii Tmin r2 JE - Vejj ( r) is commensurable with 21!', i.e. , � X = 21!' m for some m , n E Z. If the angle n �X is not commensurable with 21!', the orbit is everywhere dense in the annulus rmin :-:=; r :-:=; rmax · If lim VeJJ ( r) = lim V ( r ) = V < oo ,

J

-

irmax

r � oo

r � oo

the motion is infinite for E > V j� ( E - V ).

-

the particle goes to oo with finite velocity

1 2 It is the statement that sectorial velocity of a particle in a central field is constant .

24

1 . Classical Mechanics

A very important special case is when V(r )

-�.

=

r It describes Newton's gravitational attraction (a > 0) and Coulomb elec­ trostatic interaction (either attractive or repulsive) . First consider the case when a > 0 - Kepler ' s problem. The effective potential energy is Veff ( r ) =

a

--

r

and has the global minimum

+

M2

2 J.tr 2

at ro = - . The motion is infinite for E � 0 and is finite for Vo � E < aJ.t 0. The explicit form of trajectories can be determined by an elementary integration in ( 1 . 10) , which gives M2

M

M

r ro + C . J2�-t ( E - Vo ) Choosing a constant of integration C = 0 and introducing notation X = cos - 1 p =

ro and e

=

V1 - � ,

we get the equation of the orbit (trajectory)

!!. 1 + e cos x. ( 1.11) r This is the equation of a conic section with one focus at the origin. Quantity 2p is called the latus rectum of the orbit, and e is called the eccentricity. The choice C = 0 is such that the point with x = 0 is the point nearest to the origin (called the perihelion ) . When Vo � E < 0, the eccentricity e < 1 so that the orbit is the ellipse 1 3 with the major and minor semi-axes a = ( 1 . 12) a = 1 -P e2 = 2 I E ' b = I y"1-e2 � · Correspondingly, rmin = - rmax _P_ , and the period T of elliptic 1+e 1-e orbit is given by T= 1ra . =

p

p

,

=

IMI

�

The last formula is Kepler 's third law. When E > 0, the eccentricity e > 1 and the motion is infinite - the orbit is a hyperbola with the origin as 1 3 The statement that planets have elliptic orbits

with a

focus at the Sun is Kepler 's firs t law.

1.

25

Lagrangian Mechanics

internal focus. When E = 0, the eccentricity e = 1 the particle starts from rest at oo and the orbit is a parabola. For the repulsive case a < 0 the effective potential energy Vef f ( r) is always positive and decreases monotonically from oo to 0. The motion is always infinite and the trajectories are hyperbolas (parabola if E = 0) -

!!. =

r

with

p=

M2 aJ.L

-

- 1 + e cos x

and e =

1

2EM2 + J.La2 .

Kepler's problem is very special: for every a E lR the Lagrangian system on JR3 with ( 1 . 13)

has three extra integrals of motion wl, w2 , w3 in addition to the components of the angular momentum M. The corresponding vector W = (WI , W2 , W3 ) , called the Laplace-Runge-Lenz vector, is given by ar . W = r X M(1 . 14) -

.

=

Indeed, using equations of motion J.LT lar momentum M = J.LT X r , we get . W

= J.L r..

(1 . 15)

x

-

a ; and conservation of the angu­ r

: - + ---'---::-'--

a ( r · r) r r3 ar a ( r · r)r . = ( J.Lr · r ) r - (J.L r r ) r- -r- + __:._r-;3:-'-X

..

Using J.L ( r get

r

=

(r

X

. r)

ar r

-

"

..

·

0.

M) r = M 2 and the identity (a x b) 2 = a2 b2- ( a . b)2 , we ·

W2 = a2 +

2M 2 E J.L

--­

where

p2 a E=--2J.L r is the energy corresponding to the Lagrangian (1 . 13) . The fact that all orbits are conic sections follows from this extra symmetry of Kepler's problem. Problem 1 . 1 1. Prove all the statements made in this section.

1. Classical Mechanics

26 Problem 1 . 18 . Show that if lim VeJJ (r)

r --> 0

=

- oo ,

then there are orbits with rmin = 0 - "fall" of the particle to the center.

Problem 1 . 19. Prove that all finite trajectories in the central field are closed only when

V(r)

=

kr 2 , k > 0,

and V(r)

=

--, a

r

a >

0.

Problem 1 . 20 . Find parametric equations for orbits in Kepler's problem.

Problem 1 . 2 1 . Prove that the Laplace-Runge-Lenz vector W points in the di­ rection of the major axis of the orbit and that ! W I = ae, where e is the eccentricity of the orbit. P roblem 1.22. Using the conservation of the Laplace-Runge-Lenz vector, prove that trajectories in Kepler's problem with E < 0 are ellipses. (Hint: Evaluate W r and use the result of the previous problem. ) ·

L)

1 . 7. Legendre transform. The equations of motion of a Lagrangian sys­ tem ( M, in standard coordinates associated with a coordinate chart U C M are the Euler-Lagrange equations . In expanded form, they are given by the following system of second order ordinary differential equations:

8Li (q q) = (8L (q q ) 8q ' dt 8(ji ' ()2 L - � ( ()2 L ( q , q ) () "i() "j q + () " i f) .

d

·)

j

)

.j ( q , q. ) q ' i = 1 , . . . , n . q q q q In order for this system to be solvable for the highest derivatives for all initial conditions in TU, the symmetric n x n matrix 02 £ HL ( q, q ) = ( q, q) t. ,J. = l q· i f)q·j - L......,; j=l

.

··

{

should be invertible on TU.

8

j

}n

Definition. A Lagrangian system ( M, L) is called non-degenerate if for every coordinate chart U on M the matrix HL ( q , q) is invertible on TU.

n x n matrix HL is a Hessian of the Lagrangian function for vertical directions on TM. Under the change of standard coordinates q ' = F ( q ) and q' = ( see Section 1 .2) it has the transfor­ mation law HL ( q , q) = F* ( q f HL ( q ', q')F* ( q ) , where F* ( q )T is the transposed matrix, so that the condition det HL =I 0 does not depend on the choice of standard coordinates.

L

Remark. Note that the

F* (q ) v

1.

27

Lagrangian Mechanics

For an invariant formulation, consider the 1-form fh , defined in standard coordinates associated with a coordinate chart U C M by fh =

=

n 8L i 8L L . i dq aq dq . i=l 8q

It follows from Corollary 1 . 2 that fh is a well-defined 1-form on TM . Lemma 1 . 2 . A Lagrangian system (M, L) is non-degenerate if and only if the 2-form dfh on TM is non-degenerate. Proof. In standard coordinates, dOL

=

."" � t,J = l n

(

02 L . f) . t. 0 . J di/ q q

A dq2 + .

. 02 L . dq1 J q q

f) . t f)

·)

A dq2

,

and it is easy to see, by considering the 2n-form dO£ = dOL A · · · A dOL , n

that the 2-form dOL is non-degenerate if and only if the matrix HL is nondegenerate. D Remark. Using t he 1-form OL , the Noether integral

be written ( 1 . 16 )

as

I in

Theorem 1 .3

can

where X' is a lift to TM of a vector field X on M given by ( 1 .5 ) . It also immediately follows from ( 1.6 ) that if is an infinitesimal symmetry, then

X

( 1 . 17)

Definition. Let (U, 'P ) be a coordinate chart on M. Coordinates 1 n = (p 1 , · · · , P n , q , . . . , q )

(p , q )

on the chart T* U � IR.n x U on the cotangent bundle T* M are called standard coordinates 14 if for (p, q) E T* U and f E C 00 ( U) 8f Pi ( df) = -t. , i = 1 , . . . , n . 0 q

Equivalently, standard coordinates on T* U are uniquely characterized by the condition that (p1 , . . . , pn ) are coordinates in the fiber corresponding 1 to the basis dq , . . . , dqn for r; M, dual to the basis . . . , f) n for Tq M .

p

=

0�1 , �

14Fo!lowing tradition, the first n coordinates parametrize the fiber o f T * U and the last n coordinates parametrize the base.

1. Classical Mechanics

28 Definition. The

1- form 0 on T* M, defined in standard coordinates by

0 = L Pi dqi = pdq , n

i =l is called Liouville 's canonical 1 -form.

Corollary 1.2 shows that 0 is a well-defined 1-form on T* M. Clearly, the 1-form 0 also admits an invariant definition and

1T

:

T* M

--t

O(u) = p (1T* (u) ) , where u E T(p,q) T* M, M is the canonical projection.

: TM --t T* M is called a Legendre transform associated with the Lagrangian L if

Definition. A fibre-wise mapping TL

In standard coordinates the Legendre transform is given by rL ( q , q ) = (p, q) , where

p

=

�� (q ,

q) .

The mapping TL is a local diffeomorphism if and only if the Lagrangian L is non-degenerate.

:

--t

Definition. Suppose that the Legendre transform TL TM T* M is a diffeomorphism. The Hamiltonian function T* M IR, associated with the Lagrangian L : TM IR, is defined by

H:

--t

H

0

In standard coordinates,

TL = EL = q

��

-

--t

L.

H(p , q ) = (pq - L ( q , q) ) J p= FtioL ,

��

where q is a function of p and q defined by the equation p = (q , q) through the implicit function theorem. The cotangent bundle T* M is called the phase space of the Lagrangian system (M, It turns out that on the phase space the equations of motion take a very simple and symmetric form.

L) .

Suppose that the Legendre transform TL : T M --t T * M is a diffeomorphism. Then the Euler-Lagrange equations in standard coordinates on TM, d 8L - 8L i = 1 , . . . ' n, = dt 8i i 8qi 0 ,

Theorem 1 .4.

29

1 . Lagrangian Mechanics

are equivalent to the following system of first order differential equations in standard coordinates on T* M : 8H 8H . Pi = - {) i , q·i = � ' � = 1 ' . . . , n . UPi q .

Proof. We have

dH =

8H 8H dp + dq {)q ap {) {)L pdq + q dp - dq - � dq {)q {)q L qdp - dq q {)L 'P= aq

( (

� )I

)I

'P= aq

8L

•

Thus under the Legendre transform, . d 8L . 8H and p = q=dt {)q {)p

8L {)q

8H . {)q

- - = - = --

0

Corresponding first order differential equations on T* M are called Hamil­ ton's equations ( canonical equations ) .

Corollary 1 . 5 .

ton's equations. Proof.

For H(t)

The Hamiltonian H is constant on the solutions of Hamil­ =

H(p(t), q(t))

we have

8H . 8H . = 8H 8H - 8H = 8H dH p 0 q+ = dt 8q {)p 8q {)p 8p 8q .

0

For the Lagrangian

L=

mr2 2- - V ( r) = T - V,

r E

JR 3 ,

of a particle of mass m in a potential field V ( r) , considered in Example 1 .4, we have 8L p= = mr. 8r Thus the Legendre transform T£ TIR3 T*R3 is a global diffeomorphism, linear on the fibers, and .

:

H (p, r)

=

---.

p2 (pr - L) I ._ E. = 2 m + V ( r) = T + V. r- m

1 . Classical Mechanics

30 Hamilton's equations oH P r. = - = op m ' oH ov . p = - or = - or

are equivalent to Newton ' s equations with the force F = For the Lagrangian system describing small oscillators, considered in Example 1 .6, we have p = mq , and using normal coordinates we get

��.

Similarly, for the system of N interacting particles, considered in Example 1 .2, we have p = (PI , . . . , PN ) , where oL r a = 1 , . , N. Pa = � uTa = ma a , The Legendre transform T£ TJR3 N T*JR3 N is a global diffeomorphism, linear on the fibers, and .

:

H (p , r )

=

.

-+

(pr - L) i,.= E m

=

.

2

L :a + V (r) = T + V. =l ma N

a In particular, for a closed system with pair-wise interaction, H (p , r )

=

2

N

L + L a = l 2r::,a lS,a
N

Vab (ra - rb) ·

In general, consider the Lagrangian n

where A(q)

=

L = L ! aij ( q)
=

oL � q

=

n

L aij ( q )q1 ,

j=l

.

i

=

1,

.

We

have

. , n, .

R.n .

and the Legendre transform is a global diffeomorphism, linear on the fibers , In this case, if and only if the matrix A(q) is non-degenerate for all q E

n

H (p, q) = (pq - L(q, q ) ) i = a� = L ! aij ( q) PiPj + V(q ) , p &q i,j = l where { aij (q ) }�j =l = A-1 ( q) is the inverse matrix.

2.

31

Hamiltonian Mechanics

Problem 1 . 23 ( Second tangent bundle ) . Let 1r : TM ___. M be the canonical projection and let Tv (T M) be the vertical tangent bundle of TM along the fibers of 1r the kernel of the bundle mapping 1r* : T(TM) ___. TM. Prove that there is a natural bundle isomorphism i : TM c:::: Tv (TM) .

-

Problem 1 . 24 ( Invariant definition of the 1-form OL ) · Show that fh (v) dL( (i o 1r* ) v ) , where v E T(TM) .

=

Problem 1 .25. Give an invariant proof of ( 1 . 1 7) . Problem 1 . 26. Prove that the path 1(t) i n M is a traj ectory for the Lagrangian system (M , L) if and only if

it' (t) (dfh ) + dEL (--/ (t ) ) 0, where 'Y' ( t ) is the velocity vector of the path 1' (t) in TM. =

Problem 1 . 27. Show that for a charged particle in an electromagnetic field, considered in Example 1 . 5 , e p = mr + � A and

(

)2

e 1 p - � A + e
8L at p = 84 .

Problem 1 . 29. Give an example of a non-degenerate Lagrangian system (M, L) such that the Legendre transform T£ : T M ___. T * M is one-to-one but not onto. 2.

Hamiltonian Mechanics

2 . 1 . Hamilton's equations. With every function H : T* M --t JR. on the phase space T* .f.vf there are associated Hamilton ' s equations - a first-order system of ordinary differential equations, which in the standard coordinates on T* U has the form f)H . 8H . ( 2. 1 ) P = - a , q = ap . q The corresponding vector field XH on T* U, � - f)H � - f)H � = XH = � i i � api aq aq api ap aq aq ap '

(f)H

�) f)H

gives rise to a well-defined vector field H on T* M, called the Hamiltonian vector field. Suppose now that the vector field XH on T* M is complete, i.e., its integral curves exist for all times. The corresponding one-parameter group { 9t } t E JR of diffeomorphisms of T* M generated by XH is called the Hamiltonian phase flow. It is defined by 9t (P, q) = (p (t ) q(t) ) , where p(t) , q(t) is a solution of Hamilton's equations satisfying p(O) = p, q (O) = q .

X

,

32

1. Classical Mechanics

Liouville ' s canonical 1-form () on T* M defines a 2-form w = dO. In stan­ dard coordinates on T* M it is given by n

w

= 2:.:: dpi (\ dqi = dp (\ dq ,

i= 1

and is a non-degenerate 2-form. The form w is called the canonical sym­ plectic form on T* M. The symplectic form w defines an isomorphism J : T* (T* M) ---+ T(T* M) between tangent and cotangent bundles to T* M. For every (p, q ) E T* M the linear mapping J- 1 : T(p ,q ) T* M T(� ) T* M is ,q given by w ( u 1 , u 2 ) J-1 (u2)( u1 ) , u 1 , u 2 E T(p, q ) T* M. The mapping J induces the isomorphism between the infinite-dimensional vector spaces A 1 ( T* M) and Vect(T* M) , which is linear over C00 (T* M) . If {) is a 1-form on T* M, then the corresponding vector field J(1'J) on T* M satisfies w ( X, J ( 1J ) ) = 1'J(X) , X E Vect(T* M) , and J-1 (X) -i xw . In particular, in standard coordinates, ---+

=

=

[) [) J (dp) = oq and J (dq ) = - op ' so that XH J (dH) . =

The Hamiltonian phase flow on T* M preserves the canonical symplectic form.

Theorem 2 . 1 .

Proof. We need to prove that (gt ) *w = w. Since 9t is a one-parameter group

of diffeomorphisms, it is sufficient to show that

[

=

d ( g ) *w w 0, dt t t=O £xH =

where £xH is the Lie derivative along the vector field vector field X ,

XH. Since for every

=

£x (df) d( X( f)) ,

we compute

£ xH (dpi) so that n

i =l

= ( �;) -d

and

£xH (dqi )

= ( �=) , d

2.

33

Hamiltonian Mechanics

Corollary 2.2. CxH (O)

1 -form.

=

d(-H + O (XH ) ) ,

where () is Liouville 's canonical

The canonical symplectic form w on T* M defines the volume form 1 w 1\ · · · 1\ w ---v---" on T* M, called Liouville 's volume form. = 1 n. ""'n Corollary 2.3 ( Liouville's theorem ) . The Hamiltonian phase flow on T* M preserves Liouville 's volume form.

wn n.1

The restriction of the symplectic form w on T* M to the configuration space M is 0. Generalizing this property, we get the following notion.

2 of the phase space T* M is called a La­ grangian submanifold if dim 2 = dim M and w ly = 0.

Definition. A submanifold

It follows from Theorem 2. 1 that the image of a Lagrangian submanifold under the Hamiltonian phase flow is a Lagrangian submanifold. Problem 2 . 1 . Verify that XH is a well-defined vector field on T* M.

Problem 2 . 2 . Show that i f all level sets o f the Hamiltonian H are compact submanifolds of T* M, then the Hamiltonian vector field XH is complete.

Problem 2.3. Let n : T* M -7 M be the canonical projection, and let 2 be a Lagrangian submanifold. Show that if the mapping nl2 : 2 -7 M is a diffeomor­ phism, then 2 is a graph of a smooth function on M. Give examples when for some t > 0 the corresponding projection of 9t (2) onto M is no longer a diffeomorphism.

H on the phase space T* M there is an associated 1-form () - H dt = pdq - H dt on the extended phase space T* M x IR, called the Poincare- Cartan form. Let 1 : [t0, t1] ---+ T* M be a smooth parametrized path in T* M such that n(!(to) ) = qo and n (l( t1 ) ) = q1 , where n : T* M M is the canonical projection. By definition, the lift of a path 1 to the extended phase space T* M x lR is a path u : [to , t1] ---+ T* M x lR given by u ( t) = (l ( t) , t) , and a path u in T* M x lR is called an admissible path if it is a lift of a path 1 in T* M. The space of admissible paths in T* M x lR is denoted by P(T* M) �� : �� · A variation of an admissible path u is a smooth family of admissible paths uE , where E [-Eo , co] and uo = u , and the corresponding infinitesimal variation is q t 0<7 = E T P(T * M ) 1 , t 1 qo, o 2.2. The action functional in the phase space. With every function

---+

£

OUt:£ I O 2) . The principle of the least action in the phase space is the

(cf. Section 1 . following statement .

t:=O

17

34

1 . Classical Mechanics

Theorem 2.4 (Poincare) . The admissible path a- in T* M x lR is an extremal for the action functional t S(a-) = (pdq - H dt) = { 1 (pq - H )dt ito if and only if it is a lift of a path 1(t ) (p ( t ) , q (t ) ) in T* M, where p(t ) and q ( t ) satisfy canonical Hamilton's equations

1

u

=

. p=

{)H . {) - {)q ' q = {)H p.

Proof. As in the proof of Theorem 1 . 1 , for an admissible family (p( t, E ) , q (t, E ) , t ) we compute using integration by parts,

a-c:

(t )

=

i l tl + """' L.,. Pi J q to . n

i= l

Since J q (to) = Jq(t1 ) = 0, the path a- is critical if and only if p(t ) and q (t ) 0 satisfy canonical Hamilton's equations (2. 1 ) .

(M, L) , every path 1(t ) = ( q( t ) ) in the configuration space M connecting points qo and Ql defines an admissible path i'(t ) = (p(t) , q (t ) , t ) in the phase space T* M by setting p = If the Legendre transform T£ T M T* M is a diffeomorphism, then t t S( )' ) = f \pq - H )dt = { 1 L(!' (t) , t)dt. ito ito Thus the principle of the least action in a configuration space - Hamilton's principle - follows from the principle of the least action in a phase space. In fact, in this case the two principles are equivalent (see Problem 1 . 28) . Remark. For a Lagrangian system

:

---+

��.

From Corollary 1 .5 we immediately get the following result.

Solutions of canonical Hamilton's equations lying on the hypersurface H (p, q) = E are extremals of the functional fu pdq in the class of admissible paths lying on this hypersurface.

Corollary 2 . 5 .

a-

The trajectory 1 = ( q (r ) ) of a closed Lagrangian system (M, L) connecting points qo and q 1 and having energy E is the extremal of the functional

Corollary 2 . 6 (Maupertuis' principle) .

1 pdq 1 �� (q (r) , q (r ))q(r )dr =

2.

35

Hamiltonian Mechanics

on the space of all paths in the configuration space M connecting points qo and q1 and parametrized such that H( � (T) , q(T) ) = E. The functional

So(r) =

is called the abbreviated action1 5 .

1 pdq

Proof. Every path 1 = q( T) , parametrized such that H ( �� , q) = E , lifts to an admissible path a = (�(T), q(T), T), a :S T :S b, lying on the hypersurface H(p, q) = E 0 .

Problem 2 . 4 ( Jacobi) . On a Riemannian manifold (M, ds 2 ) consider a La­ grangian system with L (q, v) = ! l l v l l 2 - V (q) . Let E > V (q) for all q E M. Show that the trajectories of a closed Lagrangian system (M, L ) with total energy E are geodesics for the Riemannian metric ds2 = (E - V (q))ds 2 on M.

2.3. The action as a function of coordinates. Consider a non-degene­ rate Lagrangian system ( M, L) and denote by 1(t ; qo, vo) the solution of

Euler-Lagrange equations

:.!:._ 8L _ 8L

O

= dt 8q 8q with the initial conditions / (to) qo E M and )'(to) = vo E Tq0 M . Suppose that there exist a neighborhood Vo C Tv0 M of vo and t1 > to such that for all v E Vo the extremals 1(t; qo, v) , which start at time to at qo , do not intersect in the extended configuration space M x lR for times to < t < t1 . Such extremals are said to form a central field which includes the extremal ro (t) = 1(t; qo , vo ) . The existence of the central field of extremals is equivalent to the condition that for every to < t < t1 there is a neighborhood Ut C 1vi of 10(t) E M such that the mapping (2.2) Vo 3 v f-t q(t) r(t ; qo , v ) E Ut is a diffeomorphism. Basic theorems in the theory of ordinary differential equations guarantee that for t1 sufficiently close to to every extremal 1(t) for to < t < t1 can be included into the central field. In standard coordinates the mapping (2.2) is given by q f-t q(t) = r(t ; Qo , q ) . For the central field of extremals r(t; qo, q) , t0 < t < t1 , we define the action as a function of coordinates and time ( or, classical action) by t r L(r1(T))dT, S(q, t; qo , to) it o =

=

=

1 5 The accurate formulation of Maupertuis' principle is due to Euler and Lagrange.

1. Classical Mechanics

36

where i( T ) is the extremal from the central field that connects Qo and q . For given Qo and t o , the classical action is defined for t E ( t o ti ) and q E Uto
-

-

The differential of the classical action S( q, t) with fixed ini­ tial point is given by

Theorem 2 . 7.

dS = pdq - Hdt ,

��

where p = ( q , q) and H = pq L ( q , q) are determined by the velocity q of the extremal !(T) at time t . -

Proof. Let qe b e a path i n M passing through q at c = 0 with the tangent vector v E Tq M � JRn , and for c small enough let /e ( T) be the family of extremals from the central field satisfying /e ( to ) = Qo and ie ( t ) = Qe · For the infinitesimal variation 81 we have 81( t 0 ) = 0 and 81 (t ) v, and for fixed t we get from the formula for variation with the free ends ( 1 .2) that =

dS(v)

This shows that

so that

�� = L

�! -

=

=

p. Setting q ( t )

8L aq v .

=

1 ( t ) , we obtain

88 . 88 S(q(t) , t) = a q + a = L , t q dt

d

pq = -H.

0

The classical action satisfies the following nonlinear partial differential equation

Corollary 2.8. (2.4)

This equation is called the Hamilton-Jacobi equation. Hamilton's equa­ tions (2 . 1 ) can be used for solving the Cauchy problem (2 .5 ) S(q, t) lt=O = s (q) , s E C00 ( M ) , for Hamilton-Jacobi equation (2.4) by the method of characteristics. Namely, assume the existence of the Hamiltonian phase flow 9t on T* M and consider the Lagrangian submanifold .!£'

{

}

8s(q) = (p, q ) E T* M : p = aQ ,

2.

37

Hamiltonian Mechanics

a graph of the 1-form ds on M - a section of the cotangent bundle 1r : T* M ---+ M. The mapping 1r l 2' is one-to-one and for sufficiently small t the restriction of the projection 1r to the Lagrangian submanifold .2t = gt (.Z) remains to be one-to-one. In other words, there is t1 > 0 such that for all 0 � t < t 1 the mapping 7rt = 1r gt ( 1rlz ) - 1 : M ---+ M is a diffeomorphism, and the extremals / ( T , qo , tio) in the extended configuration space M x JR., where qo = (po , qo) and (po , qo ) E £ , do not intersect . Such extremals are called the characteristics of the Hamilton-Jacobi equation. o

�:

For 0 � t (2.4) -(2.5) is given by

Proposition 2 . 1 .

S(q , t)

<

=

o

h the solution S(q, t) to the Cauchy problem s (qo) +

lot L(J'(T))dT.

Here !(T) is the characteristic with 1(t) = q and with the starting point qo = !(0) which is uniquely determined by q . Proof. As in the proof of Theorem 2.7 we use formula ( 1 . 2) , where now qo depends on q , and obtain

as as aqo aL aL . aqo . (q) (qo) (q q) (qo , qo) + aq =aQo aq aq· , - a Qo · aq = p ,

since

a as (qo ) = Po = � (qo , tio ) . Setting q(t) = 1(t) we get aQo aQo

! s(q ( t ) t ) ,

so that

=

as

�! ti + ��

=

L (q , q ) ,

(p q) ,

at = - H , and S satisfies the Hamilton-Jacobi equation.

D

We can also consider the action S(q, t; qo, to ) as a function of both vari­ ables q and qo . The analog of Theorem 2.7 is the following statement.

The differential of the classical action as a function of initial and final points is given by

Proposition 2 . 2 .

dS = pdq - po dqo - H (p, q)dt + H (po , qo) dto . Problem 2 . 5 . Prove that the solution to the Cauchy problem for the Hamilton­ Jacobi equation

is

unique.

38

1. Classical Mechanics

2 . 4 . Classical observables and Poisson bracket . Smooth real-valued functions on the phase space T* M are called classical observables. The vector space coo ( T* M) is an IR-algebra - an associative algebra over lR with a unit given by the constant function 1 , and with a multiplication given by the point-wise product of functions. The commutative algebra C 00 ( T * M) is called the algebra of classical observables. Assuming that the Hamiltonian phase flow gt exists for all times , the time evolution of every observable f E C 00 (T* M ) is given by (p, q ) E T M . ft (p, q ) = f ( gt (p , q ) ) = f (p ( t ) q ( t ) ) Equivalently, the time evolution is described by the differential equation dfs +t dft d (ft ogs ) = = XH ( ft ) dt ds s = O ds s=O

=

I

I

=

, ,

t ( 8H 8fti _ 8Hi 8ft ) = 8H 8ft _ 8H 8ft i= l

opi oq

oq opi

op oq

oq op '

called Hamilton's equation for classical observables. Setting { f, g} = Xt ( g ) =

( 2.6)

o f og o f og ' op oq oq op

f, g

E C00 ( T* M ) ,

we can rewrite Hamilton's equation in the concise form df = { H, f } , dt

(2.7)

where it is understood that ( 2. 7) is a differential equation for a family of functions ft on T* M with the initial condition ft (P, q) l t =O = f (p , q ) . The properties of the bilinear mapping { , } : C00 ( T * M) x C00 ( T* M) C00 ( T * M ) are summarized below. �

The mapping { , } satisfies the following properties. ( i ) (Relation with the symplectic form}

Theorem 2 . 9 .

{ f , g} = w ( J (df) , J ( dg ) ) = w (Xj , X9 ) .

( ii ) (Skew-symmetry) ( iii ) (Leibniz rule) ( iv )

{ !, g } = - {g , !} .

{fg , h} = f { g , h}

(Jacobi identity)

+ g { f, h } .

+ {g, { h , ! } } + { h, { f , g } } = 0 for all f, g , h E C 00 ( T* M) . { !, {g , h } }

2.

39

Hamiltonian Mechanics

Proof. Property ( i ) immediately follows from the definitions of w and J in

Section 2. 1 . Properties ( ii ) - ( iii ) are obvious. The Jacobi identity could be verified by a direct computation using (2.6) , or by the following elegant ar­ gument. Observe that { !, g } is a bilinear form in the first partial derivatives of f and g, and every term in the left-hand side of the Jacobi identity is a linear homogenous function of second partial derivatives of f g , and h . Now the only terms in the Jacobi identity which could actually contain second partial derivatives of a function h are the following: ,

{ ! , { g, h} } + { g , {h , !} } =

(XJX 9 - X9X1 ) ( h ) .

However, this expression does not contain second partial derivatives of h since it is a commutator of two differential operators of the first order which 0 is again a differential operator of the first order! The observable {!, g } is called the canonical Poisson bracket of the ob­ servables f and g. The Poisson bracket map { , } : C 00 (T* M ) x C00 (T* M) C00 (T* M) turns the algebra of classical observables C 00 (T* M) into a Lie algebra with a Lie bracket given by the Poisson bracket. It has an important property that the Lie bracket is a hi-derivation with respect to the multi­ plication in C00 (T* M ) . The algebra of classical observables C00 (T* M ) is an example of the Poisson algebra a commutative algebra over lR carry­ ing a structure of a Lie algebra with the property that the Lie bracket is a derivation with respect to the algebra product. In Lagrangian mechanics, a function I on TM is an integral of motion for the Lagrangian system (M, L) if it is constant along the trajectories. In Hamiltonian mechanics, an observable I a function on the phase space T* M - is called an integral of motion ( first integral ) for Hamilton's equa­ tions (2. 1 ) if it is constant along the Hamiltonian phase flow. According to (2. 7), this is equivalent to the condition �

-

-

{ H, I}

=

0.

It is said that the observables H and I are in involution (Poisson commute) . 2.5. Canonical transformations and generating functions. Definition. A diffeomorphism g of the phase space T* M is called a

canon­ ical transformation, if it preserves the canonical symplectic form w on T* M, i.e. , g* (w) w. By Theorem 2 . 1 , the Hamiltonian phase flow 9t is a one­ parameter group of canonical transformations. =

Proposition 2 .3.

Canonical transformations preserve Hamilton's equations.

1 . Classical Mechanics

40 Proof. From g * (w) T ( T* M ) satisfies

=

w

T* ( T* M ) -

it follows that the mapping J 0

0

=

(2.8) g* J g * J. Indeed, for all X , Y E Vect ( M ) we have1 6

= g * (w) (X, Y) = w (g * (X) , g* (Y) ) so that for every 1-form 1J on M, w (X, Y)

o

g,

o

w (X, J (g * ('IJ) ) )

= g * ('IJ) (X) = 'IJ (g* (X) ) g = w (g* (X) , J ('IJ) ) which gives g * ( J (g * ('IJ) ) ) = J ('IJ) . Using (2.8) , we get

o

g,

= g. ( J ( dH ) ) = J ( (g * ) - 1 ( dH ) ) = XK , where H g- 1 . Thus the canonical transformation g maps trajectories of the Hamiltonian vector field XH into the trajectories of the Hamiltonian 0 vector field XK .

K

g. (XH )

o

=

Remark. In classical terms, Proposition 2.3 means that canonical Hamil­

ton's equations

. p

8H

. 8H q= (

8p p , q ) 8q in new coordinates ( P, Q) g (p, q ) continue to have the canonical form 8K ( P, Q) P Q), Q ( p = - 8K 8P 8Q , with the old Hamiltonian function K ( P, Q ) H (p , q ) . =-

(p , q ) ,

=

·

·

=

=

Consider now the classical case M IRn . For a canonical transformation = g (p , q ) set q ) . Since d 1\ (p q ) and n 2 dp 1\ dq on T* M � IR , the 1-form pdq - PdQ the difference between the canonical Liouville 1-form and its pullback by the mapping g - is closed. From the Poincare lemma it follows that there exists a function F (p, q ) on IR 2 n such that

( P, Q )

P P =

=

pdq -

( 2 .9)

=

,

P dQ

Q Q (p , -

PdQ

=

dF (p, q ) .

{ 8Pi }n

=

Now assume that at some pomt (Po , qo ) the n x n matnx a = � P PJ i ,j= l is non-degenerate. By the inverse function theorem, there exists a neighborhood U of (Po , qo ) in IR2n for which the functions q are coordinate functions. The function

.

. 8P P,

S ( P, q ) 16 Since

g

is

a

diffeomorphism,

g.X

is

= F (p , q ) + a

PQ

well-defined vector field on M .

2. Hamiltonian Mechanics

41

is called a generating function of the canonical transformation g in U. It follows from (2.9) that in new coordinates P, q on U, as as p= ( q) and Q = (P q) aq P, aP , . The converse statement below easily follows from the implicit function the­ orem. Proposition

point (Po , qo)

2.4.

E

Let S (P, q) be a function in some neighborhood U of a JR2n such that the n x n matrix n a2s a2s !lp!l (Po , qo ) = . i (Po , qo ) .. aP aq u uq

{

}

�

� .J = l

is non-degenerate. Then S is a generating function of a local (i. e., defined in some neighborhood of (Po , qo) in IR2 n ) canonical transformation. Suppose there is a canonical transformation (P , Q ) g (p, q) such that H (p , q) = K(P ) for some function K. Then in the new coordinates Hamil­ ton's equations take the form aK P = O, Q = (2. 10) aP ' and are trivially integrated: =

·

()

Pt

=

P( O ) ,

Q (t) = Q ( O ) + t

�� (P(O ) ) .

. ap 1s non- d egenerate, t he generatmg . functiOn . · t hat the matnx A ssummg ap S (P, q) satisfies the differential equation ·

H(�! (P, q) , q) = K(P) ,

(2. 11)

where after the differentiation one should substitute q = q (P , Q ) , defined by the canonical transformation g- 1 . The differential equation (2. 1 1 ) for fixed as it follows from (2.3) , coincides with the Hamilton-Jacobi equation for the abbreviated action So = S - Et where E = K(P ) , aso H (P, q) , q = E aq Theorem 2 . 1 0 ( Jacobi ) . Suppose that there is a function S (P , q) which depends on n parameters P = (P1 , . . . , Pn ) , satisfies the Hamilton-Jacobi equation (2. 1 1 ) for some function K(P) , and has the property that the n x n matrix is non-degenerate. Then Hamilton's equations

P,

(

a��q

.

)

a

H p = -aq ,

.

q =

.

aH ap

-

42

1 . Classical Mechanics

=

can be solved explicitly, and the functions P (p , q) ( Pl(P, q ) , . . . , Pn(P, q) ) , defined by the equations p = ( q) , are integrals of motion in involution.

�! P,

Set �! and = ;; By the inverse function the­ orem, ( P, Q ) i s a lo cal canonical transformation with the gener­ nthatg functiHamionltS.on'Its folequati lowsofrom ( 2. 1 1 ) that soiatintegral ns take the foarerm in(2.in1 0vol) . uSitinocen. s of motion 0 The sol2.u1ti0 oins calof ltheed theHamicomplete lton-Jacobiintegral.equatiAtonfirsatist glsfaynceing icondi tionsthatin Theorem t seems solavl iequati ng theon,Hamiis altmore on-Jacobi equati on,emwhithench isols avnonl inearltparti aequati l differen­ tiwhi di ffi c ul t probl i n g Hami on' s ons,e c h i s a system of ordi n ary di ff erenti a l equati o ns. It i s qui t e remarkabl that for ofmanythe probl elmston-Jacobi of classiequati cal mechani cthes onemethod can finofd separati the comploneteof ivari ntegral Hami o n by a bl e s. By Theorem 2. 1 0, thi s sol v es the correspondi n g Hami l t on' s equa­ tions. p

Proof.

=

g (p, q) =

(P, q)

(P, q) .

Q

H(p(P, Q ) , q (P, Q ) ) = K(P) , w = dP 1\ dQ ,

P1 (p, q) , . . . , Pn (p, q)

Problem 2.6. Find the generating function for the identity transformation P = p, Q = q. Problem 2. 7. Prove Proposition 2.4.

Problem 2.8. Suppose that the canonical transformation g (p, q) (P, Q) is such that locally ( Q , q) can be considered as new coordinates ( canonical transfor­ mations with this property are called free) . Prove that S1 (Q, q) = F(p, q) , also =

called a generating function, satisfies as1 p= and Bq

P

=-

as1 . BQ

Problem 2.9. Find the complete integral for the case of a particle in JR3 moving

in a central field.

notion ofbundla sympl generalization of the example ofThea cotangent e ectic manifold is a A non-degenerate, cl o sed 2-form on a mani f ol d . L is cal l e d a symplectic form, and the pair .L, ) is called a symplectic manifold. Si n ce a sympl e cti c form i s non-degenerate, a sympl e cti c mani f ol d .L i s necessari definesly even-di canonimcensial orional,entatidimon.Lon .L, andThe asnowhere in the vanicaseshi.Lng = - is called Liouvil e ' s volume fo rm. We also have the general notion of a Lagrangian submanifold. 2.6. Symplectic manifolds.

T* M.

Definition.

(

w

wn wn 1 n.

a

w

w

=

2n.

2n-form

T* M,

2.

43

Hamiltonian Mechanics

A

submanifold � of a symplectic manifold (.A, w) is called a Lagrangian submanifold, if dim � = � dim .A and the restriction of the symplectic form w to � is Definition.

0.

A

Symplectic manifolds form a category. morphism between (.AI , wi ) and ( .A2 , w2) , also called a symplectomorphism, is a mapping f .AI .A2 such that WI = J* ( w2) . When .AI = .A2 and WI w2 , the notion of a sym­ plectomorphism generalizes the notion of a canonical transformation. The direct product of symplectic manifolds (.AI , WI ) and (.A2 , w2 ) is a symplectic manifold (.AI x .A2 , 1r r { w i ) + 7r2 (w2 ) ) , where 7ri and 1r2 are, respectively, projections of .AI x .A2 onto the first and second factors in the Cartesian product. Besides cotangent bundles, another important class of symplectic man­ ifolds is given by Kahler manifolds 1 7 . Recall that a complex manifold .A is a Kahler manifold if it carries the Hermitian metric whose imaginary part is a closed ( 1 , 1 ) -form. In local complex coordinates z = (z i , . . . , zn) on .A the Hermitian metric is written as :

=

h

=

____.

n z::::: hCi13 ( z , z)dza 0 dz!3 . a,f3=I

Correspondingly, g

=

Re h = 2 L ha 13 (z , z) (dza ® dz!3 + dz!3 ® dz a ) 1

n

a,f3=I

is the Riemannian metric on .A and w =

- Im h =

� L hajj(Z, z) dzCi .

n

a,f3=I

1\

dz!3

is the symplectic form on .A ( considered as a 2n-dimensional real manifold ) . The simplest compact Kahler manifold is c_p i � 82 with the symplectic form given by the area 2-form of the Hermitian metric of Gaussian curvature 1 - the round metric on the 2-sphere. In terms of the local coordinate z associated with the stereographic projection c_p i C U { oo } , w = 2i

dz 1\ dz (1 + l z l 2 ) 2 .

�

Similarly, the natural symplectic form on the complex projective space c_pn is the symplectic form of the Fubini-Study metric. By pull-back, it defines symplectic forms on complex projective varieties. 1 7 Needless to say, not every symplectic manifold admits a complex structure , not to mention a

Kahler structure.

44

1 . Classical Mechanics

The simplest non-compact Kahler manifold is the n-dimensional complex vector space en with the standard Hermitian metric. In complex coordinates z = ( z 1 , . . . , z n ) on .rn It IS giVen by n = dz ® dz = L dza ® dza . h "-"

·

·

·

a =l

In terms of real coordinates (x, y) = (x 1 , , x n , y1 , . , yn ) on JR2n � e n , where z = x + iy, the corresponding symplectic form w = Im h has the canonical form . n w = dz 1\ dz = L dxa 1\ dy a = dx 1\ dy .

.

.

.

.

-

�

.

a =l

This example naturally leads to the following definition.

symplectic vector space is a pair ( V, w ) , where V is a vector w is a non-degenerate, skew-symmetric bilinear form on

A space over lR and

Definition. v.

It follows from basic linear algebra that every symplectic vector space V has a symplectic basis - a basis e 1 , . . , en , JI , , Jn of V, where 2n = dim V, such that w ( e i , eJ ) = w (li , fj ) = O and w ( e i , fj ) = r5j , i, j = 1 , . . . , n . In coordinates (p , q ) = (p 1 , . . . , pn , q 1 , . . . , q n ) corresponding to this basis, V � IR2n and n .

w

=

dp 1\ dq =

. . .

I: dpi i= l

1\

dq i .

Thus every symplectic vector space is isomorphic to a direct product of the phase planes IR2 with the canonical symplectic form dp 1\ dq. Introducing complex coordinates z = p + iq, we get the isomorphism v � en ' so that every symplectic vector space admits a Kahler structure. It is a basic fact of symplectic geometry that every symplectic manifold is locally isomorphic to a symplectic vector space.

Theorem 2 . 11 ( Darboux' theorem ) . Let ( .4l, w ) be a 2n-dimensional sym­ plectic manifold. For every point x E .41 there is a neighborhood U of x with local coordinates (p, q) = (P I , . , Pn , q 1 , . . , q n ) such that on U n .

. .

w = dp 1\ dq

=

I: dpi 1\ dqi . i=l

Coordinates p, q are called canonical coordinates ( Darboux coordinates) . The proof proceeds by induction on n with the two main steps stated as Problems 2 . 1 3 and 2.14.

2. Hamiltonian Mechanics

45

Ar;.4!non-degenerate 2-form for every .4! defi n es an i s omorphi s m Tx.4t by Tx .4t. Explicitly, for every Vect (.4t) and E A1 .4t) we have and ( cf. Section 2 1 In local coordinates for the coordinate chart (U,
J:

w

x E

w (u1 , u 2 ) = J-1 ( u 2 ) ( ui ) , u1 , u2 XE i) (

E

J -1 (X) = -i x( w) w (X , J (iJ ) ) = iJ( X ) . ). x = (x1 , . . . , x 2n ) w 2n w = ! L Wij ( x) dx i A dxi , i ,j=l {wij (x) } �,}= l
where tion on

is a non-degenerate, skew-symmetri c matri x -val u ed func­ Denoting the inverse matrix by we have '2n. L: r consireal stin-gvalofuaedsympl ectionc man­on i.4!,foldcalled a Acalled a The motiandios naaofpaismooth functi points on the phase space is de­ scribed by the vector field called a Hamiltonifieladn systemon .4!..4!,In canoniarecal thecoordiintegral curvesThetheyoftrajectori aareHamidescriletsoniofbaedna byvector n ates 1 the canonical Hamilton's equations (2. ) , J(dxi ) = -

Definition. (.4t, w) ,

0

2n

w ii ( x ) a ' xi j= l

i = 1, . . .

Hamiltonian system phase space,

H

Hamiltonian.

XH = J(dH) , Hamiltonian vector field. XH

(p , q)

fJH

P. = - aq '

.

((

w) , H)

fJH

q= a p

·

Suppose now that the Hamion l.4!toniassoci an vector fiwieltdh a Hami on .4!ltoniisacompl este.a one­ The a ted n i grou parameter of di ff eomorphi s ms of .4! generated by The p following statement generalizes Theorem 2 . 1 . It is sufficient to show that Using Cartan's formula and 0, we get for every Vect(.4t) , XH

Hamiltonian phase flow { gthEIR

H

XH .

Theorem 2 . 1 2. The Hamiltonian phase flow preserves the symplectic form.

Proof.

dw =

£xH w = 0 . £x = ix o d + d o ix XE £xw = (d o ix ) (w) .

1 . Classical Mechanics

46

Sithatnce Thus

ix (w) (Y)

X = XH

we have for and every Vect (.4t and the statement fol ows from

w (X, Y) ,

=

ixH ( w ) (Y) ixH ( w) = - dH ,

w ( J (dH) , Y )

=

A vector field X on and only if the 1 -form ix ( w ) is exact.

Corollary 2 . 1 3 .

.4!

=

Y

- dH (Y) . d2

=

E

0.

)

0

is a Hamiltonian vector field if ( .4!, w

A vector fiel d on a sympl e cti c mani f o l d ) is called a symplectic vector field i f the 1-form is closed, which is equivalent to

Definition.

X

ix (w)

.!L'xw = 0 .

e functi algebraons, is called withethalgebra a multofiplclassical ication giobservables. ven by the poiAssumi nThet-winsgcommutati e that productthe ofvHami l t oni a n phase fl o w 9t exists for all ti m es, the time evolution of every observable is given by x ft x = and is described by the differential equation -observabl Hamilteon's ons equatihaveon fortheclsame assicalformobservabl es.ltHami lequati ton's oequati ons for as Hami on' s ns on T* M, considered i n Secti o n Since we have the fol owing. A Poi s son bracket on the al g ebra of cl a ssi c al ob­ servables on a symplectic manifdefiold ned by is a bilinear mapping , Now Hamilton's equation takes the concise form 12 understood as a di ff erenti a l equati o n for a fami l y of functi o ns ft on with In local coordinates x1 , , x2n on the initial condition C00 (.4t') ,

( )

f

C00 (.4t')

E

f ( gt (x) ) ,

E

.4! ,

dft = XH (ft ) dt

.4!

.4!

2.3.

XH (f) = df (XH ) = w (XH , J (df) ) = w (XH , Xj ) , ( .4!, w )

Definition.

C00 (.4{)

(2.

X

C00 (.4{)

--+

C00 (.4t') ,

{ f, g } = w (X, , X9) , df dt

)

.4! ,

ft l t = O

=

=

f.

f, g

C00 (.4t')

} :

{

E

=

C 00 (.4t') .

{ H, f } ,

x

=

(

...

.4!

)

i 8 f ( x ) ag (x) { f, g } ( x ) = - L.....J w j (x) a xt axJ . i,j= l

�

The Poisson bracket { , } on a symplectic manifold ( .4t , w ) is skew-symmetric and satisfies Leibniz rule and the Jacobi identity.

Theorem 2. 14.

2. Hamiltonian Mechanics

47

first twoandproperti s areaobvious. It fol ows from the definition of a PoisTheson bracket the foermul = {! , h}} - { f, { g, h } } that the Jacobi identity is equivalent to the property Let and be symplectic vector fields. Using Cartan's formulas we get Proof.

[XJ , X9] ( h) = (X9Xf - X1X9 ) ( h )

{g,

( 2 . 1 3)

X

Y

i [x,Yj ( w ) = .Cx ( i y (w ) ) - i y ( .Cx ( w ) ) = d ( i x o i y (w ) ) + ix d ( i y (w ) ) = d ( w ( Y, X))

= iz ( w ) ,

s a Hamilitsoninon-degenerate, an vector fieldthicorrespondi ng Y]to Z, so thatc=setti(.4f)ng. Siwhere nceXf,theZ i2-form s i m pl i e s X9 and using g } we get From we immediately get the fol owing result . Ham(.4) .4 Ham ( .4 ) , f, f .4. As in the case .4 ( see Section an observabl e I - functi o n onthetheHamiphase space .4 -( (.4,is calledH)anifiinttegral of motialoonng(fitherst Hami integralltoni) foran l t oni a n system i s constant phase flow. According to this is equivalent to the condition { H, I} 0. ItFromis saithed thatJacobitheidobservabl entity forestheH andPoisIsonarebracket we get the following result . ( Poisson' s theorem ) . If {H, h } { H, h } 0, then { H, { I1 , I2} } = { { H, I1 }, I2 } - { { H, I2}, I1 } = 0. It foand, l owsbyfrom Poiscorrespondi son's theoremng Hami that litnonitegralan vector s of motifieoldns form aa LiLiee alsubalgebra f o rm g ebra i n Vect ( .4 ) . Si n ce {I , H } 0, the vector fi e l d s X1 HX totonisubmani folds .4E E E } - the level sets ofthearethetangent Hami l a n H. This defines a Li e algebra of i ntegrals of moti o n for Hamiltonian system ( (.4, H) at the level set .4 X=

Y=

w

{f ,

= w ( X f , X9 ) ,

w (X , Y)

=

[X ,

E

(2 . 1 3) .

D

(2. 13)

Corollary 2 . 1 5 . The subspace of Hamiltonian vector fields on is a Lie subalgebra of Vect (.4 ) . Th e mapping C 00 (.4) ---> given by �----+ X is a Lie algebra homomorphism with the kernel consisting of locally constant functions on = T *M

a

2 .4) ,

w) , (2.12) ,

=

( 2 . 14)

in involution ( Poisson commute ) .

The Poisson bracket of two integrals

Corollary 2 . 1 6 of motion is an integral of motion.

Proof.

=

=

D

(2. 13) ,

=

w) ,

.4

= d

{x

(

) H(x) I

:

=

=

E.

1 . Classical Mechanics

48

G befoladfinite-dimbyensisympl onal Liectomorphi e group thatsms.actsTheonLiae connected sym­ plactsectiLetonc mani al g ebra of G by vector fields A

g

(A, w)

=

Xf, ( j ) (x) g

:s l s=O f (e-sf. . x) ,

the linear mapping E Ve t ( ) is a homomorphism of Lie alandgebras, [Xf., X17] The G-acti o n is cal l e d a if Xf. are Hami ltonian vector ficonstant, elds, i.e . ,suchfor every E there is �f. E defined up to an addiif there tive that Xf. Xq, e J(d�f,) · It is called a ias homomorphi a choice of functi o ns �f. such that the l i n ear mapping i s sm of Lie algebras, Lie group G is latonian action of Gofonthe Hami onian system iAf there is a Hami suchltthat )= ) G, x E Noether theorem with) symmetries . G G By definition of the Hamiltonian action, for every �

=

3

�

�---+

Xf.

=

c

A

� ' 'f/ E g . Hamiltonian action g C00 (A) , X[f., ry] •

=

Poisson action � g --+ C00 (A) :

(2. 15)

symmetry group

Definition. ( (A, w ) , H )

Theorem 2 . 1 7 (

H (g · x

H (x

,

A

g

E

A.

) If is a symmetry group of the Hamiltonian system ( (A, w , H), then the functions �f. , � E g , are the integrals of motion. If the action of is Poisson, the integrals of motion satisfy (2.15) . �

Proof.

E

g,

0

Corollary 2 . 1 8 . Let (M, L) be a Lagrangian system such that the Legendre

transform T£ : T M T* M is a diffeomorphism. Then if a Lie group is a symmetry of (M, L) , then is a symmetry group of the corresponding Hamiltonian system ( (T*M, w) , H = EL r£ 1 ) , and the corresponding G­ action on T* M is Poisson. In particular, �f. = -If. r£1 , where If. are Noether integrals of motion for the one-parameter subgroups of generated by � E g . --+

G

G

o

o

G Let X be the vector field associatedusedwiinthTheorem the one-parameter sub­ group {e8f.} s EIR of diffeomorphisms of and l e t be its lift to We have18

Proof.

T M.

M,

(2. 16) 1 8The negative sign reflects the difference in definitions of X and Xe .

1 .3,

X'

2. Hamiltonian Mechanics

49 =

ix{ (O) = O(Xe ) ,

and i t fol l o ws from ( 1. 1 6 ) that where () i s the canoni c al Liouvil e 1-form on T* M . From Cartan's formula and ( 1. 1 7) we get = so that the G-acti weandobtai n on is Hamiltonian. Using ( 1. 1 7) and another Cartan's formula,
- ix{ ( dO) + £x{ (O) = - ix{ ( w) ,

d

=

J( d

=

=

- J (ix{ ( w) ) = Xe ,

i [xe ,X17J (O) = £xe (ix17 ( 0 ) ) + ix17 (£x{ (O) )

Xf. (
=

{
0

The Lagrangian � () forwitha respect particletointhe actimovionnofg ithen agroup centralS0fie3ld of(seeorthogonal Section 1.transf 6) isoirmati nvarioansnt ofcorrespondi the Euclindgeantospace Let be a basi s for the Li e al g ebra .so(3) the rotatiforons wi(seeth Exampl the axese gi1.v1en0 inbySectithe ovectors of the standard basi s n 1 . . These generators satisfy the commutation relations [ui , Uj] E:ijkUk , 1, 3, and E:ijk is a totally anti-symmetric tensor, 1. where Corresponding Noether integrals of motion are given by -Mi , where Example

L = mr 2 - V r

�3

�3 .

u 1 , u2 , U3

�3

e 1 , e 2 , e3

i , j, k =

( )

4)

=

2,

M1 M2 M3

=

= =

(r (r (r

p) I x p) 2 x p )a x


=

r2 p3 - r3p2 , = r3p1 - r1 p3 , = r 1P2 - r2P 1

=

c 123

=

areconvenicomponents of the angul a r momentum vector M ( Here it is on ) eFornt tothelowerHamitheltoniindiances of the coordinates ri by the Euclidean metric () we have 0. Accordi 1 7 and Corolsatilasryfy2. 18, Poisson brackets of the com­ ponentsnofg totheTheorem angular 2.momentum { Mi , Mj } -E:ijkMk, which is also easy to verify directly using �3 .

H=

p2

2m

+Vr

{ H, Mi } = =

{ f , g } (p , r)

=

(2.6) ,

of og of og op or - or op .

=

r x

p.

50

1 . Classical Mechanics

For every E lR the Lagrangian system on with Kepler's problem . �mr 2+ has three extra integralvector, s of gimotivenonby- the components W1, W2, W3 of the Lapl ace-Runge-Lenz )

Example 2 . 2 ( IR3

L

o:

� r

=

W =

p

m

x

M - o: r r

see Sectiwitohn 1. 6 . Using Poissonandbrackets from the previwe ogetus byexampl e,ght­to­ gether a strai forward computation, ) { ri , Mj }

(

=

{pi , Mj } = -E ij kPk ,

-E ij krk

where H 2 is the Hamiltonian of Kepler's problem. The Hamiif itltonihasan system w), H) , di m is called in i n dependent i n tegral s of moti o n iarenvolliuntiearlon.y Theindependent former condi t i o n means that E forthatalmostdH alhasl onlE y finHami lmany tonianzerossystemsare com­ with onepleteldegree of freedom such i t el y yonintegrabl e. oCompl eteproviseparati on ofexampl variableessofincompl the Hami ltion-Jacobi equati see Secti n d es other e tel y ntegrable HamiLetltonian systems. w) , H) be a completely integrable Hamiltonian system. Sup­ pose thatandthe tangent level setvectors= H (x) is compact are l i n earl y i n dependent for altherel exist so-cal Thenlbyed the Liouvil e-Arnold theorem, a nei g hborhood of coordinatessuchI that(hw, and
integrable

.4 =

( ( .4,

n

x

(

E

.4, .

IR� = (IR>o ) n H=

.

.4 .

{x E .4 : JdF� , . . . , JdFn

.4/

=

fl , .

in

action-angle variables : ( . . , 'Pn ) E Tn = (IR/27rZ) n . . , In ) · c.p =


=

i ( t) =

completely F1 = H, . . , Fn dF1 (x) , . . . , dFn (x) T;.4

2.5)

( ( .4,

x

2n,

i

= .

, Wn . .

.

.

=

=

i=

.

.

, Fn (x) =

=

=

fn }

.4!

. . . , In) E dl 1\ dc.p

. . . , n,

, n.

Problem 2. 10. Show that a symplectic manifold (.$t, w ) admits an almost com­ plex structure : a bundle map / : T.$( ___, T.$( such that f 2 = - id . Problem 2 . 1 1 . Give an example of a symplectic manifold which admits a com­ plex structure but not a Kahler structure.

2.

Hamiltonian Mechanics

51

(Coadjont orbits).

Problem 2 . 1 2 Let G b e a finite-dimensional Lie group, let g be its Lie algebra, and let g* be the dual vector space to g . For u E g * let .41 Ou be the orbit of u under the coadjoint action of G on g* . Show that the formula =

w (u1 , u2 )

u ( [x 1 , x 2 ] ) ,

where u 1 = ad * x 1 ( u) , u 2 = ad * x 2 (u ) E 0u , and ad * stands for the coadjoint action of a Lie algebra g on g * , gives rise to a well-defined 2-form on .4, which is closed and non-degenerate. (The 2-form w is called the Kirillov-Kostant symplectic form. ) =

Problem 2 . 1 3 . Let (.4/, w) be a symplectic manifold. For x E .41 choose a function q1 on .41 such that q1 ( x) = 0 and dq1 does not vanish at x , and set X = -Xq' · Show that there is a neighborhood U of x E .41 and a function p1 on U such that X ( q1 ) = 1 on U , and there exist coordinates p 1 , q 1 , z 1 , . . . , z 2 n- 2 on U such that f) and Y = Xp, = l. fJq

Problem 2. 14. Continuing Problem 2 . 1 3 , show that the 2-form w - dp 1 1\ dq 1 on U depends only on coordinates z 1 , . . . , z2 n -2 and is non-degenerate.

Problem 2 . 1 5 . Do the computation in Example 2.2 and show that the Lie algebra of the integrals M1 , M2 , M3 , W1 , W2 , W3 in Kepler's problem at H (p, r ) = E is isomorphic to the Lie algebra .so (4) , if E < 0, to the Euclidean Lie algebra e (3) , if E 0, and to the Lie algebra .so ( l , 3) , if E > 0. =

Problem 2 . 16 . Find the action-angle variables for a particle with one degree of freedom, when the potential V (x) is a convex function on JR. satisfying lim l x l -->oo V (x) oo . ( Hint: Define I f pdx, where integration goes over the closed orbit with H(p, x) = E . ) =

=

Problem 2 . 1 7. Show that a Hamiltonian system describing a particle i n JR.3 moving in a central field is completely integrable, and find the action-angle variables.

(Symplectic quotients).

Problem 2 . 18 For a Poisson action of a Lie group G on a symplectic manifold ( .41, w) , define the moment map P : .41 ---+ g* by P(x) (�)

=

e (x) , �

E

g, x

E

.4/,

where g is the Lie algebra of G. For every p E g* such that a stabilizer Gp of p acts freely and properly on .4/p = p-1 (p) (such p is called the regular value of the moment map) , the quotient Mp = Gp \.4/p is called a reduced phase space . Show that Mp is a symplectic manifold with the symplectic form uniquely characterized by the condition that its pull-back to .4/p coincides with the restriction to .4/p of the symplectic form w .

notion of a symplectic maniThe fold.notion of a Poisson manifold generalizes the aAskew-symmetric biliisneara manimappifoldng equipped with C00 { which satisfies the Leibniz rule and Jacobi identity. 2. 7 . Poisson manifolds.

Definition.

structure

-

ult

Poisson manifold ,

} : C00 ( ulf)

X

( ulf)

---->

C00 ( ulf)

a

Poisson

1. Classical Mechanics

52

A

alently, es iiss aaPoiPoissonssonalmani fol-d iaf Lithee alalggebra ebra such that the Liofe clbracket assiEquicalivobservabl g ebra s a hiof-derifunctivatioons).n wiItthfrespect to thethemulderitipvlatiicationoproperty n in A (athatpoinit­n wilocalse product o l o ws from coordinates x (x1 , . . . x ) on the Poisson bracket has the form .A

N

,

=

=

C00 (.A)

.A,

o;(�) . T/ij (x) {)�(x) {f, g}(x) uxt uxJ The 2-tensor T/ij (x), called a Poisson tensor, defines a global section = � L...J i,j=l N

T/

of the The vectorevolbundl e T T over u ti o n of cl a ssi c al observabl e s on a Poi s son mani f ol d is gi v en by Hamilton's equations, which have the same form as (2. 12), H for a compl e te Hami l t oni a n vector fi e l d The phase fl o w defines the A A by ( ) ( x ) ( ( x )) E A H A, A, E A. (2. 17) C ( ) } HE A, (2. 1 7), Let and19 By definition, and {H, Ut . = Ifthat ) is a Poisson manifold, then it follows from the Jacobi identity + so thatfunctiandons coincide satiat sfy 0,the(2.same diloffwserenti altheequatiuniqoueness n (2. 1 2).theorem Since these 1 7) fol from for the ordinary differential equations. Conversel y , we get the Jacobi i d enti t y for the functi o ns and by differentiating (2. 1 7) with respect to t at t 0. .A 1\ .A df dt

.A.

=

XH ( f)

9t evolution operator Ut Ut f

:

=

=

{ , !} .

XH

---+

f gt

, f

=

{ H, · }

.

Theorem 2 . 1 9 . Suppose that every Hamiltonian vector field on a Poisson manifold (.A, { , } ) is complete. Then for every E the corresponding i. e., evolution operator Ut is an automorphism of the Poisson algebra Ut ({f, g} )

=

{Ut ( f) , Ut (g) } for all f , g

Conversely, if a skew-symmetric bilinear mapping { ' } : 00 .A X C00 (.A) (.A) is such that XH = { H, are complete vector fields for all and corresponding evolution operators Ut satisfy then (.A , { , } ) is a Poisson manifold.

---+ coo

·

ft

P roof.

=

d 9t } dt (.A, { , }

ht

Ut (f ) , 9t

=

Ut (g) ,

ht

=

{ { H , ft } , 9t } + {It , { H, gt } }

{ { H, ft } , gt } { ft , 9t } t

Ut ( {f, g}) .

� dt =

ht } ·

{ ft , {H, gt } } = {H , { ft , gt } } ,

=

f, g ,

=

1 9 Here 9t is

not the phase flow!

H

0

2. Hamiltonian Mechanics Corollary

and only if

2 . 20.

53

A global section rJ of TA

Cx1 rJ

=

1\

0 for all f

TA is a Poisson tensor if A.

E

The of a Poisson algebra A is for all g E APoiPoissonssonalgebra manifofolcld assical observabl is calesleAd = consiisftstheonlcenter oflya y of l o cal constant functions ( (A) = for connected Equi v al e ntl y , a Poi s son mani f ol d i s non-degenerate i f the Poi s ­ sonan even-di tensor mrJ ensiis non-degenerate is necessari ly everywhere on Poiso that o nal mani f ol d . A non-degenerate s son tensor for every A defines an isomorphism J by center

Definition.

Z(A)

=

A : { !, g}

{!

=

(A, { , } )

Z

R

0 E A} . non-degenerate C00 (A) A) .

(A , { , } )

:

X E

r; A

-t

A

A,

Tx A

rJ(UI . u2) = U2 (J (u i ) ) , UI , U 2 x = (x i , . , x N )

E

T;A.

for the coordinate chart (U,r.p) on 1, J ( dxi ) = L ( o�j ' folds form a category. A morphi sm between andPoi( sson{ ,mani of smooth manifolds such, that} }2 ) i s a mapping for all g o g o }I = g }2 o A direct product of Poisson manifolds I , , h ) and (A2 , } ) is defi n ed by the property that natural Poiprojecti sson omani f ol d A2 , { n mapsngs. For f E andand (xi , X2 ) A2 denote,are Poirespecti ssonvmappi ctions off to and {xi } Then for ely, byE fJ!l and fJ7l 2restri ) g } x 2 uJ;l , g�;) h (x i ) + uJ;l , g��) }2 (x2 ) · Non-degenerate Poisson manifoldPois. sson manifolds form a subcategory of the category of Inwe lhave ocal coordinates

.

N

j=l

A2 ,

{f

r.p,

.

r.p

r] ij x )


:

AI

i

--+

r.p

{!,

A,

. . . ' N.

=

(AI , {

A2

f,

E

I)

C00 (A2 ) .

(A { { , 2 a (AI x , }) 7r i : AI x A2 --+ A1 1r2 AI x A2 A2 C00 (AI X A2 ) E AI X A x {x2 } x A2 . f, g C00 (AI X A , { f, ( i , X ) :

--+

=

Theorem 2 . 2 1 . The category of symplectic manifolds is (anti-) isomorphic

to the category of non-degenerate Poisson manifolds.

ng Itsto Theorem 2. 14, every sympl ectithec mani fold carries ofa Poia sympl sson eAccordi structure. non-degeneracy fol l o ws from non-degeneracy manifold.ctiDefic form. ne theConversel 2-form y,onlet (A,by } ) be a non-degenerate Poisson w (X , Y ) = J-I (Y ) (X ), X , Y Vect ( A ) , Proof.

w

A

{ ,

E

54

1 . Classical Mechanics

where isomorphi lIn ocalthecoordi nates sm .A on .A,is defined by the Poisson tensor J : T* � T.A x = (xl , . . . , xN )

w=- L

1'5,i<j 5, N

'f/ij (x) dx i

1\

ry.

dxj ,

{ TJij ( x ) }ij = l · f E X

where isnon-degenerate. the inverse matriForx toevery A let Thef 2-form wbeis skew-symmetri c and thebracketcorrespondiis nequig vector Jacobif iEdentiA, soty that for the Poisson valentfitoeld£onx1 TJ.A.0Thefor every £ 1 w 0. Since Xf we have w(X, = d X for every X Vect (.L) , so that By Cartan's formula, {TJij (x) }ij=l {

,

}

x

= Jdf,

=

Jdf)

=

f( )

=

{ !, } ·

E

dw(X, Y, Z ) = � (£xw(Y, Z ) £yw(X, Z ) + £ z w ( X Y ) -w( [X, Y] , Z ) + w([X , Z] , Y ) - w([Y, Z] , X)) , ( ) X = Xf , Y = X9 , Z = Xh , X , Y, Z E dw(Xf , X9 , Xh ) = � (w ( Xh [XJ , X9 ] ) + w(XJ, [X9 , Xh] ) + w(X9 , [Xh , Xf ] )) = � (w ( Xh X{ f,g} ) + w(Xf , X{g,h} ) + w(X9 , X{ h , f } ) ) = � ( { h, { !, g} } + {! , { g , h} } + { g , { h, f } } ) = 0. -

where

,

Vect .L . Now setting

we get

,

,

generate then vector vectorfieldspace of 1-fogen­ rms Aerate1 (.The Lthe) asexact avectormodul1-fspaceoermsoverVectA,(.Lso)thatasA,a Hami l t oni a s modul e overPoisA.sonThusmanifold 0.Aand, .A,.w)It ifols alosympl e cti c mani f o l d associ a ted wi t h the ws from the defi n i t i o ns that Poi s son mappi n gs of non-degenerate Poi s ­ son mani mani folds.folds correspond to symplectomorphisms of associated symplectic One can al s o prove thi s theorem by a strai g htf o rward compu­ taticondiontioinn local coordinates = . . . , on .A. Just observe that the 1, . . . + 0 · 0 0 . which is a coordinate form of dw 0 , follows from the condition df, f

E

Xj = Jdf dJ.JJ = ( ( { , })

0

Remark.

x

O'f/ij (x ) xz

(

OTJj z (x) xt

(x1 ,

+

xN )

_ O, OTJl i (x ) xJ =

i , j, l

=

, N,

N ik li kl '"""' TJij ( x) OTJ ( x ) + TJlj ( x ) O'I} (x ) + TJ kj ( x ) OTJ ( x) L... oxJ oxJ oxJ j= l

)

=

O,

2.

55

Hamiltonian Mechanics

whitimesch byis thea coordi inversenatematriformx 'T/iofj the usiJacobi ng identity, by multiplying it three ( x)

(rJip (x ) a'T/pk (x)

) . Letctic form= T*IRn with thewhere Poisson bracket(P{t ,, . }. .gi, pven, qlby, . the. . , qn)canon­are icoordi cal sympl e nate functions on T*IRn . The non-degeneracy of the Poin sson manifold � 6

8xJ

vii

p= l

Remark.

dp 1\ dq,

arJiP(x)

pk ( x ) = 0 8xJ 'T/ (p, q)

=

cansatibesfyfionrmul a ted as the property that the onl y observabl e g { f, pl } { f, qn } 0 { f , pn } 0, { f, q 1 } canst. ( Dual space to a Lie algebra) .

(T*IRn, { , } ) f E C00(T*IRn)

is

w =

+

f (p , q) = Problem 2 . 1 9

=

·

·

·

=

=

=

···=

=

Let g be a finite-dimensional Lie algebra with a Lie bracket [ , ] , and let g* be its dual space. For j, g E C 00 (g* ) define { f, g} ( u ) u ( [df, dg] ) , where u E g* and T: g* � g . Prove that { , } is a Poisson bracket. ( It was introduced by Sophus Lie and is called a linear, or Lie-Poisson bracket. ) Show that this bracket is degenerate and determine the center of A = C00 (g* ) . =

Problem 2 . 20. A Poisson bracket { , } on Jlt restricts to a Poisson bracket { , }0 on a submanifold ._AI if the inclusion 2 : ._AI ___.. Jlt is a Poisson mapping. Show that the Lie-Poisson bracket on g* restricts to a non-degenerate Poisson bracket on a coadjoint orbit, associated with the Kirillov-Kostant symplectic form. Problem 2 . 2 1 ( ) A finite-dimensional Lie group is called a Lie-Poisson group if it has a structure of a Poisson manifold (G, { , } ) such that

Lie-Poisson groups .

..

the group multiplication G x G ___.. G is a Poisson mapping, where G x G is a direct product of Poisson manifolds. Using a basis ih , . , 8n of left-invariant vector fields on G corresponding to a basis x 1 , . . . , X n of the Lie algebra g, the Poisson bracket { , } can be written as n = r /i (9)8di 8j h , { fr , h } (9) ,j = 1 i where the 2-tensor 1Jij (9 ) defines a mapping 1J : G ___.. A 2 g by 1J(9) L� j = l r/i (9 ) x i ® Xj . Show that the bracket { , } equips G with a Lie-Poisson structure i f and only if the following conditions are satisfied: ( i ) for all g E G, n z �iik (9) = (r"t (9) az1Jjk (g) + 'f/j l (g)Bz'f/ki (9) + 'f/k (9)az'f/i j (9) ) =l l n cfp'TJPJ (9)'TJkl (g) + cfpTJPk (9)'TJil (g) + c7p'TJPi (9) TJjl (9) = O, + l, p =1

L

=

L

(

L

)

where [x , Xj ] = I:Z= 1 ct xk ; ( ii ) the mapping 'fJ is a group 1-cocycle with the adjoint i action on A 2 g , i .e. , 'f/ (9 1 92 ) A d - 1 92 · "7 (91 ) + "7 (92 ) , 91 , 92 E G . =

1. Classical Mechanics

56

Problem 2 .22. Show that the second condition in the previous problem trivially

holds when Tf is a coboundary, ry (g) = -r + Ad- 1 g · r for some r = L:: �j =l rij xi ®Xj E A2 g , and then the first condition is satisfied if and only if the element

e ( r ) = [r 12 , T!J + T2J] + [r 1 3 , T2J] E A3 g is invariant under the adjoint action of g on A 3 g . Here r1 2 = L:: �j =l rij xi ® Xj ® 1 , r 1 3 = L:: �j=l rij xi ® 1 ® Xj , and r23 = L:: � j l r i1 1 ® Xi ® Xj are corresponding elements in the universal enveloping algebra Ug of a Lie algebra g. In particular, G is a Lie-Poisson group if e ( r ) 0, which is called the classical Yang-Baxter equation. Problem 2 . 23. Suppose that r = L:: � j =l rij x i ® Xj E A 2 g is such that the matrix {rij } is non-degenerate, and let {rij } be the inverse matrix. Show that r satisfies the classical Yang-Baxter equation if and only if the map c : A2g -> C, defined by c ( x, y ) = L:: �j =l Tij uivj , where x 2.:�1 uix i , y 2.:�1 vi x i , is a non-degenerate Lie algebra 2-cocycle, i.e., it satisfies =

=

=

=

c (x , [y , z] ) + c (z, [x, y] ) + c (y, [z , x] ) = 0 ,

x, y, z E g.

Toprocess complofete the for­ mulatioInn physi of clacssis,cbyal mechani c s, we need to descri b e the measurement ofchagiclvaesssinumeri cal system oneuesunderstands theobservabl resulteofs. aThephysiexperi cala experi m ent whi c al val foronsclforassithecal m ent consi s ts of creati n g certai n condi t i system, andTheit icondi s alwaystionsassumed that these condi teioans can beof repeated overif and over. of the experi m ent defi n the system ng theseecondi onssystem. results in probability distributions for the values ofrepeati alMathemati l observabl scallofy, theatistate of classical ob­ servables on the phase spaceJ.L onisthethealassigebragnment A &'(R) , where &(JR.the) istotal a setmeasure of probabiof lityismeasures on RBorel - Borel measures on such that 1. For every subsetthe valuetheof quanti 1 i s a probabi l i t y that i n the state t y the observabl ongs tois givByen bydefithenitiLebesgue-Sti on, the eltjes integral of an observabl e f ien fthebelstate Ell- (!) = I: where J.LJ (>.) ( f shoul is a did ssatitribsutify othen functi the measure The correspondence fol owionngofnatural propertidJ.LeJs.· I E ll- ( !) for f the subalgebra of bounded observables. Ell- ( 1) 1, where 1 is the unit in 2.8. Hamilton's and Liouville's representations.

measure­

ment.

state

A

.A 3

0 � J.L J (E)

f

r--+

J.L J

E <;;;

J.L

E.

expectation value

J.L

AdJ.L J (>.) ,

Sl.

82.

= J.L J ( - oo , >.))

I <

=

r--+

oo

J.L 1

E

C00 (..A)

E

R

�

=

Ao -

A.

R

lR

2.

57

Hamiltonian Mechanics

E lR

and j, E and exist. then for every Borel subset h with smooth ( cp- 1 (E) It folthatows from property and the definition of the Lebesgue-Stieltj es integral S3.

S4.

For all if both

a,

b

g

A,

EJJ- (af + bg) EJJ- ( g)

Ell- ( !)

If fi = 'P o E � JR,

f.L 11

S4

( 2 .18)

2::

=

aEJJ- (f) + bEJJ- (g) ,

'P

:

lR ----> JR,

= 1-Lh

(E) ) .

Inposiparti 0 fors onalthel f subalA, sogebrathat the states define normalized, tAssumi ive,cullinanearr,g thatfunctitheonalfuncti o nal extends to the space of bounded, pi e ce­ 2 wisonee conticannuous functitheonsdisontributioandn fusatinctisfioensfrom ( . 18) for measurable functions recover the expectation values by the formula (8 ( >.. - f ) ) , 1-Lt ( >.. ) where 8 (x) is the Heavyside step function, { 1 x > 0.O, 8 (x) = , Indeed, on of theelitnjestervalintegral( we>..)get Using (2.le18)t andbe thethe deficharacteri nition ofsticthefuncti Lebesgue-Sti EJJ- (! 2 )

E

vii ,

t.p,

Ao .

Ell-

= E ll-

0, X :S

x

-

Ell- ( 8 ( >.. - f ) ) =

I: x (s)df.Lt (s ) vii

=

oo ,

C

R

f.Lt (( - oo , >.. ) ) = 1-Lt ( >.. ) .

Every probabi l i t y measure df.L on defines the state on by assigning20 toforward everyofobservabl e f adj.Lprobabi litythemeasure onItlRis defianpush­ the measure on by mappi n g f ed by function f.L(f-1 (E) ) for every Borel subset � JR, and has the distribution 1-Lt (E )

where that

vii

=

vii>, (!)

=

1-L t ( >.. ) {x

E

=

vii

:

E

1-LU- 1 ( - oo , >.. ) ) f (x)

f.L 1 =

<

>.. } .

=

:

A

f* ( f.L )

vii

---->

R

j.A;. (f) df.L,

-

It follows from the Fubini theorem

(2.19) 20 There should be no confusion in denoting the state and the measure by 11-·

58

1 . Classical Mechanics

It turns out that probabi l i t y measures on .A are essential y the only es of states. Namelasserts y, forthata loforcallevery y compact topol oeargicalfuncti spaceonal.A[theon Ritheexampl esz-Markov theorem posi t i v e, l i n space Cc ( .A) of continuous functions on .A wi t h compact support, there is a unique regular Borel measure on .A such that ( f = JJ(I for all f E Cc (.A) . This leads to the fol owing definition of states in classical mechanics. The set of states S for a Hamiltonian system with the phase space .A is the convex set of all probabisupported lity measures onts .A.EThe states correspondi n g to Di r ac measures at poi n are calledAll other statesandarethecallphase space .A is also called the ed A process of measurement in classical mechanics is the correspondence � to every observabl e f E and state S assigns a probability whi c h measure on lR - a� push-forward ofy 0the� measure on .A by For every Borel subset lR the quanti t � 1 i s the probabi l i t y J that state theonresul fgivisenifornbythea system set E.inThethe expectati valuteofofaanmeasurement observable off ithen aobservabl state ies In physics,ofpureeverystates are characteri zgiedvesbya havi ndefig thenedproperty that a measurement observabl e al w ays wel l resul t . Mathe­ matically this can be expressed as fol ows. Let 0 ( ) be the of the observable in the state dJ.L

fdJ.L

)

l

Definition.

f!IJ (.A)

dJ.Lx

states 2 1 .

.A

x

pure states,

space of

mixed states.

AX

J.L 1

s

3

E

J.LJ = f* (J.L) E f!IJ (IR) ,

( ! , J.L)

J.L

A

E

J.L ( E )

J.L

dJ.L

f.

J.L

(2. 19) .

e1; ( f)

variance

=

EJ.L ( ! - EJ.L (f) ) 2 = EJ.L (f 2 ) - EJ.L ( f) 2 f J.L.

�

Lemma 2 . 1 . Pure states are the only states in which every observable has zero variance.

It0 fifolonlowsy ifrom the Cauchy-Bunyakovski Schwarz i n equal i t y that f is constant on the support of a probability measure stateInwipartith cular, a of pure(1 -states and0 1, .A, is a mixed so that > 0 for every observable f such that Proof.

CT� (f) dj.L .

=

f

mixture

CT� ( f)

dj.L = a. dJ.Lx +

21 The space of pure states, to be precise.

dJ.Lx

a.)dJ.Ly ,

dJ.Ly , x , y E

< a. <

f (x) f= f(y ) .

0

59

2. Hamiltonian Mechanics

For a system consisting of few interacting particles (say, a motion of planets in celestial mechanics) it is possible to measure all coordinates and momenta, so one considers only pure states. Mixed states necessarily appear for macroscopic systems, when it is impossible to measure all coordinates and momenta22 . Remark. As a topological space, the space of states .41 can be reconstructed from the algebra A of classical observables. Namely, suppose for simplicity that .41 is compact. Then the C-algebra C = C(.41) of complex-valued continuous functions on .41 - the completion of the complexification of the IR-algebra A of classical observables with respect to the sup-norm is a commutative C* -algebra. This means that C(.4t') is a Banach space with respect to the norm 1 1 ! 1 1 supx E .4t' i f (x) i , has a C-algebra structure (associative algebra over C with a unit) given by the point-wise product of functions such that I I ! · g i l :::; l l f ll l l g l l , and has a complex anti-linear anti­ involution: a map * : C -+ C given by the complex conjugation f* ( x ) f (x) and satisfying II ! · ! * I I = 1 1 ! 1 1 2 . Then the Gelfand-Naimark theorem asserts that every commutative C*-algebra C is isomorphic to the algebra C(.4t') of continuous functions on its spectrum - the set of maximal ideals of C - a compact topological space with the topology induced by the weak topology on C* , the dual space of C . =

=

We conclude our exposition of classical mechanics by presenting two equivalent ways of describing the dynamics - the time evolution of a Hamil­ tonian system ( ( .41, { , } ) , H) with the algebra of observables A = c oo ( .41) and the set of states S = &' (.41) . In addition, we assume that the Hamil­ tonian phase flow 9t exists for all times, and that the phase space .41 carries a volume form dx invariant under the phase fiow23 . Hamilton's Description of Dynamics. States do not depend on time, and time evolution of observables is given by Hamilton's equations of motion, d/L dt

1L

= 0,

E

S,

and

The expectation value of an observable EJJ. ( ft)

=

j.4t'

f o 9t d/L =

df = {H, ! } , f E A. dt f in the state IL at time t

j.4t'

is given by

f ( gt ( x ) ) p( x ) dx ,

where p( x) = d!L is the Radon-Nikodim derivative. In particular, the expecdx tation value of f in the pure state d!Lx corresponding to the point x E .41 is f ( gt ( x) ) . Hamilton ' s picture is commonly used for mechanical systems consisting of few interacting particles. 22 Typically, a macroscopic system consists of N 10 2 3 molecules. Macroscopic systems are studied in classical statistical mechanics . 2 3 It is Liouville's volume form when the Poisson structure on A is non-degenerate. �

1 . Classical Mechanics

60

Liouville's Description of Dynamics. The observables do not depend

on time

= 0 , f E A, and states dJ.1 (x ) = p(x ) dx satisfy Liouville ' s equation df dt

dp dt

=

- { H, p} , p(x ) dx

��

E

S.

Here the Radon-Nikodim derivative p(x ) = and Liouville's equation are understood in the distributional sense. The expectation value of an observ­ able f in the state Jl at time t is given by E JJ.t ( f)

=

j-11

f (x ) p( g -t (x ) ) dx .

Liouville ' s picture, where states are described by the distribution func­ tions p(x ) - positive distributions on .A corresponding to probability mea­ sures p(x ) dx - is commonly used in statistical mechanics. The equality E JJ. ( ft ) = E JJ.t ( f) for all f E A, J.1 E S, which follows from the invariance of the volume form dx and the change of variables, expresses the equivalence between Liouville ' s and Hamilton's descriptions of the dynamics. 3. Notes and references Classical references are the textbooks [Arn89] and [LL76] , which are written, respectively, from mathematics and physics perspectives. The elegance of [LL76] is supplemented by the attention to detail in [Gol80J , another physics classic. A brief overview of Hamiltonian formalism necessary for quantum mechanics can be found in [FY80] . The treatise [AM78] and the encyclopaedia surveys [AG90] , [AKN97] provide a comprehensive exposition of classical mechanics, including the history of the subject , and the references to classical works and recent contributions. The textbook [Ste83J , monographs [DFN84] , [DFN85] , and lecture notes [Bry95] contain all the necessary material from differential geometry and the theory of Lie groups, as well as references to other sources. In addition, the lectures [God69J also provide an introduction to differential geometry and classical mechanics. In particular, [God69] and [Bry95] discuss the role the second tangent bundle plays in Lagrangian mechanics ( see also the monograph [YI73] and [Cra83] ) . For a brief review of integration theory, including the Riesz-Markov theorem, see [RSSOJ ; for the proof of the Gelfand-Naimark theorem and more details on C* -algebras, see [Str05J and references therein. Our exposition follows the traditional outline in [LL76] and [Arn89] , which starts with the Lagrangian formalism and introduces Hamiltonian formalism through

3.

Notes and references

61

the Legendre transform. As in [Arn89] , we made special emphasis on precise math­ ematical formulations. Having graduate students and research mathematicians as a main audience, we have the advantage to use freely the calculus of differential forms and vector fields on smooth manifolds. This differs from the presentation in [Arn89] , which is oriented at undergraduate students and needs to introduce this material in the main text. Since the goal of this chapter was to present only those basics of classical mechanics which are fundamental for the formulation of quan­ tum mechanics, we have omitted many important topics, including mechanical­ geometrical optics analogy, theory of oscillations, rotation of a rigid body, per­ turbation theory, etc. The interested reader can find this material in [LL76] and [Arn89] and in the above-mentioned monographs. Completely integrable Hamil­ tonian systems were also only briefly mentioned at the end of Section 2.6. We refer the reader to [AKN97] and references therein for a comprehensive exposition, and to the monograph [FT07] for the so-called Lax pair method in the theory of inte­ grable systems , especially for the case of infinitely many degrees of freedom. In Section 2. 7, following [FY80] , [DFN84] , [DFN85] , we discussed Poisson manifolds and Poisson algebras. These notions, usually not emphasized in standard exposition of classical mechanics, are fundamental for understanding the meaning of quantization - a passage from classical mechanics to quantum mechanics. We also have included in Sections 1 . 6 and 2 . 7 the treatment of the Laplace-Runge-Lenz vector, whose components are extra integrals of motions for the Kepler problem24 . Though briefly mentioned in [LL76] , the Laplace-Runge-Lenz vector does not ac­ tually appear in many textbooks, with the exception of [Gol80] and [DFN84] . In Section 2 . 7, following [FY80] , we also included Theorem 2 . 19, which clarifies the meaning of the Jacobi identity, and presented Hamilton's and Liouville's descrip­ tions of the dynamics. Most of the problems in this chapter are fairly standard and are taken from various sources, mainly from [Arn89] , [LL76] , (Bry95] , [DFN84] , and (DFN85] . Other problems indicate interesting relations with representation theory and sym­ plectic geometry. Thus Problems 2 . 12 and 2 .20 introduce the reader to the orbit method [Kir04] , and Problem 2 . 1 8 - to the method of symplectic reduction ( see [Arn89] , [Bry95] and references therein ) . Problems 2.2 1 - 2.23 introduce the reader to the theory of Lie-Poisson groups ( see [Dri86] , [Dri87] , [STS85] , and [Tak90] for an elementary exposition ) .

2 4 We will see in Chapter 3 that these extra integrals are responsible for the hidden 80 (4) symmetry of the hydrogen atom.

Chapter

2

Basic P rinciples of Q uant um Mechanics

We recall the standard notation and basic facts from the theory of self­ adjoint operators on Hilbert spaces. Let £ be a separable Hilbert space with an inner product ( , ) , which is complex-linear with respect to the first argument, and let A be a linear operator in £ with the domain D(A) � £ - a linear subset of £. An operator A is called closed if its graph r (A) = { ( 'P, A 'P) E .Yt' x .Yt' : 'P E D (A) } is a closed subspace in £ x £. If the domain of A is dense 1 in £, i.e. , D (A) = .Yt', the domain D (A* ) of the adjoint operator A* consists of 'P E £ such that there is 'fJ E £ with the property that ( A'lj; , 'P ) = ( 'lj;, 'fJ) for all 'lj; E D ( A) , and the operator A* is defined by A* 'P = "' · An operator A is called sym­ metric if (A'lj; , 'P) = ( 'lj;, A 'P) for all 'tJ, 'lj; E D (A) . By definition, the regular set of a closed operator A with a dense domain D (A) is the set p(A) = { A E C I A-AI : D (A) ---+ .Yt' is a bijection with a bounded inverse2 } ,

E p(A) , the bounded operator R;. (A) = (A - AI)-1 is called the resolvent of A at A. The regular set p(A) C C is open and its complement a(A) = C \ p(A) is the spectrum of A. The subset ap (A) of a(A) consisting of eigenvalues of A of finite multiplicity is called the point spectrum.

and for A

1 We consider only linear operators with dense domains. 2By the closed graph theorem, the last condition is redundant.

63

2. Basic Principles of Quantum Mechanics

64

operatorc andA isD (A) = D(A*(or) , and for suchifoperators A = A*. Equivalently, isymmetri sAnsymmetri O'(A) A c operator A is called if its closure = A** i s sel f-adjoi nt. For a symmetri c operator A the fo l owing conditions are equivalent: (i) A is essential y self-adjoint. (ii) Ker(A* + Ker(A* {0} . (ii ) Im(A ) Im(A - = £. Aoperator symmetriA ics operator and self-adjoint. An 3 Aif wi(A.n (A) 0, where .Xn (A) areclasseiigfenval valently, an basioperator and onluesyofif A*forA.everyEquiorthonormal s {en }�A= l for (££) is of trace self- adjoint

Hermitian)

A

C

essentially self- adjoint

ii) =

+ il =

-

il)

R

A

=

il)

�

'ljJ

2

·

E

compact

E

'if

trace class 00

< 00 ,

E

00

L I (A en , en ) I

<

2

�

=

oo .

n= l

a permutati thiSinscecondi tion canoben ofreplanaorthonormal ced by basis is again an orthonormal basis, 00

L )Aen , en )

n=l

< oo

3Non-negative, to be precise. 4This is true only for complex Hilbert spaces. 5The sets with compact closure. 6Ideal '6'(£) is the only closed two-sided ideal of .Z(£).

65

1 . Observables, states, and dynamics

for every orthonormal basis {en}�=1 for .Ye. The trace of a trace class oper­ ator A is defined by A = L (Aen , en) , n= 1 and does not depend on the choi c e of an orthonormal basi s { en }�=1 for .Ye. A posit ive operator A E .Z(.Ye) is of trace class i f 00

Tr

00

n =1 { en } �= 1

foroperators some orthonormal basi s for .Ye. The space Y1 of trace class and is ini.dYeealis(vaonBanach algebra with theidealnorm) in JJAIIthe 1C*-algJA*A, a two-si d ed Neumann-Schatten ebra .Z(.Ye). The property AB BA for all A Y'1 , B E .Z(.J'e) is calAB,ledBtheE cycl i c property of the trace. If A E Y1 , the mapping l A (B) .Z(.Ye), is a continuous linear functional on .Z(.Ye), so that Y1 .Z(.Ye)*-the dual Banach space to .Z(.Ye). However, Y1 i- .Z(.Ye)*, butof Banach rather spaces. Y1 'if(£)* , and the mapping A lA i s an isomorphism .Z (.Ye) Yt . Similarly, the mapping B lB gives an isomorphism An operator A E .Z(.Ye) is called a Hilbert-Schmidt operator if AA* E ..9'1 . Equivalently, an operator E is a Hiforlb£ert-Schmidt operator if and only if for some orthonormal basis {en}�=1 Tr

= Tr

= Tr

=

E

Tr

c

=

�---t

=

A

�---t

2 (£)

00

n= 1

..9'2 of Hi l b ert-Schmidt operators i n is a Hilbert space AB * . The Hi l b ert-Schmidt space ..9'2 is wialThestohavector thetwo-siinnerspace product (A, ded ideal in the C*-algebra .Z(.Ye). Quantum mechani cs cannot studies betheadequatel microworly descri d - btheed byphysiclacssial clalawsmechan­ at an atomi c scal e that iexperi cs. Theencesproperti e s of the mi c roworl d are so di ff erent from our everyday that itcliasssinocalsurpri se thatcs andits laclwsassiseem toectrodynami contradictcthes cannot com­ mon sense. Thus mechani c al el expl a i n the stabi l i t y of atoms and mol e cul e s. Nei t her can these theori e s rec­ le dioffnerent propertieands ofiltisght,partiitsclwave-l ike behavi orphoto-el in interference ands­ dionci ffsionracti phenomena, e -l i k e behavi o r i n e ctri c emi and scattering by free photons. The fundamental difference between B) 2 = Tr

1.

Observables, states, and dynamics

£

2. Basic Principles of Quantum Mechanics

66

the microworld and the perceived world around us is that in the microworld every experiment results in interaction with the system and thus disturbs its properties, whereas in classical physics it is always assumed that one can neglect the disturbances the measurement brings upon a system. This imposes a limitation on our powers of observation and leads to a conclusion that there exist observables which cannot be measured simultaneously. We will not discuss here these and other basic experimental facts, re­ ferring the interested reader to physics textbooks. Nor will we follow the historic path of the theory. Instead, we show how to formulate quantum me­ chanics using the general notions of states, observables, and time evolution, described in Section in Chapter There we have seen that commutativ­ ity of the algebra of observables A results in its realization as an algebra of functions on the topological space - the space of states - and thus brings us to the realm of classical mechanics. Therefore in order to get a realiza­ tion of observables and states which is different from classical mechanics, we must assume that the C* -algebra associated with observables is no longer commutative. A fundamental example of a non-commutative C* -algebra is given by the algebra of all bounded operators on a complex Hilbert space, and it turns out that it is this algebra which plays a fundamental role in quantum mechanics! Here we formulate the basic principles of quantum mechanics using pre­ cise mathematical language. At this point it should be noted that one can not verify directly the principles lying at the foundation of quantum me­ chanics. Nevertheless, the validity of quantum mechanics, whenever it is applicable, is continuously being confirmed by numerous experimental facts which perfectly agree with the predictions of the theory 7 .

2. 8

1.

1 . 1 . Mathematical formulation. The following axioms constitute the

basis of quantum mechanics.

A l . With every quantum system there is associated an infinite-dimensional separable complex Hilbert space £, in physics terminology called the space of states 8 . The Hilbert space of a composite quantum system is a tensor product of Hilbert spaces of component systems.

observables sz1 of a quantum system with the Hilbert space consists of all self-adjoint operators on £. The subset J<1Q sd n .Z(.Yf') of bounded observables is a vector space over R A2. The set of

.Yf'

=

A3. The set of states Y of a quantum system with a Hilbert space .Yf' consists of all positive (and hence self-adjoint) trace class operators M with 7This refers to non-relativistic phenomena at the atomic scale. 8The space of pure states, to be precise.

1 . Observables, states, and dynamics

67

=

Tt M 1 . Pure states are projection operators onto one-dimensional sub­ spaces of £. For '1/J E £, 1 1 '1/JII = 1 , the corresponding projection onto C'ljJ is denoted by P..p . All other states are called mixed states 9 • A4. A process of measurement is the correspondence .9/

X

Y

3

(A, M )

r>

f.tA

E

&' (!R) ,

which to every observable A E .9/ and state M E Y assigns a probability measure f.t A on R For every Borel subset E � !R, the quantity 0 :::; JtA (E) :::; 1 is the probability that for a quantum system in the state M the result of a measurement of the observable A belongs to E. The expectation value (the mean-value) of the observable A E .9/ in the state M E Y is where JtA ( >-.) measure f.t A ·

(A I M)

=

JtA ( ( - oo , >-.))

=

I:

).. dp,A ( >-.) ,

is a distribution function for the probability

The set of states Y is a convex set. According to the Hilbert-Schmidt theorem on the canonical decomposition for compact self-adjoint operators, for every M E Y there exists an orthonormal set { '1/Jn } �= l in .Yt' (finite or infinite, in the latter case N = oo ) such that =

N

L CXn P..Pn and Tr M = L CXn = 1 , N

n= l n=l where an > 0 are non-zero eigenvalues of M. Thus every mixed state is a convex linear combination of pure states. The following result characterizes the pure states.

M

(1.1)

1.1.

state M E Y is a pure state if and only if it cannot be represented as a non-trivial convex linear combination in Y .

Lemma

A

Proof. Suppose that P..p = a M1

+ ( 1 - a) M2 , 0 < a < 1 , and let .Ye = C'ljJ EB Jri b e the orthogonal sum decomposition. Since M1 and M2 are positive operators, for r.p E Jri we have a ( M1 r.p, r.p ) :::; (P..pr.p, r.p ) = 0, so that ( M1 r.p, r.p) = 0 for all r.p E Jri and by (0. 1 ) we get M1 l .m = 0. Since M1 is self-adjoint, it leaves the complementary subspace C'ljJ invariant, and 0 from Tt M1 = 1 it follows that M1 = P..p . Therefore, M1 = M2 = P..p . 9In physics terminology, the operator M is called the density operator.

68

2. Basic Principles of Quantum Mechanics x ..7

icit constructi ontheorem of the correspondence iszesbasedthe onfundamental theExplgeneral spectral of von Neumann, whi c h emphasi role the self-adjoint operators play in quantum mechanics. measure on is a mapping .96' IR 2bounded £ ofoperators theA 0'-alprojection-valued gonebra£,.96'(IR) of Borel subsets of i n to the al g ebra of satisfying the following properties. Borel subset = and � IR,= is an orthogonal projection, i.Fore . , every = 0, I, the i dentity operator on £. For every disjoint union of Borel subsets,n = l-oo im i=l n � n=l in the strong topology on 2 (£ . Si m i l a rl y , a projecti o n-val u ed measure on i s a mappi n g n &6'(IR ) 2 £ , satisfying the same properti e s It follows from that for all .96'(R) . Wivaluthedevery projecti o n-val u ed measure on we associ a te a projecti o n­ function A)), thecallefdoltheowiprojection-valued ng properties. resolution of the identity. It is characterized by min lim = 0, >..l-imoo = . lim resol ution measure of the idention ty probabidefilintesy measure a distribwhen ution functiForoneveryBy the £the of the bounded polarization identity d

lR

Definition. ( )

P (E)

PM2. P (0) PM3.

P (E)2 P (IR) = 00

9 (1R)

lR

P :

( )

�

P (E) P (E) * .

E P (E)

PMl .

�

"" P(Ei)

P (E)

)

IRn PM1-PM3.

Remark. �

(

)

PM1-PM3

P (EI ) P (E2 )

( 1 .2 )

=

P (E1

n

E2 )

E1 , E2

E

R

P

P (A) = P ( ( - oo ,

PDL

P (A) P (J.L)

=

P(

PD2.

>.. - - oo

P (A)

{ A , J.L } ) . P (A)

I

PD3.

P (J.L)

J.t- >..

=

P (A) .

J.t<>..

I I 'P I I

=

( P (A)
R

P (A) (

= � { ( P (A) ( cp + 1/;) ,

P

:

69

1 . Observables, states, and dynamics

so that ( P (A)cp, 't/J) corresponds to a complex measure on lR a complex linear combination of measures. A measurable function f on lR is said to be finite almost everywhere (a. e.) with respect to the projection-valued measure P if it is finite a.e. with respect to all measures ( P'ljJ , 't/J) , 'ljJ E £. For separable £ a theorem of von Neumann states that for every projection-valued measure P there exists c,o E .Jit' such that a function f is finite a.e. with respect to P if and only if it is finite a.e. with respect to the measure ( Pep, e,o) . The next statement is the celebrated general spectral theorem of von Neumann. -

Theorem 1 . 1 (J. von Neumann) . For every self-adjoint operator A on the Hilbert space £ there exists a unique projection-valued resolution of the identity P (A ) , satisfying the following properties.

(i) D(A)

=

{

and for every c,o

c,o

E

E

£:

D (A) Ac,o

=

I:

A 2d ( P ( A ) cp, e,o)

<

00 } '

I:

A dP(A)cp,

I:

l f ( A) I 2 d ( P (A)cp, e,o) <

I:

f (A)d P (A)cp ,

defined as a limit of Riemann-Stieltjes sums in the strong topology on £ . The support of the corresponding projection-valued measure P coincides with the spectrum of the operator A : A E o- ( A ) if and only if PA ( (A - c, A + c )) '/= 0 for all c > 0. (ii) For every continuous function f on IR, f (A) is a linear operator on £ with a dense domain D ( f ( A) ) =

{

c,o

E

£:

defined for c,o E D ( f (A) ) as f (A)c,o =

00} '

and understood as in part (i) . The operator f(A) satisfies f (A) *

=

f(A) ,

where J is the complex conjugate function to f, and the operator f ( A ) is bounded if and only if the function f is bounded on o- (A) . For bounded on o- (A) continuous functions f and g , f ( A ) g (A) c,o =

I:

f (A)g ( A )d P ( A )cp,

c,o

E

£.

70

2. Basic Principles of Quantum Mechanics

( iii ) For every measurable function f on � ' finite a. e. with respect to the projection-valued measure P, f (A) is a linear operator on £ defined as in ( ii ) , where the integral for f(A)c.p is now understood in the weak sense: for

=

l: f (>.)d( P (>.) c.p, '1/J )

- a Lebesgue-Stieltjes integral with respect to a complex measure. The correspondence f f (A) satisfies the same properties as in ( ii ) , where the integrals are understood in the weak sense. ( iv ) A bounded operator B commutes with A, that is, B (D( A )) c D(A) and AB = BA on D (A) , if and only if it commutes with P(>.) for all >. and, therefore, B commutes with every operator f(A) . ( v ) For every projection-valued resolution of identity P (>.) the operator A on £, defined as in part ( i ) , is self-adjoint. �--->

Using the spectral theorem, the correspondence (A, M) lated in A4, can be explicitly described as follows.

�--->

/-LA , postu­

A5. The probability measure ItA on � ' which defines the correspondence sz1 x Y --> &' ( � ) , is given by the Born-von Neumann formula ( 1 .3) tt A ( E ) = Tr PA ( E ) M, E E 36' (�) , where PA is a projection-valued measure on � associated with the self-adjoint operator A.

It A on � can be considered as a "quan­ tum push-forward" of the state M by the observable A ( cf. the discussion in Section 2.8 in Chapter 1 ) . Remark. The probability measure

From the Hilbert-Schmidt decomposition { 1 . 1 ) we get

lt A ( E ) = L an ( PA ( E ) 'l/Jn , '1/Jn ) = L an ii PA ( E ) 'l/Jn il 2 � L an = 1 , n=l n=l n=l so that indeed 0 � tt A ( E ) � 1 . Denote by ItA (>.) the distribution function of the probability measure /-LA , ItA (>.) = ( PA (>.)'IjJ, '1/J ) for M = P'I/J . N

N

N

Suppose that an observable A E sz1 and a state M are such that (AI M) < oo and Im M D(A) . Then AM E J''i and (A I M) Tr AM. In particular, if M = P'I/J and 'ljJ E D(A) , then (A I M) = (A'ljJ, '1/J) and ( A 2 1 M) = II A 'I/J II 2 .

Proposition 1 . 1 .

�

=

E

Y

1.

71

Observables, states, and dynamics

Proof. Let { en }�= l be an orthonormal basis for 00

£. Since

n=l

we get for every E E 86'(IR) ,

00

/-L A ( E ) = L 1-tn ( E ) , n=l where 1-Ln are finite measures on defined by /-Ln ( E ) Since

lR

{ f d�-tA = { f d�-tn JJR n = l }"JR for every function f integrable with respect to the measure /-LA , it follows from the spectral theorem that

f

Thus AM E Yi and (A I M} = '1/J E D(A) ,

(AIM} =

'fr AM.

In particular, when M = P'I/J and

i: >.d (PA (>.)'lj;, 'lj;)

=

(A'l/;, '1/J ) .

Finally, from the spectral theorem and the change of variables formula we get

I I A 'I/J I I 2

= I: A2 d( PA (>.) 'lj;, = fooo '1/J)

Corollary 1 . 2 .

'fr AM.

Proof. Since

we get that e n 1.1.

If (A I M} , (A 2 I M}

< oo ,

>. d( PA 2 (>. ) '1j; , '1/J )

then AM E

I: A2d�-tn (A) I: A2d�-tA (A) :S

E

=

(A2 I M}

=

.7'!

(A2 I M } .

0

and (A I M}

< oo ,

D(AM) , and the result follows from the proof of Proposition 0

Remark. It is convenient to approximate an unbounded self-adjoint oper­

ator A by bounded operators A n = fn (A) , where fn = X[ - n , n] a char­ acteristic function of the interval [ -n, n] . Assuming that (AI M} exists, we have -

2. Basic Principles

72

of Quantum Mechanics

Definition. Self-adjoint operators A and B commute if the corresponding

projection-valued measures PA and PB commute, PA (EI ) PB (E2) = PB (E2) PA (EI ) for all EI , E2

E

�(�) .

The following results, which follow from the spectral theorem, are very useful in applications.

The following statements are equivalent. (i) Self-adjoint operators A and B commute . (ii) For all >-., J.L E Im >-., lm J.L =I= 0,

Proposition 1 . 2 .

C,

R>. (A)RJJ- (B) = RJJ- (B)R>. (A) .

(iii) For all u, v

E

�'

(iv) For all u E �' the operators eiuA and B commute . Slightly abusing notation 1 0 , we will often write [A, B] = AB - BA = 0 for commuting self-adjoint operators A and B.

Let A = {AI , . . . , A n } be a finite set of self-adjoint, pair­ wise commuting operators on .Yt' . There exists a unique projection-valued measure PA on the Borel subsets of �n having the following properties. (i) For every E = EI X · · · X En E �(�n ) , PA ( E ) PA 1 (EI ) . . . PAn (En) · (ii) In the strong operator topology,

Proposition 1 . 3 .

=

Ak =

r ).. k dPA , }TR n

k = 1 , . . . ' n,

where ).. k is the k-th coordinate function on �n , Ak ( XI , . . . , Xn )

Xk ·

=

(iii) For every measurable function f on �n , finite a. e. with respect to the projection-valued measure PA, f (AI , . . . , An ) is a linear opera­

tor on .Yt' defined by

{ }TR n where the integral is understood in the weak operator topology. The f ( AI ,

. . . , An )

=

f d PA ,

f---+

correspondence f ! ( A I , . . . , An) satisfies the same properties as in part (ii) of the spectral theorem. 1 0 In general, for unbounded self-adjoint operators A and B the commutator [A, B] = AB-B A is not necessarily closed, i.e. , it could be defined only for cp = 0 .

1 . Observables, states, and dynamics

73

The support of the projection-valued measure PA on lRn is called the joint spectrum of the commutative family A = {AI , . . . , An } · Remark. According to von Neumann's theorem on a generating operator, for every commutative family A of self-adjoint operators ( not necessarily

finite ) on a separable Hilbert space .Yt' there is a generating operator - a self-adjoint operator R on .Yt' such that all operators in A are functions of R.

It seems natural that simultaneous measurement of a finite set of ob­ servables A = { AI , . . . , An } in the state M E Y should be described by the probability measure /-LA on lRn given by the following generalization of the Born-von Neumann formula: (1.4) J.LA ( E) Tr( PA1 (EI ) . . . PAn (En ) M) , E = EI x · · · x En E �(JRn ) . However, formula ( 1 . 4) defines a probability measure on lRn i f and only if PA 1 (EI ) . . . PAn (En ) defines a projection-valued measure on JRn . Since a product of orthogonal projections is an orthogonal projection only when the projection operators commute, we conclude that the operators AI , . . . , An should form a commutative family. This agrees with the requirement that simultaneous measurement of several observables should be independent of the order of the measurements of individual observables. We summarize these arguments as the following axiom. =

A6. A finite set of observables A = {A1 , . . . , An } can be measured simul­ taneously (simultaneously measured observables ) if and only if they form a

commutative family. Simultaneous measurement of the commutative family A C S2l in the state M E Y is described by the probability measure /-LA on lRn given by J.LA( E ) = Tr PA ( E ) M , E E �(JRn ) , where PA is the projection-valued measure from Proposition 1 .3. Explicitly, n PA ( E ) = PA 1 (E1 ) . . . PAn ( En ) for E E1 X X En E �(JR ) . For ev­ n ery Borel subset E <; lR the quantity 0 ::; /-LA ( E) ::; 1 is the probability that for a quantum system in the state M the result of the simultaneous measurement of observables A 1 , . . . , An belongs to E. =

·

·

·

The axioms Al-A6 are known as Dirac-von Neumann axioms. Problem 1 . 1 . Prove property ( 1 .2) . Problem 1 . 2 . Prove that the state M is a pure state if and only if Tr M 2

=

1.

Problem 1 . 3 . Prove that the Born-von Neumann formula ( 1 .3) defines a proba­ bility measure on JR, i.e., f..l A is a a-additive function on JB ( JR ) . Problem 1 .4. Prove all the remaining statements in this section.

74

2. Basic Principles of Quantum Mechanics

1 . 2 . Heisenberg's uncertainty relations. The variance of the observ­ able A in the state M, which measures the mean deviation of A from its expectation value, is defined by

a� ( A ) = ( (A - (AJ M ) I) 2 J M ) = ( A 2 J M ) - (AJ M ) 2 � 0, provided the expectation values ( A 2 J M ) and (A J M ) exist. It follows from Proposition 1 . 1 that for M = P.,p , where '1/J E D (A) , a� (A)

=

J J (A - ( A J M ) J) 'I/J I I 2 = J J A'I/J J J 2 - (A'!jJ , 'IjJ ) 2 .

For A E d and M E Y the variance O"M (A) = 0 if and only eigenspace for the operator A with the eigenvalue a ( A J M ) . In particular, if M = P.,p, then '1/J is an eigenvector of A, A'ljJ a'I/J . Lemma 1 .2. if Im M is an

=

=

Proof. It follows from the spectral theorem that a� (A)

=

1: (.X - a) 2 d!LA (.X) ,

so that aM (A) 0 if and only if the probability measure /LA is supported at the point a E IR, i.e. , ILA ( {a}) = Since ILA({a}) Tr PA ({a})M and TrM = 1 , we conclude that this is equivalent to Im M being an invariant subspace for PA ( {a} ) , and it follows from the spectral theorem that Im M D is an eigenspace for A with the eigenvalue a. =

1.

=

Now we formulate generalized Heisenberg 's uncertainty relations.

Proposition 1 . 4 (H. Weyl ) . Let A, B E d and let M = P.,p state such that '1/J E D (A) n D (B) and A'ljJ, B'I/J E D (A) n D ( B ) . a� (A) a� (B)

�

be the pure Then

� ( i [A, BJ J M ) 2 .

The same inequality holds for all M E Y, where by definition ( i [A, B] J M )

limn._oo ( i [An , Bn ] J M ) . Proof. Let M

=

P.,p . Since [A - (A J M ) J, B - ( B J M ) J] = [A, B] ,

it is sufficient to prove the inequality ( A 2 J M) ( B 2 J M )

We have for all o: E IR , l l (A + i o: B) 'I/J l l 2

�

� ( i [A , B ] JM)2.

=

o: 2 ( B 'IjJ , B 'I/J ) - io: ( A'I/J , B 'I/J ) + io: (B 'I/J , A 'ljJ ) + (A'IjJ , A 'ljJ ) = o: 2 (B 2 '1/J , '1/J) + o: ( i [A , B ] 'I/J , '1/J) + ( A2 '1jJ , '1/J) � 0,

so that necessarily 4(A 2 '1jJ, 'I/J) (B 2 '1jJ, 'I/J )

�

( i [A, B ] 'I/J , 'l/J) 2 .

=

1 . Observables, states, and dynamics

75

The same argument works for the mixed states. Since o-1t- (A) o-L (B)

= n---+ limoo o-L (An ) o-1t- (Bn )

( see the remark in the previous section ) , it is sufficient to prove the inequality for bounded A and B. Then using the cyclic property of the trace we have for all a E JR, 0 ::; Tr( (A + iaB) M (A + iaB) * ) = Tr((A + iaB) M(A - iaB) ) =a2 Tr BMB + ia Tr BMA - ia Tr AMB + Tr AM A

=a2 Tr B2 M + a Tr(i [A, B] M) + Tr A2 M,

so that 4 Tr A2M Tr B 2 M ;::: Tr(i [A, B] M)2 .

D

Heisenberg's uncertainty relations provide a quantitative expression of the fact that even in a pure state non-commuting observables cannot be measured simultaneously. This shows a fundamental difference between the process of measurement in classical mechanics and in quantum mechanics. 1. 3 . Dynamics. The set d of quantum observables does not form an alge­ bra with respect to an operator product 11 . Nevertheless, a real vector space do of bounded observables has a Lie algebra structure with the Lie bracket

i [A, B]

=

A, B E dQ .

i(AB - BA) ,

Remark. In fact, the C*-algebra 2 (£) of bounded operators on £ has

a structure of a complex Lie algebra with the Lie bracket given by a com­ mutator [A , B] = AB - BA. It satisfies the Leibniz rule [AB , C]

=

A [B , C] + [A , C] B ,

so that the Lie bracket is a derivation of the C*-algebra 2(£) . In analogy with classical mechanics, we postulate that the time evolution of a quantum system with the space of states £ is completely determined by a special observable H E d, called a Hamiltonian operator ( Hamiltonian for brevity ) . As in classical mechanics, the Lie algebra structure on do leads to corresponding quantum equations of motion. Specifically, the analog of Hamilton's picture in classical mechanics ( see Section in Chapter 1) is the Heisenberg picture in quantum mechanics, where the states do not depend on time

2. 8

dM ---;tt = O,

M

E

Y,

1 1 The product of two non-commuting self-adjoint operators is not self-adjoint.

2. Basic Principles

76

of Quantum Mechanics

and bounded observables satisfy the Heisenberg equation of motion dA

dt = {H, A}n, A E d0 ,

(1 .5) where ( 1 .6)

is the quantum bracket - the !i-dependent Lie bracket on do . The positive number li, called the Planck constant, is one of the fundamental constants in physics 12 . The Heisenberg equation (1.5) is well defined when H E do . Indeed, let U(t) be a strongly continuous one-parameter group of unitary operators associated with a bounded self-adjoint operator H, ( 1 .7) U ( t) e - K tH , t E R It satisfies the differential equation =

ili

( 1 .8)

��t) = HU(t)

d

=

U(t)H,

so that the solution A ( t) of the Heisenberg equation of motion with the initial condition A ( O) = A E do is given by (1 .9) A ( t) = U(t)- 1 AU(t) . In general, a strongly one-parameter group of unitary operators (1.7) , as­ sociated with a self-adjoint operator H, satisfies differential equation (1 .8) only on D (H) in a strong sense, that is, applied to cp E D(H) . The quantum dynamics is defined by the same formula ( 1 .9) , and in this sense all quantum observables satisfy the Heisenberg equation of motion ( 1 .5). The evolution operator Ut d d is defined by Ut (A) = A ( t) U (t) - 1 AU(t) , and is an automorphism of the Lie algebra do of bounded observables. This is a quantum analog of the statement that the evolution operator in classical mechanics is an automorphism of the Poisson algebra of classical observables (see Theorem 2 . 19 in Section 2.7 of Chapter 1 ) . B y Stone ' s theorem, every strongly-continuous one-parameter group of unitary operators 1 3 U(t) is of the form ( 1 . 7) , where :

{'P

=

--+

U (t) - I

}

U(t) - I cp.

cp exists and Hcp = ili lim £ : t--+0 lim t--+0 t The domain D (H) of the self-adjoint operator H, called the infinitesimal generator of U ( t ) , is an invariant linear subspace for all operators U ( t) .

D ( H)

=

E

t

12The Planck constant has a physical dimension of the action ( energy x time ) . Its value x 1 0-27 erg x sec, which is determined from the experiment, manifests that quantum mechanics is a microscopic theory. 1 3 According to a theorem of von Neumann, on a separable Hilbert space every weakly mea­ surable one-parameter group of unitary operators is strongly continuous.

n

=

1 . 054

1.

77

Observables, states, and dynamics

We summarize the presented arguments as the following axiom. A7 (Heisenberg's Picture) . The dynamics of a quantum system is de­ scribed by the strongly continuous one-parameter group U ( t) of unitary op­ erators. Quantum states do not depend on time, Y 3 M � M (t) = M E Y , and time dependence of quantum observables is given by the evolution op­ erator Ut , d 3 A � A(t) = Ut (A) = U (t) - 1 AU(t) E d. Infinitesimally, the evolution of quantum observables is described by the Heisenberg equation of motion ( 1 .5), where the Hamiltonian operator H is the infinitesimal generator of U (t) . The analog of Liouville's picture in classical mechanics ( see Section 2.8 in Chapter 1) is Schrodinger 's picture in quantum mechanics, defined as follows. AS (Schrodinger's Picture) . The dynamics of a quantum system is de­ scribed by the strongly continuous one-parameter group U ( t) of unitary operators. Quantum observables do not depend on time, d 3 A � A ( t) = A E d, and time dependence of states is given by the inverse of the evolution operator ut- 1 = U- t , ( 1 . 10) Y 3 M � M(t) = U- t (M) = U (t) MU (t) - 1 E Y. Infinitesimally, the evolution of quantum states is described by the Schrodinger equation of motion dM = - {H, M}h, dt

(1.11)

ME

Y,

where the Hamiltonian operator H is the infinitesimal generator of U ( t ) . Proposition

equivalent.

1.5.

Heisenberg and Schrodinger descriptions of dynamics are

Proof. Let J.l A ( t ) and ( J.lt ) A be, respectively, probability measures on JR. asso­ ciated with (A( t) , M) E d x Y and ( A, M (t) ) E d x Y according to A3-A4, where A (t) = Ut (A) and M (t) = U- t (M) . We need to show that J.l A(t) = (J.lt)A · It follows from the spectral theorem that PA (t ) = U (t) - 1 PA U ( t) , so that using the Born-von Neumann formula ( 1 .3) and the cyclic property of the trace, we get for E E �(JR.) , J.lA (t) ( E ) = Tr PA(t) ( E ) M

=

Tr(U(t) - 1 PA ( E ) U ( t ) M )

= Tr( PA ( E ) U (t)MU( t) -1 ) = Tr PA ( E ) M (t) Corollary 1 .3. ( A(t) j M ) = ( A j M ( t ) ) .

=

( J.lt ) A ( E ) .

0

In analogy with classical mechanics ( see Section 1 . 4 of Chapter 1 ) , we have the following definition.

2. Basic Principles of Quantum Mechanics

78

J2l

is a quantum integral of motion ( or a constant of motion) for a quantum system with the Hamiltonian H if in Heisenberg's picture dA(t) 0 Definition. An observable

A

E

=

dt It follows from Proposition 1 .2 that

0

A

J2l

is an integral of motion if and only if it commutes with the Hamiltonian H, so that, in agreement with (1.5),

E

{ H, A} n = 0.

This is a quantum analog of the Poisson commutativity property, given by formula (2. 14) in Section 2.6 of Chapter 1. It follows from ( 1 . 1 1 ) that the time evolution of a pure state M = P'l/J is given by M(t) = P'l/J ( t ) where '1/J (t) = U(t)'ljJ. Since D(H) is invariant un­ der U(t) , the vector '!jJ ( t ) = U(t)'ljJ satisfies the time-dependent Schrodinger >

equation

d'I/J dt = H'I/J with the initial condition '1/J ( O ) = '1/J. Definition. A state M E Y is called stationary for a quantum system with Hamiltonian H if in Schrodinger's picture dM(t) = O. dt The state M is stationary if and only if [M, U(t)] = 0 for all t, and by Proposition 1 .2 this is equivalent to {H , M}n = 0, in agreement with ( 1 . 1 1 ) . The following simple result is fundamental. Lemma 1 . 3 . The pure state M P'l/J is stationary if and only if '1/J is an eigenvector for H, H'ljJ = A.'I/J, and in this case ( 1 . 12)

in

=

U(t)P'l/; = P'l/J U(t) that '1/J is a common eigenvector for unitary operators U(t) for all t, U(t)'ljJ = c(t)'ljJ, l c(t) l = 1. Since U(t) is Proof. It follows from

a strongly continuous one-parameter group of unitary operators, the contin­ uous function c(t) = (U(t)'ljJ, '1/J) satisfies the equation c(t1 + t2) = c(ti)c(t2) for all t1, t2 E JR., so that c(t) e- Pt for some )... E R Thus by Stone's theorem '1/J E D (H) and H'ljJ = A.'ljJ. 0 =

79

2. Quantization

In physics terminology, the eigenvectors of H are called bound states. The corresponding eigenvalues are called energy levels and are usually denoted by E. The eigenvalue equation H'lj; E'lj; is called the stationary Schrodinger =

equation.

Problem 1 . 5 . Show that if an observable A is such that for every state M the expectation value (AI M(t)) does not depend on t, then A is a quantum integral of motion. ( This is the definition of integrals of motion in the Schrodinger picture. ) Problem 1 .6. Show that a solution of the initial value problem for the time­ dependent Schrodinger equation ( 1 . 12) is given by '!f; (t)

=

j_: e-*t>.. dP(>-.)'1/J,

where P(>-.) is the resolution of identity for the Hamiltonian H.

Problem 1 . 7. Let D be a linear subspace of £, consisting of Carding vectors 1/;J

=

/_: f (s)U( s)'!f; ds ,

f E Y(IR) , 'If;

E

.Yt',

where Y (IR) is the Schwartz space of rapidly decreasing functions on R Prove that D is dense in .Yt' and is invariant for U(t) and for the Hamiltonian H. (Hint: Show that U(t) 'l/Jt = '1/J t, E D, where ft ( s) = f( s - t) , and deduce H'l/lt �'1/Jt' ·) =

2.

Quantization

To study the quantum system one needs to describe its Hilbert space of states £ and the Hamiltonian H - a self-adjoint operator in £ which defines the evolution of a system. When the quantum system has a classical analog, the procedure of constructing the corresponding Hilbert space £ and the Hamiltonian H is called quantization. Definition. Quantization of a classical system ( (vll , { , } ) , He ) with the Hamiltonian function 1 4 He is a one-to-one mapping Qli : A -+ J2l from the set of classical observables A = C 00 (..4'l) to the set J2l of quantum

observables - the set of self-adjoint operators on a Hilbert space £. The map Q n depends on the parameter n > 0, and its restriction to the subspace of bounded classical observables Ao is a linear mapping to the subspace J210 of bounded quantum observables, which satisfies the properties p m � Q h 1 ( Q n (h ) Q n (h) + Q n ( h ) Q n ( !I ) ) = h h and

,.-+0

14 Notation He is used to distinguish the Hamiltonian function in classical mechanics from the Hamiltonian operator H in quantum mechanics.

2. Basic Principles of Quantum Mechanics

80

The latter property is the celebrated correspondence principle of Niels Bohr. In particular, He Qn, ( H ) = H the Hamiltonian operator for a quantum system. f--+

-

e

Remark. In physics literature the correspondence principle is often stated

in the form

,

[ ]

c::

li

li i { } as ,

�

0.

Quantum mechanics is different from classical mechanics, so that the correspondence f Q n,( f) cannot be an isomorphism between the Lie alge­ bras of bounded classical and quantum observables with respect to classical and quantum brackets. It becomes an isomorphism only in the limit li � 0 when, according to the correspondence principle, quantum mechanics turns into classical mechanics. Since quantum mechanics provides a more accurate and refined description than classical mechanics, quantization of a classical system may not be unique. f--+

Definition. Two quantizations Q�1) and Q�) of a given classical system ( ( .4', { , } ) , He ) are said to be equivalent if there exists a linear mapping %'n, : A � A such that Q � ) = Q�1 ) o %'n, and lim %'n, = id.

n,_,o

For many "real world" quantum systems - the systems describing actual physical phenomena - the corresponding Hamiltonian H does not depend on a choice of equivalent quantization, and is uniquely determined by the classical Hamiltonian function He . 2 . 1 . Heisenberg commutation relations. The simplest classical system

with one degree of freedom is described by the phase space R 2 with coor­ dinates p, q and the Poisson bracket { , } associated with the canonical symplectic form w = dp A dq. In particular, the Poisson bracket between classical observables p and q - momentum and coordinate of a particle ­ has the following simple form: ,

{p, q}

(2 . 1 )

=

1.

It is another postulate o f quantum mechanics that under quantization clas­ sical observables p and q correspond to quantum observables P and Q self-adjoint operators on a Hilbert space £, satisfying the following prop­ erties. CRl . There is a dense linear subset D C Yt' such that P : D � D and

-

Q : D � D.

CR2. For all '1/J E D,

(PQ - QP ) 'Ij;

=

-ili'lj;.

2. Quantization

81

£ which commutes with P and Q is a multiple of the identity operator I. Property CR2 is called the Heisenberg commutation relation for one

CR3. Every bounded operator on

degree of freedom. In terms of the quantum bracket ( 1 . 6 ) it takes the form

I, which is exactly the same as the Poisson bracket ( 2. 1 ) . Property CR3 is a quantum analog of the classical property that the Poisson manifold (R2 , { , } ) is non-degenerate: every function which Poisson commutes with p and q is a constant ( see the last remark in Section 2.7 of Chapter 1 ) . The operators P and Q are called, respectively, the momentum operator and the coordinate operator. The correspondence p � P, q � Q with P and Q satisfying CR1-CR3 is the cornerstone for the quantization of classical systems. The validity of (2.2) , as well as of quantum mechanics as a whole, is confirmed by the agreement of the theory with numerous experiments. (2 . 2 )

{ P, Q } n

=

Remark. It is tempting to extend the correspondence p P, q � Q to all observables by defining the mapping f (p, q) f ( P, Q) . However, this approach to quantization is rather naive: operators P and Q satisfy (2.2 ) and do not commute, so that one needs to understand how f ( P, Q) - a "function of non-commuting variables" - is actually defined. We will address this problem of the ordering of non-commuting operators P and Q in Section 3 .3. �

�

It follows from Heisenberg's uncertainty relations ( see Proposition 1 .4) , that for any pure state M = P'I/J with 1/J E D, n O"M (P) aM (Q) 2: 2 ·

This is a fundamental result saying that it is impossible to measure the coordinate and the momentum of a quantum particle simultaneously: the more accurate the measurement of one quantity is, the less accurate the value of the other is. It is often said that a quantum particle has no observed path, so that "quantum motion" differs dramatically from the motion in classical mechanics. It is now straightforward to consider a classical system with n degrees of freedom, described by the phase space R 2n with coordinates p = (Pl , . . . , Pn ) and q = (q1 , . . . , qn ) , and the Poisson bracket { , } , associated with the canonical symplectic form w = dp 1\ dq. The Poisson brackets between clas­ sical observables p and q - momenta and coordinates of a particle - have the form (2 . 3) {Pk , Pz } = o , {q k , q 1 } = o , { pk > q 1 } = 8L k , l = 1 , . . . , n .

2.

82

Basic Principles of Quantum Mechanics

Corresponding momenta and coordinate operators = (Pt , , Pn ) and Q = ( Q1, . , Q n ) are self-adjoint operators that have a common invariant dense linear subset D c £, and on D satisfy the following commutation relations:

P

. .

. . .

These relations are called Heisenberg commutation relations for n degrees of freedom. The analog of CR3 is the property that every bounded operator on £ which commutes with all operators and is a multiple of the identity operator I . The fundamental algebraic structure associated with Heisenberg com­ mutation relations is the so-called Heisenberg algebra.

P

Q

Definition. The Heisenberg algebra � n with n degrees of freedom is a Lie algebra with the generators e1 , . . , en , ft , . . . , fn , c and the relations (2. 5 ) [ e k , c] = O, [fk , c] = O , [e k , Jzl = 8fc, k, l = 1 , . . . , n. .

The invariant definition is the following. Let (V, w ) be a 2n-dimensional symplectic vector space considered as an abelian Lie algebra, and let g be a one-dimensional central extension of V by a Lie algebra 2-cocycle given by the bilinear form w. This means that there is an exact sequence of vector spaces 0 JR. g v 0, (2.6) and the Lie bracket in g is defined by (2.7) [x , y] w ( x , y ) c , where x, fj are the images in V of elements x , y E g , and c is the image of 1 under the embedding JR. � g, called the central element of g . A choice of a symplectic basis e1 , . . . , e n , ft , . . . , fn for V (see Section 2.6 of Chapter 1) establishes the isomorphism g � � n , and relations (2.5) are obtained from the Lie bracket (2.7) . By Ado's theorem, the Heisenberg algebra � n is isomorphic to a Lie subalgebra of a matrix algebra over R Explicitly, it is realized as a nilpotent subalgebra of the Lie algebra g (n + 2 of ( n + 2) x ( n + 2) matrices with the elements 0 u l u2 un 0: 0 0 0 0 Vt n 0 0 V2 0 0 (2.8) L ( uk fk + vk e k ) + o:c -+

-+

-+

-+

=

k=l

=

0 0 0 0

0 0

0 0

Vn

0

2.

83

Quantization

Remark. The faithful representation fJ n __,. g (n + 2 , given by (2 .8 ) , is clearly reducible: the subspace V = { x ( x 1 , . . . , Xn +2 ) E � n + 2 : Xn +2 0} is an invariant subspace for fJn with the central element c acting by zero. However, this representation is not decomposable: the vector space �n +2 cannot be =

=

written as a direct sum of V and a one-dimensional invariant subspace for fJ n · This explains why the central element c is not represented by a diagonal matrix with the first n + 1 zeros, but rather has a special form given by ( 2.8 ) .

Analytically, Heisenberg commutation relations (2 . 5 ) correspond to an irreducible unitary representation of the Heisenberg Lie algebra fJ n · Recall that a unitary representation p of fJn in the Hilbert space £ is the linear mapping p fJn i.rd - the space of skew-Hermitian operators in £ ­ such that all self-adjoint operators i p(x ) , x E fJ n , have a common invariant dense linear subset D c £ and satisfy :

__,.

p ( [x , y] ) cp = ( p ( x ) p (y) - p( y) p ( x ) ) cp,

x, y

E

fJ n , cp E D .

Formally applying Schur's lemma we say that the representation p is irre­ ducible if every bounded operator which commutes with all operators i p(x ) is a multiple of the identity operator I. Then Heisenberg commutation rela­ tions (2 .5) define an irreducible unitary representation p of the Heisenberg Lie algebra fJn in the Hilbert space £ by setting =

=

l , . , n, p(c) = - iltl. Since the operators pk and Qk are necessarily unbounded (see Problem 2. 1 ) , the condition PkP, cp PzPk cp for all cp E D does not necessarily imply (see Problem 2 . 2 ) that self-adjoint operators Pk and P1 commute in the sense of the definition in Section 1.1. To avoid such "pathological" representations, we will assume that p is an integrable rep­ resentation, i.e., it can be integrated (in a precise sense specified below) to an irreducible unitary representation of the Heisenberg group H n - a connected, simply-connected Lie group with the Lie algebra fJ n · Explicitly, the Heisenberg group is a unipotent subgroup of the Lie al­ gebra SL(n + 2 , �) with the elements un 0: 1 u l u2 0 1 0 0 Vl 0 V2 0 0 1 (2 . 9 )

p ( fk )

-i Pk ,

p( e k ) = - iQ k ,

k

.

=

g=

0 0 0 0

0 0

1 0

Vn

1

.

2. Basic Principles of Quantum Mechanics

84

The exponential map exp : �n .- Hn is onto, and the Heisenberg group is generated by two n-parameter abelian subgroups exp uX = exp

(t fk) k=l

uk

,

exp v Y = exp

(t ) k=l

V

k e k , u, v E !Rn ,

and a one-parameter center exp a c , which satisfy the relations (2. 10)

Hn

exp uX exp vY = exp ( -uvc) exp vY exp uX,

n

uv = L: uk vk ·

k=O

Indeed, it follows from (2.5) that [uX, vY] = -uvc is a central element, so that using the Baker-Campbell-Hausdorff formula we obtain exp uX exp vY = exp ( - � uvc) exp ( uX + vY ) , exp vY exp uX = exp ( � uvc) exp ( uX + vY ) . In the matrix realization, the exponential map is given by the matrix expo­ nential and we get eux = I + uX, evY = I + vY, and eac = I + ac, where I is the ( n + 2) x ( n + 2) identity matrix. Let R be an irreducible unitary representation of the Heisenberg group Hn in the Hilbert space £ - a strongly continuous group homomorphism R : Hn .- %' ( £ ) , where %' ( £ ) is the group of unitary operators in £. By Schur ' s lemma, R ( e a c ) = e-i >.a I, >. E R Suppose now that >. = n, and define two strongly continuous n-parameter abelian groups of unitary operators U ( u ) = R ( exp uX ) , V ( v ) = R ( exp vY ) , u, v E !Rn . Then it follows from (2. 10) that unitary operators U ( u ) and V ( v ) satisfy commutation relations U ( u ) V ( v ) = e i nuv v ( v ) U ( u ) , (2. 1 1 ) called Weyl relations. Let P = ( P1 , . . . , Pn ) and Q = (Q1 , . . . , Qn ) be, respectively, infinitesimal generators of the subgroups U ( u ) and V ( v ) , given by the Stone theorem,

I

pk = i 8 U(:) 8u u= o

k = 1,

.

.

.

, n.

Taking the second partial derivatives of Weyl relations (2 . 1 1) at the origin u = v = 0 and using the solution of Problem 1 . 7 in the previous section, we easily obtain the following result. .-

Let R : Hn %' ( £ ) be an irreducible unitary representation of the Heisenberg group Hn in £ such that R ( eac) = e-ina I, and let P ( P1 , . . . , Pn ) and Q = ( Q 1 , . . . , Q n ) be, respectively, infinitesimal generators of the strongly continuous n-parameter abelian subgroups U ( u ) and V ( v ) .

Lemma 2 . 1 .

=

2.

85

Quantization

Then formulas (2.9) define an irreducible unitary representation Heisenberg algebra �n in .Yt' .

p

of the

The representation p in Lemma 2 . 1 i s called the differential o f a represen­ tation R, and is denoted by dR. The irreducible unitary representation p of IJn is c alled integrable if p = dR for some irreducible unitary representation R of Hn .

Remark. Not every irreducible unitary representation of the Heisenberg algebra is integrable, so that Weyl relations cannot be obtained from the Heisenberg commutation relations. However, the following heuristic argu­ ment (which ignores the subtleties of dealing with unbounded operators) is commonly used in physics textbooks. Consider the case of one degree of freedom and start with { P, Q } n = I . Since the quantum bracket satisfies the Leibniz rule we have, for a "suitable" function f, {!( P) , Q }h = f ( P ) . In particular, choosing f (P) = e- iu P = U(u ) , we obtain U(u)Q - Q U (u) = nuU(u) or U(u)QU(u)-1 = Q + nul. '

This implies, for a "suitable" function g,

U(u) g(Q) = g(Q + nu i)U( u ) , and setting g(Q) = e- ivQ

=

V(v) , we get the Weyl relation.

We will prove in Section 3. 1 that all integrable irreducible unitary repre­ sentations of the Heisenberg algebra IJ n with the same action of the central element c are unitarily equivalent. This justifies the following mathemati­ cal formulation of the Heisenberg commutation relations for n degrees of freedom. A9 (Heisenberg's Commutation Relations) . Momenta and coordinate operators P (Pb . , Pn ) and Q = (Q1 , . . . , Q n ) for a quantum particle with n degrees of freedom are defined by formulas (2 . 9) , where p is an integrable irreducible unitary representation of the Heisenberg algebra IJ n with the property p( c ) = - ilil. =

.

.

Problem 2 . 1 . Prove that there are n o bounded operators o n the Hilbert space £ satisfying [A, B] = I . Problem 2.2. Give an example of self-adjoint operators A and B which have a common invariant dense linear subset D E D , but eiA and eiB do not commute.

cp

C

£ such that ABcp

=

BAcp for all

2. Basic Principles of Quantum Mechanics

86

Problem 2 . 3 . Prove Lemma 2 . 1 . ( Hint: As in Problem 1 . 7, let D be the linear set of Garding vectors

�f

=

{ f(u, v ) U(u)V(v)� dnudnv, }JR2n

f E Y(IR2n ) , � E £,

where Y (IR2n ) is the Schwartz space of rapidly decreasing functions on IR 2 n . )

2 . 2 . Coordinate and momentum representations. We start with the case of one degree of freedom and consider two natural realizations of the Heisenberg commutation relation. They are defined by the property that one of the self-adjoint operators P and Q is "diagonal" (i.e. , is a multiplication by a function operator in the corresponding Hilbert space) . In the coordinate representation, £ L 2 (JR, dq ) is the L 2 -space on the configuration space lR with the coordinate q, which is a Lagrangian subspace of JR2 defined by the equation p 0. Set =

D (Q)

and for cp

E

=

{

=

cp E

.Yt' :

i:

q2 i cp (q) i2dq

<

oo }

D ( Q) define the operator Q as a "multiplication by q operator" , (Qcp) (q)

=

qcp(q) , q E JR,

justifying the name coordinate representation. The coordinate operator Q is obviously self-adjoint and its projection-valued measure is given by (P(E ) cp) (q ) = XE (q) cp(q ) , (2. 12) where XE is the characteristic function of a Borel subset E � R Therefore supp P = lR and a (Q) = R Recall that a self-adjoint operator A has an absolutely continuous spec­ trum if for every � E £, 11'1/!11 = 1 , the probability measure v'l/1 , v'I/J ( E ) = (PA (E)'Ij;, '1/!), E E �(JR) , is absolutely continuous with respect to the Lebesgue measure on R

The coordinate operator Q has an absolutely continuous spec­ trum JR, and every bounded operator B which commutes with Q is a function of Q, B = f(Q) with f E L00 (1R) .

Lemma 2 . 2 .

Proof. It follows from (2. 12) that v'I/J (E)

=

JE l 'l/J (q) l 2 dq, which proves the

first statement. Now a bounded operator B on £ commutes with Q if and only if B P (E) = P (E) B for all E E �(JR) , and using (2.12) we get (2. 13) B ( X E 'P) = XE B (cp ) . Choosing in (2. 13) E = E1 and cp = X E2 , where E1 and E2 have finite Lebesgue measure, we obtain B ( x El XE2 ) B ( xE nE2 ) = XEl B ( xEJ = X E2 B ( xE ) , ·

=

1

1

87

2. Quantization

so that denoting !E = B ( xE ) we get supp !E fE1 l E1 nE2

=

�

E, and

fE2 l E1 nE2

for all E1 , E2 E �(JR.) with finite Lebesgue measure. Thus there exists a measurable function f on JR. such that f i E = !E I E for every E E � ( JR. ) with finite Lebesgue measure. The linear subspace spanned by all XE E L 2 (JR) is dense in L 2 (JR) and the operator B is continuous, so that we get ( B r.p ) (q) = f (q) r.p (q) for all

0 Since B is a bounded operator, f E £C>O (JR) and I I B II = l l f l l oo · Remark. By the Schwartz kernel theorem, the operator B can be repre­ sented by an integral operator with a distributional kernel K(q, q') . Then the commutativity BQ Q B implies that, in the distributional sense, =

= 0, so that K is "proportional" to the Dirac delta-function, i.e. , (q - q') K(q, q')

K (q, q')

with some f books.

E

=

f (q)8(q - q') ,

L00 (JR) . This argument is usually given in the physics text­

Remark. The operator Q has no eigenvectors - the eigenvalue equation Qr.p = A
has no solutions in L 2 (JR) . However, in the distributional sense, this equation for every A E JR. has a unique ( up to a constant factor ) solution 'P>.. (q) = 8 ( q - A ) , and these "generalized eigenfunctions" , also called eigenfunctions of the continuous spectrum, combine to a Schwarz kernel of the identity operator I on L 2 (JR) . This reflects the fact that operator Q is diagonal in the coordinate representation. Remark. Normalization of the eigenfunctions of the continuous spectrum 'P>.. (q) can be also determined by the condition that for every A E JR. the

function

satisfies

(2.14) X(>..o ,>.. ) - the

Here Ao cf>>.. J I 2 =

E

.6. .

(2.14). Indeed, in our case W>.. = characteristic function of the interval ( Ao , A) - so that l i ef>>..+ � ­

JR. is fixed and does not enter

2.

88

Basic Principles of Quantum Mechanics

=

For a pure state M P'I/J , l l 7jJ I I = 1 , the corresponding probability mea­ sure J..l Q on lR is given by J..l Q ( E)

=

V1f; ( E)

=

fe l 7jJ (q) l 2 dq,

E

E

.@ (JR) .

Physically, this is interpreted that in the state P'I/J with the "wave function" 'ij!(q) , the probability of finding a quantum particle between q and q + dq is l 7jJ (q) l 2dq. In other words, the modulus square of a wave function is the probability distribution for the coordinate of a quantum particle. The corresponding momentum operator P is given by a differential operator p = � .!!:__ i dq

with D (P) = W1•2 (JR) - the Sobolev space of absolutely continuous func­ tions f on lR such that f and its derivative f ' (defined a.e.) are in L2 (JR) . The operator P is self-adjoint and it is straightforward to verify that on D = Cgo (JR) , the space of smooth functions on lR with compact support, QP - PQ = ifil. e quation

Remark. The operator P on £ has no eigenvectors - the eigenvalue

Pep = p ep, p E lR,

has a solution

ep (q) = const x e* pq which does not belong to L2(JR) . We will see later that the family of nor­

m alized eigenfunctions of the continuous spectrum epp (q)

i

1 = �e-,; Pq

combines to a Schwartz kernel of the inverse fi-dependent Fourier transform operator, which diagonalizes the momentum operator P. In the distribu­ tional sense, epp (q)c.ppr (q)dq = 8(p - p1 ) .

I:

Remark. As for the case of the coordinate operator, the normalization of the eigenfunctions of the continuous spectrum cpp ( q) of the momentum operator can be determined from the condition (2. 14) . Indeed,

so that

2.

89

Quantization

Using the elementary integral, we obtain 1

(q) (q) 2 � I I p+A - <1>p l l

so that

c =

=

k·

2 c2 n

�

oo �� q -- dq -oo q 2

=

27r c2 /i,

Proposition 2 . 1 . The coordinate representation defines an irreducible, uni­ tary, integrable representation of the Heisenberg algebra. Proof. To show that the coordinate representation is integrable, let U (u) = e - ivQ be the corresponding one-parameter groups of uni­ tary operators. Clearly, (V( v) cp)'ljJ (q) = e- ivqcp (q) and it easily follows from the Stone theorem (or by the definition of a derivative) that (U(u)cp) (q) = cp ( q - nu) , so that unitary operators U ( u) and V ( v) satisfy the Weyl rela­ tion (2. 1 1 ) . Such a realization of the Weyl relation is called the Schrodinger representation.

e- iuP and V (v)

=

To prove that the coordinate representation is irreducible, let B be a bounded operator commuting with P and Q. By Lemma 2.2, T = J (Q) for some f E £00(�). Now commutativity between T and P implies that TU(u)

=

u E �'

U(u)T for all

which is equivalent to f (q - nu) = f (q) for all q , u E � ' so that f a.e. on R

=

0

canst

To summarize, the coordinate representation is characterized by the property that the coordinate operator Q is a multiplication by q operator and the momentum operator P is a differentiation operator,

Q = q and P = nz dqd

--:- - .

Similarly, momentum representation is defined by the property that the momentum operator P is a multiplication by p operator. Namely let £ = £2(�, dp) be the Hilbert £2-space on the "momentum space" � with the coordinate p , which is a Lagrangian subspace of � 2 defined by the equation q = 0. The coordinate and momentum operators are given by

Q

=

in

!:_dp

and P

=

p,

and satisfy the Heisenberg commutation relation. As the coordinate repre­ sentation, the momentum representation is an irreducible, unitary, integrable representation of the Heisenberg algebra. In the momentum representation, the modulus square of the wave function '1/J(p ) of a pure state M = P'I/J ,

2.

90

Basic Principles of Quantum Mechanics

// 1/1 1 1 = 1 , is the probability distribution for the momentum of the quan­ tum particle, i.e. , the probability that a quantum particle has momentum between p and p + dp is /1/J(p) / 2 dp . Let ffn : L 2 (JR) L 2 (JR) be the n-dependent Fourier transform operator, defined by <j;(p) = ffn(
� joo


=

n e-*pq
By Plancherel's theorem, ffn is a unitary operator on L2 (JR) , ffnffn = ffnffn = I, and Q = ffnQffn- 1 , P = ffnPffn-1 , so that coordinate and momentum representations are unitarily equivalent. In particular, since the operator P is obviously self-adjoint, this immediately shows that the operator P is self-adjoint. For n degrees of freedom, the coordinate representation is defined by setting £ = L2 (JRn , dn q ) , where dn q dq 1 · · · dqn is the Lebesgue measure on lRn , and �

�

=

Here ]Rn is the configuration space with coordinates q - a Lagrangian sub­ space of JR2 n defined by the equations p = 0. The coordinate and momenta operators are self-adjoint and satisfy Heisenberg commutation relations. Projection-valued measures for the operators Q k are given by =

( P k ( E)

. ; l ( E) ( q )

where E E �(JR) component, k = 1, . , n . Correspondingly, the projection-valued measure P for the commutative family Q = (Q 1 , . . . , Q n ) ( see Proposition 1 .3) is defined on the Borel subsets E � JRn by ( P ( E )
91

2. Quantization

operator for some f E L00(1Rn ) . The proof repeats verbatim the proof of Lemma 2.2. For a pure state M P'fjJ , 11'¢11 = 1 , the modulus square l'¢(q) l 2 of the wave function i s the density of a joint distribution function J.L Q for the commutative family Q, i.e. , the probability of finding a quantum particle in a Borel subset E � 1Rn is given by =

J.L Q (E)

=

L l'¢(q) l 2dnq.

The coordinate representation defines an irreducible, unitary, integrable representation of the Heisenberg algebra � n · Indeed, n-parameter groups of unitary operators U(u) = e-iuP and V(v) = e-iv Q are given by (U(u)cp) (q) cp(q - nu) , (V(v)cp) (q) = e-iv q cp(q) , and satisfy Weyl relations (2. 1 1 ) . The same argument as in the proof of Proposition 2. 1 shows that this representation of the Heisenberg group Hn , called the Schrodinger representation for n degrees of freedom, is irreducible. In the momentum representation, £ L 2 (IRn, dnp) , where dnp dp1 · · · dpn is the Lebesgue measure on IRn, and Q = in = in , , in 0 n , P = p = (p1 , . . . , pn ) · 0 1 P Here IRn is the momentum space with coordinates p - a Lagrangian sub­ space of IR2n defined by the equations q = 0. The coordinate and momentum representations are unitarily equivalent by the Fourier transform. As in the case n = 1, the Fourier transform §h : L2 (1Rn) --. L2 (1Rn) is a unitary operator defined by =

=

=

:

( � ... � )

cp (p) = §n, (cp) (p) = (2 7r n) - n/2 r e - k pq cp(q) � q

}� n lim (27rn) -n/2 r e- k pqcp(q) � q, N->oo jj q j � N where the limit is understood in the strong topology on L 2 (1Rn ) . As in the case n = 1 , we have -1 -1 Qk = §n,Qk §n, , Pk §n, Pk §n, , k 1 , . , n. In particular, since operators Pr, . , Pn are obviously self-adjoint, this im­ mediately shows that P1 , . . . , Pn are also self-adjoint. Remark. Following Dirac, physicists denote a vector '¢ E £ by a ket vector 1'¢) , a vector cp E £* in the dual space to £ ( £* � £ is a complex anti­ linear isomorphism ) by a bra vector (cp l , and their inner product by (cpl'¢) . In standard mathematics notation, ('¢, cp) = (cpl'¢) and (A'¢, cp) = (cpiA I '¢) , =

�

�

=

..

=

..

92

2.

Basic Principles of Quant um Mechanics

where A is a linear operator. From a physics point of view, Dirac's notation is intuitive and convenient for working with coordinate and momentum rep­ resentations . Denoting by l q) = <5 ( q - q') and I P ) = ( 2 n1i) - n l2 ekpq the set of generalized common eigenfunctions for the operators Q and P, respectively, we formally get Q l q) = q l q) , P i p ) = PIP) , where operators Q act on q', and

(ql�) = }rR.n 6(q - q')�(q')dn ql �(q) , (P I�) = (211"n)- n l2 }R.f n e- k pq�(q)dn q -J; (p) , =

=

as well as ( q l q ' ) = <5 (q - q' ) , (PIP' ) = <5 (p - p' ) . Though in our exposition we are not using Dirac's notation , these ormulas would help the interested reader translate the notation used in physics textbooks to standard math­ ematics notation. "

"

f

3. 2 that every Lagrangian subspace of the

symplectic vector space JR2n with the canonical symplectic form w = dp l\ dq gives rise to an integrable, unitary, irreducible representation of the Heisen­ berg algebra � n · This is the simplest example of the real polarization, which for a given symplectic manifold ( .4', w ) is defined as an integrable distri­ bution {£'x } x E .A of Lagrangian subspaces £'x of tangent spaces Tx .4'. The notion of a polarization plays a fundamental role in geometric quantization: it allows us to construct ( under certain conditions ) the Hilbert space of states J't' associated with the classical phase space ( .4', w) . In the linear case .4' = JR2 n every Lagrangian subspace £' in JR2n gives rise to a real polarization by using the identification TxlR2n :::: JR2n. In particular, for the coordinate representation, £' is given by the equation q = 0, and for the momentum representation - by the equation p = 0. The corresponding Hilbert J't' space consists of functions on JR2n which are constant along the fibers of the polarization. Remark. We will show in Section

Problem 2.4. Give an example of a non-integrable representation of the Heisen­ berg algebra.

Problem 2.5. Prove that there exists cp E £ = L2 (JR:, dq) such that the vectors P(E)cp, E E �(JR:) , where P is a projection-valued measure for the coordinate

operator Q, are dense in £.

P roblem 2.6. Find the generating operator for the commutative family Q ( Q 1 , . . . Qn ) . Does it have a physical interpretation? ,

=

Problem 2. 7. Find the projection-valued measure for the commutative family P = (P1 , . . . , Pn ) in the coordinate representation.

93

2. Quantization

2.3. Free quantum particle. A

free classical particle with one degree of freedom is described by the phase space JR. 2 with coordinates p, q and the Poisson bracket (2 . 1 ) , and by the Hamiltonian function p2 (2.15)

Hc (P,

q) = 2m ·

The Hamiltonian operator of a free quantum particle with one degree of freedom is p2 Ho

= 2m '

and in coordinate representation is given by /t2 d2

Ho = - - - . .Ye =

2m dq2

L2 ( JR., dq ) with D ( Ho ) W 2 , 2 ( JR. ) the Sobolev space of functions in L2 (JR.) , whose generalized first and second derivatives are in L 2 (JR.) . The operator Ho is positive with absolutely continuous spectrum [0, oo ) of multiplicity two. Indeed, let SJo = L2 (JR.>o , o = (0, oo ) , which are square-integrable with respect to the measure dO" (.A) = � d.A , ll w ll 2 = (I1/J1 (.>.. W + I1/J2 ( .>.. W ) dO" ( .A ) < SJ o = w ( .A ) = It is a self-adjoint operator on

{

(����D

:

=

oo } .

fooo

It follows from the unitarity of the Fourier transform that the operator SJo , o/t'o L2 (JR., dq) :

--+

%'o (1/J) ( .A) =

w ( .A )

=

(t��)) ,

is unitary, %'0* %'o = I and o/t'o %'0* Io , where I and Io are , respectively, identity operators in .Ye and SJ o . The operator o/t'o establishes the isomor­ phism L2(JR., dq ) :::::' S) 02, and since in the momentum representation H0 is a multiplication by 2!n p operator, the operator o/t'o Ho %'0- 1 is a multiplication by .>.. operator in SJo . Remark. The Hamiltonian operator H0 has no eigenvectors - the eigen­ value equation Ho 1/J = .>.. 1/J has no solutions in L 2 ( JR. ) However, for every >. = 2!n k 2 > 0 this differential equation has two linear independent bounded solutions =

.

-1- e±* k q ' k > 0. � In the distribut ional sense, these eigenfunctions of the continuous spectrum combine to a Schwartz kernel of the unitary operator o/t'o , which establishes 1/J(±) k (q) =

2.

94

Basic Principles of Quantum Mechanics

the isomorphism between £ = L2 (ffi., dq) and the Hilbert space n o , where Ho acts as a multiplication by .A operator. The normalization of the eigen­ functions of the continuous spectrum is also determined by the condition

( 2 .1 4 )

:

. 1 ( k(+)+ �

li�o � llwi�� - w i±l ll 2 = 1 ' li�o �

where

w i±)(q) = Jk� 7/J�±)(q)dp.

\Ji

- '11

(+) (-) (-) ) k ' \Ji k+� - \Ji k = 0,

The Cauchy problem for the Schrodinger equation for a free particle,

( 2 .16)

in d:;t ) H07j;(t), 7/J(O) = 7/J, =

is easily solved by the Fourier transform. Indeed, in the momentum repre­ sentation it takes the form

.n fJ;J; (pt , t) = !!___ 2m_ .'�'�.(p , t) ' ,(f;(p, 0 ) = ,(f;(p), fJ ,(f;(p , t) e - � t ,(f;(p).

t

so that

. 2

( 2 .16) is given by 1 - 1 00 e*Pq;j;(p, t)dp = -1- 1 00 eix(p, q , t )t;j; (p)dp, ( 2 .17) 7/J(q, t) = ..j'j;ih, - 00 ..j'j;ih, - 00 where =

In the coordinate representat i on, the solution of

( 2 .17)

p2 pq t) q ( p , x , = - 2m + t .

( 2 .16)

Formula describes the motion of a quantum particle, and admits the following physical interpretation. Let initial condition 7/J in be such that its Fourier transform ;j; Fn_('lj;) is a smooth function supported in a neighborhood Uo of Po E JR. \ { 0 } , 0 1. Uo , and =

i: l,(f; (p) l 2dp = 1 .

Such states are called "wave packets" . Then for every compact subset E c JR. we have 17/J(q, t)l 2 dq = 0. lim iti--+oo JE Since 17/J(q , tWdq = 1

( 2 .18)

r

/_:

( 2 .18) -

for all t, it follows from that the particle leaves every compact subset of JR. as It! --+ oo and the quantum motion is infinite. To prove observe that the function x(p, q, t) the "phase" in integral representation

( 2 .18),

( 2 .17)

2.

95

Quantization

- has the property that 1 � 1 > C > 0 for all p E Uo , enough l t l . Integrating by parts we get

'1/J (q , t ) = _1_ 1

V2ift, Uo

i_ (], ap 0

so that uniformly on E,

( ) ;jJ (p) 8x(p,q,t ) 8p

ekx(p,q,t)tdp,

-

O ( l t l 1 ) as ltl --+ oo . By repeated integration by parts, we obtain that for every n E on E,

'1/J (q, t)

E and large

ekx(p,q,t ) t,(jJ (p) dp

{Ti

= -� it v 2;:

qE

=

N,

uniformly

so that '1/J(q, t) = O( l t l -00) . To describe the motion of a free quantum particle in unbounded regions, we use the stationary phase method. In its simplest form it is stated as follows. The Method of Stationary Phase. Let J, g E C00 (1R) , where f is real­ valued and g has compact support, and suppose that f has a single non­ degenerate critical point xo, i.e. , f'(xo) = 0 and f" (xo) of. 0. Then

1

(

271" 00 e i Nf(x) g(x ) dx = N l f" (xo ) l -oo as N --+ oo .

)

1

2

ei Nf(xo ) +¥-sgnf" (xo ) g(xo) + 0

(_!_) N

Applying the stationary phase method to the integral representation (2. 17) (and setting N = t) , we find that the critical point of x(p, q, t) is Po = 7 with x " (po) = - rk i- 0, and

'1/J(q, t) = [9;jJ (�q ) e i2X12 - ¥ + o(t - 1 ) = '1/Jo (q, t) + O (t-1 ) as t --+ the wave function '1/J(q, t) is supported on ! Uo - a domain oo .

Thus as t --+ oo , where the probability of finding a particle is asymptotically different from zero. At large t the points in this domain move with constant velocities v = � ' p E Uo. In this sense, the classical relation p = mv remains valid in the quantum picture. Moreover, the asymptotic wave function '1/Jo satisfies

2.

96

Basic Principles of Quantum Mechanics

and, therefore, describes the asymptotic probability distribution. Similarly, setting N = - l tl , we can describe the behavior of the wave function '1/J(q, t) as t - oo . �

Remark. We have lim l t l -+ oo '1/J (t)

for every


= 0 in the weak topology on £. Indeed, E £ we get by Parseval ' s identity for the Fourier integrals, ('1/J (t) ,
1: ;jJ (p)cp(p)e- � dp,

and the integral goes to zero as l tl

�

oo

by the Riemann-Lebesgue lemma.

A free classical particle with n degrees of freedom is described by the phase space IR2 n with coordinates p = (PI , . . . , Pn ) and q = (q1 , . . . , qn ) , the Poisson bracket ( 2 . 3) , and the Hamiltonian function 1 2 Hc (P, q) = p2 2 · m 2m (PI + · · + Pn ) · 2 The Hamiltonian operator of a free quantum particle with n degrees of freedom is p2 1 2 2 . . Ho = = 2m 2 m ( PI + . + Pn ) , and i n the coordinate representation is =

.::\

where

Ho =

=

- - .::\ ,

n,2

2m

(�)2 = ( � ) 2 [)ql

oq

+

.

( )

2 ..+ � {) qn

is the Laplace operator 15 in the Cartesian coordinates on !Rn . The Hamil­ tonian Ho is a self-adjoint operator on £ = L 2 (1Rn , dnq) with D (Ho ) = W 2 , 2 ( 1Rn ) - the Sobolev space on !Rn. In the momentum representation, p2

Ho = 2m - a multiplication by a function operator on £ = £ 2 (!Rn , dnp) . The operator Ho is positive with absolutely continuous spectrum [0, oo ) of infinite multiplicity. Namely, let sn- l = { n E !Rn : n2 = 1 } be the ( n - 1 )­ dimensional unit sphere in !Rn , let dn be the measure on sn-l induced by the Lebesgue measure on !Rn , and let (J

= {f :

sn - l �

c:

r l f (n) l 2 dn IIIII� = }sn-1

< 00

}.

1 5 It is the negative o f the Laplace-Beltrami operator of the standard Euclidean metric on

2. Quantization

97

Let SJ6n ) = L2 ( IR >o , � ; dan ) be the Hilbert space of �-valued measurable functions 1 6 W on IR>o = ( 0 , oo ) , square-integrable on IR> o with respect to the measure dan (.\) = (2 m .\) � �� ' n6n ) =

{ w : IR>o - � . ll 'll ll 2 = 100 l l w ( .X) II �

da ( .\ ) < n

00} .

When n = 1 , SJ�1 ) = SJo - the corresponding Hilbert space for one degree of freedom. The operator %'o L2 (IRn , dnq) n 6n) , %'o (7/;) (.X)

:

=

w (.X) ,

--->

w (.X) (n) = -0 ( -12m5.. n) ,

is unitary and establishes the isomorphism L2 (1Rn , dnq) � SJ 6n ) . In the mo­ mentum representation Ho is a multiplication by !n p2 operator, so that the 2 operator %'o Ho %'0- 1 is a multiplication by ). operator in SJ 6n) . Remark. As in the case n = 1 , the Hamiltonian operator Ho has no eigen­ vectors - the eigenvalue equation has no solutions in L2 (1Rn ) . However, for every ). > 0 this differential equa­ tion has infinitely many linearly independent bounded solutions

( 27rn) - � e * � nq , parametrized by the unit sphere sn- l . These solutions do not belong to L2 (1Rn ) , but in the distributional sense they combine to a Schwartz kernel of the unitary operator %'o, which establishes the isomorphism between £ = L2 (1Rn , dnq) and the Hilbert space Sj�n ) , where Ho acts as a multiplication by ). operator. 7/Jn (q)

=

As in the case n = 1 , the Schrodinger equation for free particle, d (t) in 7/J = dt

nO ·'· 'f' (t) '

7/; ( 0 )

=

7/; ,

is solved by the Fourier transform

(2 1rn) -n1 2 { e * Cpq- � tl -0 (p ) dnp. }R.n For a wave packet, an initial condition 7/; such that its Fourier transform -0 = §h( 7/;) is a smooth function supported on a neighborhood Uo of Po E IRn \ { 0 } such that 0 rt Uo and 7/J (q, t)

=

{ i -0 (P) i 2 dnp = 1 , }R.n

1 6 That is , for every f E � the function (!, >It) is measurable on R > O ·

98

2. Basic Principles of Quantum Mechanics

the quantum particle leaves every compact subset of �n and the motion is infinite. Asymptotically as ltl oo , the wave function 1/J(q, t) is different from 0 only when q = !ftt, p E Uo . -->

Problem 2.8. Find the asymptotic wave function for a free quantum particle with

n

degrees of freedom.

2.4. Examples of quantum systems. Here we describe quantum sys­

tems that correspond to the classical Lagrangian systems introduced in Sec­ tion 1.3 of Chapter 1 . In Hamiltonian formulation, the phase space of these systems, except for the last example, is a symplectic vector space JR2n with canonical coordinates p, q and symplectic form w = dp 1\ dq.

Example 2 . 1 ( Newtonian particle ) . According to Section 1 . 7 in Chapter 1 , a classical particle in IRn moving in a potential field V ( q ) is described by

the Hamiltonian function

p2 Hc (P, q ) = V (q) . 2m +

Assume that the Hamiltonian operator for the quantum system is given by p2 H = - + V:' 2m

with some operator V, so that coordinate and momenta operators satisfy Heisenberg equations of motion (2. 19)

P = {H, P}n, Q = {H, Q}n.

To determine V , we require that the classical relation q !!.._ between the m velocity and the momentum of a particle is preserved under the quantization, i.e. , =

Since { P2 , Q}n = 2P, it follows to [V, Q k ]

p . Q = -. m from ( 2 . 19)

that this condition is equivalent . . .

= 0, k 1 , , n. It follows from Section 2 . 2 that V is a function of commuting operators V (Q) . Thus the Hamiltoni an Q 1 , . . . , Qn , and the natural choice 1 7 is V operator of a Newtonian particle is =

=

p2 H= + V (Q) , 2m

1 7 Confirmed by the agreement of the t heory with the experiments.

2.

99

Quantization

in agreement with H = Hc (P, Q) 1 8 . In coordinate representation the Hamil­ tonian is the Schrodinger operator

(2.20)

H

= - 2 m � + V (q) h2

with the real-valued potential V(q) .

Remark. The sum of two unbounded, self-adjoint operators is not neces­

sarily self-adjoint, and one needs to describe admissible potentials V ( q) for which H is a self-adjoint operator on £ 2 (IR.n , dn q) . If the potential V ( q) is a real-valued, locally integrable function on IR.n , then the differential operator (2.20) defines a symmetric operator H with the domain C3 (JR.n ) - twice continuously differentiable functions on IR.n with compact support. Poten­ tials for which the symmetric operator H has no self-adjoint extensions are obviously non-physical. It may also happen that H has several self-adjoint extensions 19 . These extensions are specified by some boundary conditions at infinity and there are no physical principles distinguishing between them. The only physical case is when the symmetric operator H admits a unique self-adjoint extension, that is, when H is essentially self-adjoint . In Chapter 3 we present necessary conditions for the essential self-adjointness. Here we only mention the von Neumann criterion that if A is a closed operator and D (A) £, then H = A* A is a positive self-adjoint operator. =

2.2

(Interacting quantum particles) . In Lagrangian formalism, a closed classical system of N interacting particles on JR.3 was described in Example 1 .2 in Section 1 .3 of Chapter 1. In Hamiltonian formalism, it is described by the canonical coordinates r = ( r 1 , . . . , rN ) , the canonical momenta p = (P I , . . . , pN ) , ra , Pa E JR.3 , and by the Hamiltonian function Example

(2.2 1 )

Hc ( P , r

) = L 2�2a + V(r) , N

a =l

where ma is the mass of the a-th particle, a = 1 , . . . , N (see Section 1 . 7 in Chapter 1 ) . The corresponding Hamiltonian operator H in the coordinate representation has the form N fi2 (2.22) H = - 2: - �a + V(r) . a =l 2m a

In particular, when V (r) =

L

l� a
V (ra - rb) ,

18In the special case Hc (P, q) f ( p) + g(q) the problem of the ordering of non-commuting operators P and Q does not arise. 19This is the case when the defect indices of H are equal and are non-zero. =

100

2. Basic Principles of Quantum Mechanics

the Schrodinger operator (2.22) describes the N-body problem in quantum mechanics. The fundamental quantum system is the complex atom, formed by a nucleus of charge N e and mass M, and by N electrons of charge -e and mass m. Denoting by R E JR3 the position of the nucleus, and by r1, . . . , rN the positions of the electrons and assuming that the interaction is given by the Coulomb attraction, we get for the Hamiltonian function (2.21) N � p2 N N e2 e2 p Hc (P , p , R , r ) + L L L = 2M a=1 2m a=1 I R - ra l 1-.:;a
+

in

=

-

2

�

-

=

= -

e -:-- ----:-

-

-

-

=

Example 2 . 3 (Charged particle in an electromagnetic field) . A classical

particle of charge e and mass m moving in the time-independent electro­ magnetic field with scalar and vector potentials cp(r) and A(r), r E IR3, is described by the Hamiltonian function Hc (P, r)

(P - .:_A) 2 +ecp(r) = _.!..._ 2m

(see Problem 1 . 2 7 in Section 1 . 7 of Chapter 1 ) . The corresponding classical velocity vector v { He , r} is given by

=

c

e

v = p - -A, c

2 0Ignoring the fact that electron has spin, see Chapter 4. 21 In the case of hydrogen-1 or protium; it includes one or more neutrons for deuterium, tritium, and other isotopes.

2.

101

Quantization

and its components

= (vi , v2 , v3) have non-vanishing Poisson brackets: e e e {vi , v2} = - -2- B3 , {v2 , v3} = - -2-BI , {v3 , vi } = - -2- B2 , m e m e m e v

where B = (B1 , B2 , B3) are components of the magnetic field B = curl A. The Hamiltonian operator of a quantum particle is (2 . 24)

H=

-

1 2m

(

P - -A

e

e

) 2 + ecp (r)

- the Schrodinger operator of a charged particle in an electromagnetic field. The corresponding quantum velocity vector V = { H , Q} n is given by the same formula as in the classical case, V = P - -A,

e

and its components

V =

e

(VI , V2 , V3) have non-vanishing quantum brackets: e e e B3 , {V2 , V3 } n = - { VI , V2 }n = - BI , {V3 , VI } n = - B2 2 2 m e m e m2 e .

Thus in the presence of a magnetic field the three components of a quantum velocity operator no longer commute and cannot be measured simultane­ ously. Example 2.4 (Free quantum particle on a Riemannian manifold) . The phase space of a classical particle of mass m 1 moving on a Riemann­ ian manifold ( M, g) is the cotangent bundle T* M, and the corresponding Hamiltonian function is given by H (p , x ) = � gJ..LZI (x)pJ.Lpv , where gJ.Lll (x) is the inverse of the metric tensor 9J.L v (x) , and (p, x) (pi , , pn , XI , . . . , x n ) =

c

=

...

are standard coordinates on T* M (see Section 1 . 7 of Chapter 1 ) . The Hilbert space of the quantum system is .Ye = L 2 (M, dJ.L) , where df-L ..Ji{X)dnx, g(x) det(gJ.Lv (x)) , is the measure associated with the density of the Rie­ mannian metric (Riemannian volume form if M is oriented) . When canonical coordinates (p, x) are only locally defined on T* M, it is not possible to con­ struct corresponding operators P and Q. Still, one can always define the Hamiltonian operator by 1 a a where �9 = ..Ji(X) 9J..L v _ (2 . 2 5 ) ax v ..Ji(X) axJ.L =

=

-

(

)

is the Laplace-Beltrami operator of the Riemannian metric g on M. Note that in this case there is a non-trivial problem of the ordering of non­ commuting operators in the quantization of Hc (P , x ) , which arises if in a

2.

102

Basic Principles of Quantum Mechanics

coordinate chart on M one replaces canonical coordinates p and x by P and Q. The formula A

L.J.g

-

-g

f.ll/ (

X ) {)

0{)2

x f.l x v

+g

f.J,CT

shows that local expressions defined by

( X ) rl/ ( X ) 0 0 ,.. xv "CT

(2.26)

combine to a well-defined self-adjoint operator H on £ given by (2.25) . Thus even when M = 1Rn and canonical coordinates (p , x ) are globally defined on T*IRn , the correct formula for He ( P , Q ) - the one which extends to general Riemannian manifolds - is given by ( 2 . 26) , where the second term represents the "quantum correction" to the naive expression g11 v ( Q)P11Pv . 2 . 5 . Old quantum mechanics. The formulation of quantum mechanics presented here goes back to 1925-1927, and is due to Heisenberg, Schrodin­ ger, Born, Jordan, and Dirac. It replaced the old quantum theory, proposed in 1913 by Bohr, which was based on Rutherford ' s planetary model of the atom. In the old theory, the energy levels of a one-dimensional quantum sys­ tem correspond to the closed orbits of the associated classical Hamiltonian system which satisfy the Bohr- Wilson-Sommerfeld quantization rule ( BWS rule )

f pdq

=

2 1r n(n +

!),

where n is a non-negative integer, and integration goes over the closed or­ bit in the phase space IR 2 . Bohr-Wilson-Sommerfeld quantization rules also apply to completely integrable Hamiltonian systems with several degrees of freedom ( see Section 2.6 of Chapter 1 ) . Namely, let F1 = He , . . . , FN be N independent integrals of motion in involution. The BWS quantization rules are pdq = 21rn(n7 + l ind 1 ) ,

i

where integration goes over all 1-cycles r in the Lagrangian submanifold A = { (p , q) E IR2 N Hc (P, q) = E, Fz (p , q) = Ez , . . . , FN(P , q) = EN } , and ind r E Z is the so-called Maslov index of a cycle r in A. In the one­ dimensional case the closed orbit is topologically a circle and its Maslov index is 2 . It will be shown in Section 6.3 of Chapter 3 that, in general, Bohr­ Wilson-Sommerfeld quantization rules only give asymptotics of the energy levels as n 0. However, for integrable systems with extra symmetry, such as the harmonic oscillator and the Kepler problem, the BWS quantization rules determine the energy levels exactly. We will show this in the next section for the harmonic oscillator, and in Section 5 . 1 of Chapter 3 - for the Kepler problem. :

----7

103

2. Quantization

2.6. Harmonic oscillator. The simplest classical system with one degree

of freedom, besides the free particle, is the harmonic oscillator. It is de­ scribed by the phase space JR2 with the canonical coordinates p, q, and the Hamiltonian function

(2.27)

( see Sections 1 . 5 and 1 . 7 in Chapter 1 ) . Hamilton ' s equations

{ He , P } = - mw 2 q , q = { He , q } = .!!. m .. with the initial conditions Po , qo are readily solved, (2.28) p(t) Po cos wt - mwqo sin wt, 1 . (2.29) q (t ) qo cos wt + -po smwt, p=

= =

mw

and describe the harmonic motion. As in Section 2.6 of Chapter 1 , it is convenient to introduce complex coordinates on the phase space on IR2 C,

(2.30) We have

z =

v 2w (wq + ip) , 1

�

z=

1 (wq - ip) . v 2w

:::::::

�

{ z , z } -mi ,

He (z, z ) = mw jzj 2 , so that Hamilton ' s equations decouple, z = { He , z } -iwz, z = { He , z } = iwz, and are trivially solved, (2.32) z(t) = e-iwt zo, z- = e iwt zo. Here 1 1 . (wqo zo = v� + zpo) , zo = � (wqo - ipo) . v 2w 2w

(2.31)

=

=

For the quantum system, the corresponding Hamiltonian operator is

p2 mw2Q2 2 ' and in the coordinate representation .Ye L2(IR , dq) operator with a quadratic potential, n,2 d2 mw2 q2 H = - - -2 + -2m dq 2 H = 2m +

=

it is a Schrodinger

The quantum harmonic oscillator is the simplest non-trivial quantum sys­ tem, besides the free particle, whose Schrodinger equation can be solved ex­ plicitly. It appears in all problems involving quantized oscillations, namely in molecular and crystalline vibrations. The exact solution of the harmonic

1 04

2.

Basic Principles of Quantum Mechanics

oscillator, described below, has remarkable 22 algebraic and analytic proper­ ties. Temporarily set m = 1 and consider the operators

k (w Q + i P ) , k Q - i which are quantum analogs of complex coordinates The operators 2 and are defined on W • where and it is easy t o show t at is the adjoint operator to and so a* =

a=

(2.33)

(w

P) ,

(2.30) . a a* 1 ( JR) n W1•2 (JR) , W1•2 (JR) = §(W1•2 (JR) ) , h a* a a** = a , that a is a closed operator. From the Heisenberg commutation relation (2.2) we get the canonical commutation relation

( 2.34 )

•

•

on W 2 2 ( JR ) n W 2 2 ( 1R ) . Indeed,

[a, a * ] =

I

aa * =

p2 + w2 Q2 + iw P Q [ , ]=

* a a =

p2 + w 2 Q 2 p 2 + w2 Q 2 iw [P , - 2 w fi Q] = 2wfi 2wfi

and

2wfi

2wfi

•

Q

p2 + w 2 2 2wfi

1

+ 2 I,

1

- 2I,

so that (2.34) holds on W 2 2 (JR ) n W2•2 (JR) , where W2•2 (JR) = §(W 2 • 2 (JR ) ) , and ( aa * - � I ) . H = wfi ( a * a + �I) = In particular, it follows from the von Neumann criterion that the Hamilton­ ian operator H is self- adjoint . The operators a , a * and N = a * a satisfy the commutation relations

wfi

= a * , [ a , a * ] = I. These commutation relations correspond to the irre c!_ucible unitary repre­ sentation of a four-dimensional solvable Lie algebra � associated_ with the Heisenberg algebra � = � 1 . introduced in Section 2 . 1 . Namely, � is a Lie algebra with the generators e , J, h, and c, where e, f, c satisfy the relations of the Heisenberg algebra � ' and [ h , e ] = - J, [ h , f ] = [h , c] = 0. The irreducible integrable representation p of the Heisenberg algebra � (see ( 2.9 ) in Section 2 . 1 ) extends to a unitary representation of the Lie algebra (2 . 3 5 )

[ N, a]

= -a,

[N, a * ]

w2e,

22 The algebraic structure of the exact solution of the harmonic oscillator plays a fundamental role in quantum electrodynamics and in quantum field theory in general.

2. Quantization

105

6 by setting

P2 + 2 Q2 . + t I. 2i Remark. In invariant terms, the Lie algebra 6 is a one-dimensional right extension of the Heisenberg algebra � , 0 � � � � � lR � 0 . It is defined by the �-valued Lie algebra 1-cocycle r E Z 1 ( � , � ) - a derivation of � ' given on generators by r (e) J, r( f ) = - w 2 e , r ( c) = 0. Explicitly, if h E 6 is such that h = 1 E lR under the projection 6 � JR , then identifying elements in � with their images under the embedding � '----+ 6, we have [x + ah, y + /1h] = [x, y] - ar(y) + /1r (y) , x, y E �· p(h)

=

;

�

- iwN =

=

It is due to this Lie-algebraic structure of commutation relations (2.35) that Heisenberg equations of motion for the harmonic oscillator can be solved exactly. Namely, we have

. {H, a }

a

so that

=

a( t )

1i =

- t. w a ,

.*

a

e -iwt ao ,

= { H a* }

a * (t )

,

, . 1i = twa

*

e iw t a0 .

Comparing with ( 2 . 32 ) we see that solutions of classical and quantum equa­ tions of motion for the harmonic oscillator have the same form! Next, using commutation relations (2.35) and positivity of the opera­ tor N, we will solve the eigenvalue problem for the Hamiltonian H of the harmonic oscillator explicitly by finding its energy levels and corresponding eigenvectors . We will prove that the eigenvectors form a complete system of vectors in £, so that the spectrum of the Hamiltonian H is the point spectrum. This is a quantum mechanical analog of the fact that classical motion of the harmonic oscillator is always finite. The algebraic part of the exact solution is the following fundamental result. =

=

Suppose that there exists a n o n- zero 'ljJ E D ( an ) nD ( ( a* ) n ) , n = 1 , 2, . . . , such that H'ljJ = A. 'l/J . Then the following statements hold. ( i ) There exists 'l/Jo E £, 11'!/Jo ll 1 , such that H 'l/Jo = � liw 'l/Jo . Proposition 2.2.

=

106

2. Basic Principles of Quantum Mechanics (ii) The vectors

1/Jn =

( a* ) n Ci 1/Jo E v n!

£,

n = 0, 1 , 2,

.

.

.

,

are orthonormal eigenvectors for H with the eigenvalues liw (n + ! ) , H 'I/Jn = liw ( n + ! )1/Jn ·

(iii) Restriction of the operator H to the Hilbert space £a - a closed

subspace of £, spanned by the orthonormal set { 1/Jn };;,o=O - is es­ sentially self-adjoint.

(2.35) as Na a ( N I) and N a * = a* (N and putting >. liw ( J-t + � ) , we get for all n 2: 0, Nan'I/J = (J-t - n ) an'I/J and N ( a * ) n'I/J = (J-t + n) (a* t 'I/J. (2.36) Since N 2: 0 on D(N ) , it follows from the first equation in (2.36) that there ano 'I/J £ exists no 2: 0 such that an 1/J =1- 0 but ano+l 'I/J = 0. Setting 1/Jo ll ano 'I/J I I we get (2.37) a'I/Jo = 0 and N'I/Jo = 0. Since H = liw ( N + this proves part (i). To prove part (ii) , we use commutation relations [a, (a* t ] n(a* t- l , (2.38) which follow from (2.34) and the Leibniz rule. Using (2.37)-(2.38) , we get (2.39 ) so that 2 1 1 1 11/Jn ll = ( a * 1/Jn- 1 , 1/Jn ) = ('I/Jn- 1 , a'I/Jn) = 111/Jn- d = · · · 11 1/Jo ll = 1 . Proof. Rewriting commutation relations =

=

-

+ I) ,

E

=

°

�I) ,

=

Vn

Vn

2

=

2

From the second equation in (2.36) it follows that N'I/Jn n'I/Jn , so 1/Jn are normalized eigenvectors of H with the eigenvalues nw (n + � ) . The eigenvec­ tors 1/Jn are orthogonal since the corresponding eigenvalues are distinct and the operator H is symmetric. Finally, part (iii) immediately follows from the fact that, according to part (ii) , the subspaces Im (H ± iJ ) I £o are dense in 0 £a, which is the criterion of essential self-adjointness. =

Remark. Since the coordinate representation of the Heisenberg commuta­

tion relations is irreducible, it is tempting to conclude, using Proposition 2.2, that £a = £. Namely, it follows from the construction that the linear span of vectors 1/Jn - a dense subspace of £a - is invariant for the opera­ tors P and Q. However, this does not immediately imply that the projection

107

2. Quantization

operator ITo onto the subspace £a commutes with self-adjoint operators P and Q in the sense of the definition in Section 1 . 1 . Using the coordinate representation, we can immediately show the exis­ tence of the vector '1/Jo in Proposition 2.2, and prove that £a = .Yt'. Indeed, equation a'l/Jo 0 becomes a first order linear differential equation =

( � + wq) '1/Jo li

so that and

II'I/Jo ll 2

=

=

0,

/fn j_: e - Jt q2 dq

=

1.

The vector '1/Jo is called the ground state for the harmonic oscillator. Corre­ spondingly, the eigenfunctions

1 ( � (wq :q )) n '1/Jo are of the form Pn (q)e q2 , where Pn (q) are polynomials of degree The following result guarantees that the functions { '1/J � O form an orthonormal - li

'1/Jn (q) = -

w

�

n} =

basis in L 2 (IR, dq) . Lemma 2.3. Proof. Let

The functions qne- q 2 , n

=

0, 1 , 2,

f E L2 (IR, dq) is such that

/_: f(q)qne-q2 dq

The integral

F(z)

n.

=

=

. . . , are complete in L2 (JR., dq) .

0, n = 0, 1 , 2, . . . .

/_: J(q) eiqz-q2dq

is absolutely convergent for all z E C and, therefore, defines an entire func­ tion. We have

p(n) (O) = in

j_: J(q)qne-q2 dq = 0,

n = 0, 1 , 2, . . .

,

so that F(z) 0 for all z E C. This implies the function g(q) = f(q)e -q2 E L 1 (IR) n L 2 (JR) satisfies § ( g ) = 0, where § is the "ordinary" (li 1 ) Fourier 0 transform. Thus we conclude that g = 0. =

=

2 . Basic Principles of Quantum Mechanics

108

The polynomials Pn are expressed through classical Hermite-Tchebyscheff polynomials Hn , defined by Hn (q ) = ( - 1t eq

Namely, using the identity

e

£._

dn e -q 2 = 2 dqn

2

dn d qn

( -

= ...

e-q

2

,

(

0, 1 , 2, . . . .

] ) [e e -q2 ( ) e - -f -1t

d qdq

=

n =

£._

n d -1 dqn - 1 n d qdq 2

2

we obtain

We summarize the obtained results as follows. Theorem 2 . 1 .

The Hamiltonian H=

n2 d2 - 2m dq 2

+

mw2q2 2

of the quantum harmonic oscillator with one degree of freedom is a self­ adjoint operator on .Yt' = L2 (ffi., dq) with the domain D ( H) = W2,2 (ffi.) n W2,2 ( ffi.) . The operator H has pure point spectrum H 'l/Jn

=

A.n 'l/Jn ,

n=

0, 1 , 2, . . . ,

with the eigenvalues A n liw ( n + ! ) . Corresponding eigenfunctions '1/Jn form an orthonormal basis for .Yt' and are given by - �q 2 H rmw Jmw 1 (2.40) '1/Jn (q ) = y � V2nnf e n yhq , =

where

Hn ( q)

(

)

are classical Hermite- Tchebyscheff polynomials.

Proof. Consider the operator H defined on the Schwartz space Y (ffi.) of rapidly decreasing functions. Since the operator H is symmetric and has a complete system of eigenvectors in Y(ffi.) , the subspaces Im(H ±ii) are dense in .Yt' , so that H is essentially self-adjoint. The proof that its self-adjoint closure ( which we continue to denote by H) has the domain W2,2 (1R) n 0 W2•2 (ffi.) is left to the reader. Remark. The energy levels of the harmonic oscillator can be also ob­

tained from the Bohr-Wilson-Sommerfeld quantization rule. Indeed, the cor­ responding classical orbit with the energy E is an ellipse in the phase plane

109

2. Quantization

given by the equations q(t) E = �mw 2 A2 (see Section

A cos ( w t + ) , p(t) = -mwA sin(wt+a), where 1.5 in Chapter 1). Thus a

=

f pdq JJ dp l\ dq =

=

?TmwA 2

=

�E,

and the BWS quantization rule gives the energy levels En =

nw

(n + ! ) .

Remark. Since the energy levels of the Hamiltonian H are equidistant

by nw, the quantum harmonic oscillator describes the system of identical "quanta" with the energy nw. The ground state J O ) = 'lj;o , in Dirac's notation, is the vacuum state with no quanta present and with the vacuum energy ! nw, and the states Jn) = 'lj;n consist of n quanta with the energy nw(n + � ) . According t o (2.39), the operator a * adds one quantum t o the state J n) and is called a creation operator, and the operator a destroys one quantum in the state Jn) and is called an annihilation operator. An example of the harmonic oscillator illustrates the dramatic difference between the motion in quantum mechanics and in classical mechanics. The classical motion in the potential field V(q) = ! mw2q2 is finite: a particle with energy E moves in the region J w qJ whereas there is always a non-zero probability of finding a quantum particle outside the classical region. Thus for the ground state energy E = � nw this probability is

::; /fj,

fh

1/q/�y

;:nw

J 1j;o(q) J 2 dq = 2

r;;;. v 1T

1,00 e -x dx 2

1

�

0.1572992070.

The classical harmonic oscillator with n degrees of freedom is described by the phase space � 2n with the canonical coordinates p, q, and the Hamil­ tonian function n mwj2q2j p2 Hc ( P, q) = + 2m JL 2 ' =l where w1 , . . . , wn > 0 ( see Sections 1 . 3 and 1.7 in Chapter 1 ) . The corresponding Hamiltonian operator is n mw 2Q2 p2 H = - + '"' J J 2m j6 2 =l

and in the coordinate representation .Yt' L 2 (JRn , dn q) is a Schrodinger operator with quadratic potential, n mw 2 q 2 n,2 H = - - � + '"' J J . 2m 6 2 =

j =l

1 10

2. Basic Principles of Quantum Mechanics

The Hamiltonian H is a self-adjoint operator with D ( H) = W 2 • 2 ( 1Rn ) n W 2 • 2 (!Rn ) and a pure point spectrum. The corresponding eigenfunctions '1/Jk ( q ) = 'l/Jk 1 ( ql ) '1/Jkn (qn ) , where k = ( k 1 , . . . , kn ) and '1/JkJ (qj ) are eigenfunctions (2.40) with w = Wj , form an orthonormal basis for L2 ( 1Rn , dn q) . The corresponding energy levels are given by A k = /iw 1 ( k 1 + � ) + + liwn ( kn + � ) . The spectrum of H is simple if and only if liw 1 , . . . , liwn are linearly inde­ pendent over Z. The highest degeneracy case is w1 = · · · = Wn = w , when the multiplicity of the eigenvalue •

·

Ak

·

•

•

·

n

= liw L ( kj + � ) j=l

is the partition function Pn ( l k l ) - the number of representations of the integer l k l k1 + · · · + kn as a sum of n non-negative integers. Setting m 1 and introducing the operators 23 1 1 (2.4 1 ) aj = � (w Qj + iPj ) , aj = � (w Qj - iPj ) , j = 1 , . . . , n , =

=

v 2wn

v 2wn

we get canonical commutation relations for creation and annihilation oper­ ators for n degrees of freedom, (2 . 42 ) [aj , al] = O, [aj , ai] = O, [aj , al] = 8jtl, j, l = 1 , . . . , n , which generalize relation (2.34 ) for the one degree of freedom. The operators aj , aj and Nj = aj aj , j = 1 , . . . , n, satisfy commutation relations (2 . 43 ) [ Nj , al ] = -8jt at , [ Nj , al] = 8jta/ , j, l 1 , . . . , n. In particular, the operator =

N

satisfies and

=

[N , aj] = -aj , H = liw(N + � I ) .

n

n

j=l

j=l

L Nj = l: aj aj [ N, aj] = aj ,

j = 1 , . . . , n,

Commutation relations (2 .42 ) and (2.43) correspond to the irreducible unitary representation of the solvable Lie algebra 6 n with 3n + 1 generators ej , fj , hj , and c, where ej , fj , c satisfy the relations in the Heisenberg algebra IJ n , and [hj , e l] = -8jt !t , [hj , Jl ] = 8jtWf et , [hj , c ] = 0, j , l = 1 , . . . , n.

23 Here, using the standard Euclidean metric on lltn , we lowered the indices for QJ .

2. Quantization

111 (HIM) � � liw

Show that

Problem 2.9.

for every

M

E

!/ , where H is the

Hamiltonian of the harmonic oscillator with one degree of freedom .

q(t)

Let

Problem 2 . 1 0 .

+ a)

A cos (wt

=

be the classical trajectory of the har­

ability measure on lR supported at the point monic oscillator with m =

and the energy

1

function .

Problem 2 . 1 1 . l q l :::;

=

0 :::; a :::; 211', f..L ( q) = 7r A2 - 2 q

combination of the measures f..L o: , with the distribution function

E

0�,

Show that when

oo

n ---.

� w2 A2 ,

and let f..L o: be the prob­

is the probability measure on lR

q (t) .

Show that the convex linear

where

B( q )

is the Heavyside step

fi ---. 0 such that liw(n + � )

and

= � w2 A2

remains fixed, the envelope of the distribution function l '¢n (q) l 2 on the interval

f..L ( q)

A coincides with the classical distribution function

problem.

(Hint:

Prove the integral representation

2 e - q Hn (q)

from the previous

r= 2 n J7i J e - y y cos (2 qy - � n 1l' ) dy , o

2n + l =

and derive the asymptotic formula

'¢n (q) = when

cos V� ; y'A 2 - q2

fi ---> 0

1

and

!i(n +

Problem 2.12.

�)

{ 2n ( w

"'

� wA2 ,

=

q A2 s in- 1 _ + q JA 2 - q 2 - � A2 1!' A 2

lql

< A.)

Complete the proof of Theorem

2. 1 .

Problem 2 . 1 3 ( The N-representation theorem ) . '¢ N'¢n = n'¢n . )

the L 2 -convergent expansion

!/ (JR) . (Hint :

Use

Problem 2. 14.

=

L::':"= o cn'¢n ,

Show that the operators

where

Eij

commutation relations of the Lie algebra sl( n,

=

C) .

+

) 0(1)}

'¢ E !/(JR) . Show that = ('¢, '¢n) , converges in

Let

Cn

ai aj , i, j = 1 , . . . , n,

satisfy the

2 . 7. Holomorphic representation and Wick symbols. Let f2

�

{

C

�

{c,. }�=O

:

ll c ll 2

�

t. 00 } l c. l 2 <

be the Hilbert £2 -space. The choice of an orthonormal basis { 'lj;n }�= O for L2 (�, dq) , given by the eigenfunctions (2.40) of the Schrodinger operator for the harmonic oscillator, establishes the Hilbert space isomorphism L 2 ( �, dq) c::

£2 '

L 2 (� , dq )

3

00

'lj; = �:::>n 'lj;n n= O

1---+

c = { Cn }�=O

where Cn =

('lj; , 'lj;n ) =

/_: 'lj; (q) 'lj;n (q)dq ,

E

£2 ,

112

2 . Basic Principles of Quantum Mechanics

since the functions 'lj;n are real-valued. Using (2.39) we get 00

00

00

a *'lj; = L Cn a * 'lj;n = L vn-+1 Cn 'lj;n+l = L Vn Cn -l'lj;n , 1j; E D(a* ) , n=O n=O n= l and 00

00

00

a'lj; = l: cn a'lj;n = L vn en'lj;n-1 = L Jn + 1 en+l 'lj;n , 1j; E D(a) , n=O n=O n=l so that in £2 creation and annihilation operators a * and a are represented by the following semi-infinite matrices: 0 0 0 0 0 y'1 0 0 0 0 J2 0 y'1 0 0 0 0 0 0 a* = 0 J2 0 0 a= J3 0 0 J3 0 0 0 0 0 As a result, N = a* a

=

0 0 0 0

0 0 1 0 0 2 0 0

0 0 0 3

so that the Hamiltonian of the harmonic oscillator is represented by a diag­ onal matrix, This representation of the Heisenberg commutation relations is called the representation by occupation numbers, and has the property that in this representation the Hamiltonian H of the harmonic operator is diagonal. Another representation where H is diagonal is constructed as follows. Let !fl be the space of entire functions f (z) with the inner product (2.44) ( ! , g ) = � { f (z)g(z)e-lzl 2 d2z, 1r lc where d2 z = � dz 1\ dz is the Lebesgue measure on C � JR2 . It is easy to check that !fl is a Hilbert space with the orthonormal basis zn fn (z) = Cj ' n = 0 , 1 , 2, . . . . v n! The correspondence £2

3

c = { Cn } �=O """' f (z) =

00

L Cnfn (z) E !fl

n=O

1 13

2. Quantization

establishes the Hilbert space isomorphism £2 � . The realization of a Hilbert space £ as the Hilbert space � of entire functions is called a holo­ morphic representation. In the holomorphic representation, �

d

(

d

�) ,

a* = z, a = d ' and H = nw z d + z z and it is very easy to show that a* is the adjoint operator to a. The mapping 00

00

n==O

n==O

establishes the isomorphism between the coordinate and holomorphic rep­ resentations. It follows from the formula for the generating function for Hermite-Tchebyscheff polynomials, n ) ( L Hn q n . = e 2qz-z2 ' n ==O that the corresponding unitary operator U : £ ---> � is an integral operator

;

oo

U'ljJ(z) =

with the kernel

(2.45)

U ( z , q) =

i: U(z , q)'l/J(q)dq

'1/Jn ( q ) fn (z) = 4 /mW e ';;;;' q2 - ( J!¥q- �zr f v� n==O

Another useful realization is a representation in the Hilbert space {g of anti-holomorphic functions f (z) on C with the inner product

given by

( !, 9 ) = .!.

r f (z)g(z) e-lz i2 d2 z,

1r lc

a*

=

-

z,

a= d

dz .

Remark. Holomorphic and anti-holomorphic representations correspond to

different choices of a complex polarization of the symplectic manifold IR2 . By definition, a complex polarization of a symplectic manifold ( At, w ) is an integrable distribution on At of complex Lagrangian subspaces of the complexified vector spaces TxAl" ®JR C. As the notion of real polarization, the notion of complex polarization plays a fundamental role in geometric quantization. In particular, holomorphic representation corresponds to the complex Lagrangian subspace in TxlR2 ®JR C � C 2 given by the equation z = 0, where z and z are complex coordinates on C 2 . ( Note that here z is not a complex conjugate to z!) The equation z = 0 defines a complex Lagrangian subspace which corresponds to the anti-holomorphic representation.

1 14

2. Basic Principles of Quantum Mechanics

The anti-holomorphic representation is used to introduce the so-called Wick symbols of the operators. Namely, let A be an operator in � which is a polynomial with constant coefficients in creation and annihilation operators a* and a. Using the commutation relation (2.34) , we can move all operators a * to the left of the operators a, and represent A in the Wick normal form as follows: (2.46)

A=

L Az m (a* )1am . l,m

By definition, the Wick symbol A (z, z) of the operator A is A (z, z ) =

(2.47)

L Azm z1 z m . l,m

It is a restriction of a polynomial A (v , z) in variables v and z to v = z. In order to define Wick symbols of bounded operators in �, we consider the family of coherent states ( or Poisson vectors ) v E �' v E C, defined

by

They satisfy the properties

and f (v) = ( ! , v ) , f E �' v E C. Indeed, the first property is trivial, whereas the "reproducing property" immediately follows from the formula a v = v v

(2.48)

(2 .49)

v ( z)

00

= L fn (v) fn ( z) , n=O

where fn (z) fn ( z ) , n = 0 , 1 , 2 , . . . , is the orthonormal basis for � ­ We also have 1 ( 2 . 50) ( j, g ) - - ( f, v ) ( g, v ) e - lvl2 d v . 7r c Now for the operator A in the Wick normal form (2.46) we get, using the first property in (2.48) , =

_

( A z , v )

Therefore,

=

=

2

1

L Az m ( ( a* )1am z , v ) L Azm ( am z , a1 v )

l,m A ( v , z) ( z , v ) ·

A (v , z)

since by the reproducing

=

l,m

(A z , v ) - = e -vz (A z , v ) , ( Z l V ) property, ( z , v ) = z ( v) = =

evz.

2.

115

Quantization

The Wick symbol A(z, z ) of a bounded operator A i n the Hilbert space !» is a restriction to v = z of an entire function A (v , z ) in variables v and z, defined by Definition.

In the next theorem, we summarize the properties of the Wick symbols. Theorem 2 . 2 . Wick symbols of the bounded operators on !» have the fol­

lowing properties.

( i ) If A (z, z ) is the Wick symbol of an operator A, then for the Wick symbol of the operator A* we have A* (z, z )

=

A (z , z ) and

A* ( v, z ) = A (z, v) .

( ii ) For f E

( iii )

!» ,

(A f ) (z)

A

= .!_1r { A (z, v ) f (v) e - v ( v -z) d2 v . lc

real-analytic function A (z, z) is a Wick symbol of a bounded op­ erator A in !» if and only if it is a restriction to v = z of an entire function A(v , z ) in variables v and z with the property that for every f E !» the integral in part ( ii ) is absolutely convergent and defines a function in !» . ( iv ) If A1 (z, z ) and A2 (z, z) are the Wick symbols of operators A1 and A2 , then the Wick symbol of the operator A = A1A2 is given by A (z , z)

Proof.

= .!_1r lc{ A1 (z, v) A2 (v, z ) e- ( v - z ) (v-z) d2 v .

We have

A * (v , z ) = e - v z (A*
which proves ( i ) . To prove ( ii ) , we use the reproducing property to get (A f ) (z) = (A f,
.!_ { f (v) (A*
which proves ( ii ) . Property ( iii ) follows from the definition and the uniform boundness principle, which is needed to show that the operator A on !» ,

1 16

2. Basic Principles of Quantum Mechanics

defined by the integral in (ii) , is bounded. We leave the standard details to the reader. To prove (iv) , by using (2.50) and (i) we get A (z, z) = e- l z l2 (A 1A2 z , z ) e-l z l2 (A 2 z , A iz ) =

= �1r lc{ ( A2 z , v ) (Ai z , v)e -
0 � { A I ( z v)A2 (v, z ) e -
=

A matrix symbol A (z, z) of a bounded operator A in the Hilbert space !!) is a restriction to v = z of an entire function A ( v z ) in variables v and z , defined by the following absolutely convergent series: ,

00

L (A /m , fn)fn ( v) fm(z ) . m, n =O The matrix and Wick symbols are related as follows. A (v, z) =

(2.51)

Lemma 2 .4. For a bounded operator A in the Hilbert space !!),

A(v, z) = ev z A(v, z) . Proof. Using (2.49) we get

A (v, z) =

�

m�/

n ( v ) fm ( z )

1 (Afm ) (U)fn (u)e- 1•1' d2u

� { (A z ) (u)v (u) e - l ul2 d2 u = (Az , v) = e v z A(v , z ) . 1r lc Changing the order of summation and integration is justified by the absolute 0 convergence. =

Corollary 2 . 3 . If A 1 (z, z) and A 2 (z, z ) are matrix symbols of operators A1

and A2 , then the matrix symbol of the operator A = A1A 2 is given by A (z, z)

=

� f A 1 (z, v) A 2 ( v, z) e -1v1 2 d2 v. 1r lc 0

Proof. The proof immediately follows from part (iv) of Theorem 2.2 and

Lemma 2.4.

It is straightforward to generalize these constructions to n degrees of freedom. The Hilbert space �n defining the holomorphic representation is

117

2. Quantization

the space of entire functions f (z) of n complex variables z = ( z1 , . . . , zn) with the inner product

�r

( !, g ) = n f (z) g (z)e-1.:12 d2n z < oo , 1T len where l z l 2 = zt + · · · + z� and d2 n z = d2 z1 · · · d2 zn is the Lebesgue measure on en ::::: IR. 2 n . The functions - z;nl . . . z;;-n m l , . . . , mn = 0, 1 , 2, . . . , fT1t ( z ) ' vm l ! . . . mn !

where m = (mi . . . . , mn ) is a multi-index, form an orthonormal basis for �n · Corresponding creation and annihilation operators are given by 8 aj = 8Zj ,

j = l , . . . , n.

The Hilbert space �n of anti-holomorphic functions f (z) on e n is defined by the inner product (2.52) and the creation and annihilation operators are given by

The coherent states are v (z) = e v z , where vz = v1 z1 satisfy the reproducing property

+ ··· +

VnZn, and

f (v) = (!, v ) , f E �n , v E en . The Wick symbol A(z, z ) of a bounded operator A on �n is defined as a restriction to v = z of an entire function A(v, z) of 2n variables v = (vb · · · , vn ) and z = (zb · · · , zn ) , given by A(v, z) = e - v.: (A z , v) ·

We have and the Wick symbol A(z, z ) of the operator A = A1 A 2 is given by A(z, z ) =

� l{en A1 (z, v)A2 ( v , z)e -(v-z)(v-z) d2n v ,

7r

where AI (z, z ) and A 2 (z, z ) are the Wick symbols of the operators A1 and

A2 .

1 18

2. Basic Principles of Quantum Mechanics

The matrix symbol A(z, z ) of a bounded operator A on �n is defined as a restriction to v z of an entire function A ( v, z) of 2n variables v = (v 1 , . . . , vn) and z = ( z1 , . . . , zn ) , given by the absolutely convergent series =

A(v, z) =

00

L (Ajk , frn) frn (v) fk (z) ,

k,rn=O

where k = (k1 , , kn ) , m = (m 1 , . . , mn ) are multi-indices, and frn (z) frn (z) . The matrix and Wick symbols of a bounded operator A are related by .

.

.

.

A(v, z)

=

=

e vz A(v, z ) .

Remark. The anti-holomorphic representation is very useful in quantum mechanics and especially in quantum field theory, where it is called holo­ morphic representation ( with respect to the variables z). Slightly abusing terminology, in Part 2 we will also call it holomorphic representation. Problem 2 . 1 5 . Find an explicit formula for the unitary operator establishing the Hilbert space isomorphism �n � L2 (1Rn , dn q) . Problem 2.16. Prove that for the bounded operator A the functions A( v , z ) and A( v, z ) are entire functions of 2n variables. Problem 2 . 1 7. Let A be a trace class operator on �n with the Wick symbol A (.i, z ) . Prove that

Problem 2 . 18. Show that the Wick symbol A(z, z ) of the product A = A1

•

•

•

At

is given by A ( z, z )

where

1 1 At (.Z, Zt - I )

7r

= 1

z0 = Zt

CL

=

z , Zt

=

.

.

. At ( .Zt ,

z

) exp

{ kL=t zk ( zk - t l

-

Zk )

}d2 Zt

.

.

.

d2 Zt- I .

z, and Ak (z, z ) are the Wick symbols of the operators Ak .

3 . Weyl relations

Let R be an irreducible unitary representation of the Heisenberg group H n in Hilbert space £. It follows from Schur ' s lemma that R( e a c) = eina I for some li E JR., where I is the identity operator on £. If li 0, the n-parameter abelian groups of unitary operators U(u) = R(eux) and V(v) R(e vY) commute, so using Schur's lemma once again we conclude that there exist p, q E JRn such that U(u) = e -iupi and V (v) = e-iv ql. Therefore, in this case the irreducible representation R is one-dimensional and is parametrized by a vector (p , q ) E JR2n . When n =/= 0, unitary operators U( u ) and V(v) satisfy the Weyl relations =

=

( 3. 1 )

U (u) V(v)

=

einuvv (v) U(u) ,

3.

Weyl relations

1 19

which admit the Schrodinger representation, introduced in Section 2 . 2 . It turns out that every unitary irreducible representation of the Heisenberg group Hn is unitarily equivalent either to a one-dimensional representation with parameters (p , q ) E JR2n , or to the Schrodinger representation with some n # 0. The physically meaningful case corresponds to n > 0, and representation with -n is given by the operators u- 1 (u) = U ( - u) and V( v) . 3 . 1 . Stone-von Neumann theorem. Here we prove the following funda­

mental result.

( Stone-von Neumann theorem ) . Every irreducible unitary representation of the Weyl relations for n degrees of freedom, U (u)V(v) = eiliuvv (v) U(u) , is unitarily equivalent to the Schrodinger representation.

Theorem 3 . 1

Proof. It is sufficient to consider n 1 , the general case similarly. Set 2 U(u)V(v) . S(u, v) = e- The unitary operator S( v ) satisfies ( 3.2 ) S(u, v)* = S( -u, -v), and it follows from the Weyl relation that =

n

> 1 is treated

i1iuv

u,

( 3.3 ) S ( u 1 , vi)S(u 2 , v2) = e � ( u1v2 -u2 v1 ) S(u1 + u2 , VI + v2 ). Define a linear map W : L 1 (JR2) £ ( £ ) , called the Weyl transform, by ----+

{ f(u, v)S(u, v)dudv . W( f) = 2 7r JJR2 Here the integral is understood in the weak sense: for every 'lj; 1 , 'I/J2 E £, 1

{ f(u v)( (u v)'I/J , )dudv . JJR2 , S , I 'l/J2 The integral is absolutely convergent for all 'I/J1 , 'I/J2 E £, and defines a bounded operator W (!) satisfying ( W( /)'1/JI , 'I/J2) =

1 2 7r

I I W(f) l l :S 27r l l f l l v · The Weyl transform has the following properties. WTl . For f E L 1 (1R2 ) , W ( f) * = W(f *) , where f *(u, v) = f( -u,-v). 1

120

2. Basic Principles of Quantum Mechanics

WT2. For f E L 1 ( JR2 ) , S(ul , VI ) W (f)S(u 2 , V2 )

where f (u , v)

=

=

w(/) ,

e � { (u l - U2 ) v- (vl -V2)u+u 1 V2 -U 2 Vt} f (u Ul - U2 , v

-

WT3. Ker W {0} . WT4. For JI , h E L 1 ( JR 2 ) ,

VI - V2 ) .

=

W ( fl ) W ( h ) = W ( fl * n h ) ,

where

* n h ) (u, v) = -

1 r e � ( uv' - u' v) fl (u - u' , v - v' ) h ( u' , v' ) du' dv ' . 2 7f J'R_2 The first two properties follow from the definition and (3. 2 ) and (3.3) . To prove WT3 , suppose that W (f) = 0. Then for every 'l/JI , 'I/J2 E Yt' and all u' , v' E lR we have

(!I

0

=

=

-21 J'R_{ 2

( W (f) S(u ' , v ' ) 'I/Jl , S(u' , v ' ) 'I/J2 ) 7f

n ei ( uv ' - u'v) f (u, v) (S(u, v ) 'I/J I , 'I/J2 ) dudv .

Therefore f (u, v) (S(u, v) 'I/JI , 'I/J2 ) 0 a.e. on JR 2 , so that f 0. To prove WT4, we compute ( W ( fl ) W ( h ) 'I/J l , 'I/J 2 ) = ( W ( h ) 'I/J I , W ( fl ) * 'I/J2 ) 1 r (u , V2 ) (S(u2 , V2 ) '1/Jl , W ( JI ) * 'I/J2 ) du2dV2 2 7f J'R_2 h 2 =

=

=

=

=

{ h (u 2 , v2 ) ( W ( JI )S(u 2 , v2 )'1/J1 , 'I/J2 ) du2dv2 _..!._ 27r J'R_2

� 12 12 fl (ul - u2 , v1 - v2 ) h (u2 , v2 ) ·

(2 ) 2 ·

e � ( u 1 v2 -u2v l ) (S(u l , V ) '1/J , 'I/J ) du1 dV 1 dU 2 dV2 1 1 2

{ (!I * n h ) (u, v) (S(u, v) ,P 1 , 'I/J2 ) dudv. = _..!._ 27f J'R_2 We have fi * n h E L 1 (JR 2 ) , and it follows from the associativity of the operator product and property WT3 that the linear mapping *n : L l ( JR 2 )

defines a new associative product

L l (JR 2 ) --+ Ll ( JR 2 ) on L 1 ( JR 2 ) ,

x

!I * n ( h * n h ) (!I * n h ) * n h When n = 0, the product * n becomes L 1 ( lR2 ) . =

for all JI , h , h E L 1 ( JR2 ) . the usual convolution product on

121

3. Weyl relations

For every non-zero 't/J E .Yt' denote by � the closed subspace of .Yt' spanned by the vectors S(u, v ) 't/J for all u, v E R The subspace � is invari­ ant for all operators U(u ) and V(v) . Since the representation of the Weyl relation is irreducible, � = .Yt' . There is a vector 't/Jo E .Yt' for which all inner products of the generating vectors S(u, v ) 't/Jo of �0 can be explicitly computed. Namely, let fo (u , v) = n e -%
and set Then W0

Wo = W (fo ) . =

Wo and Wo S(u, v) Wo =

e -% (u2 +v2 )Wo.

In particular, W6 = W0, so that Wo is an orthogonal projection. Indeed, using WT4, we have Wo S(u, v)Wo = W ( fo * n fo) ,

where

Jo ( u' , v' ) = e�(uv' - u' v ) fo (u' - u, v ' - v ) .

By "completing the square" we obtain where

( I uI , v ' )

' ' Uo * n io) (u' , v ' ) = ne - %< u2 + v 2 +u 2 +v 2 ) I (u' , v ' ) ,

1 e _!!.{(2 u11 -u-u'+iv+iv' )2 + (v11-v-v1-iu-iu'?} du "d - 3_ 271' JR 2 _

11

v .

Shifting contours of integration to Im u" = -v - v ' , Im v" = u + u' and substituting � = u" - u - u' + iv + iv', rt v" - v - v' - iu - iu', we get =

I (u' , v ' ) = � { e - �Ce+T72 )d�drt = 1 . 2 71' JJR 2 Now let £a = Im Wo - a non-zero closed subspace of .Yt' by property WT3. For every 't/J1 , 't/J2 E £a we have Wo't/J1 = 't/J1 , Wo't/J2 = 't/J2 , and (S(u b v 1 )'t/Jb S(u2 , v2 )'t/J2) = (S(u b v 1 )Wo't/Jb S(u 2 , v2 ) Wo't/J2 ) (WoS( -u 2 , -v2 ) S(u1 , v 1 )Wo't/JI , 't/J2 ) =

=

e �(u l vz- uz vl)(WoS(ul - U2 , VI - V2)Wo't/JI , 't/J2 )

2. Basic Principles of Quantum Mechanics

1 22

This implies that the subspace £a is one-dimensional. Indeed, for every 'I/J 1 . 'I/J2 E £a such that ('1/Jl , 'I/J2 ) = 0, we have that the corresponding sub­ spaces Y'e,p1 and £1/12 are orthogonal . Since £1/J £ for every non- ero '1/J , at least one of the vectors 'I/J1 , 'I/J2 is 0. Let £a = C '1/Jo , 11'1/Jo ll 1 , and set 'I/Jo,(3 = S(a, /3)'1/Jo , a, /3 E R The closure of the linear span of the vectors 'I/Jo, (3 for all a , f3 E R is £. We have =

and

z

=

( (3 o /' = e il!2 u -v ) 'I/Jo +u, (3 +v · S ( u, v ) "'f'o,(3

Next we consider the Schrodinger representation of the Weyl relation in

£2 (JR, dq) :

( U (u) rp) (q) = cp (q - !iu) , ( V (v) rp ) ( q) = e-iv qcp (q) , =

( S (u , v ) cp ) (q )

ill

.

e 2 uv -wq cp (q - liu) .

For the corresponding operator Wo we have (W0 rp ) ( q) = =

1 � 1oo v ; 00 1 1 00 oo

11 ( 2+ 2 ) iii . e - 4 u v e 2 uv -w q cp(q - nu) dudv

n

27r

JR2

= --

V/ii

2 ( ) e - % {u + u- h 2 } cp (q - !iu)du

-

-

' _ .l. ( 2 + ' 2 ) e 211 q q cp ( q ) dq I ,

so that Wo is a projection onto the one-dimensional subspace of L2 (R, dq) spanned by the Gaussian exponential cpo (q) 4� e- frt q2 , l lrpo l l 1 . Let V 7rli =

Cfl o , (3

=

S (a, f3) cpo .

=

Since the Schrodinger representation i s irreducible, the closed subspace spanned by the functions Cflo , (3 for all a , f3 E lR is the whole Hilbert space L2 (1R, dq) . (This also follows from the completeness of Hermite-Tchebyscheff functions, 2 since Cfl o,(3 (q) = � e %- of3- if3q - 2� ( q -oo ) . ) We have - il! (oc5 - (3-y) - l!4 { (o --y) 2 + ((3-c5) 2 } ( Cfl o , (3 , Cfl-y,8 ) - e 2

and S ( u, v ) Cfl o,(3

=

o) e il!2 (u(3- v Cfl o +u, (3 + v ·

3.

123

Weyl relations

ut/ ( '!j; )

=

n

L Ci
E

L 2 (IR, dq) .

i =l Since inner products between the vectors '1/Ja , ,B coincide with the inner prod­ ucts between the vectors 'Pa , ,B , we have

so that tJt/ is a well-defined unitary operator between the subspaces spanned by the vectors {'1/Ja, ,B } a ,,BEIR and {
.

.

.

.

.

Proof. It follows from the Stone theorem that in the Schrodinger represen­ tation the generators P and Q are momenta and coordinate operators. 0 Remark. For

n degrees of freedom

S ( u, v) e - � uv u (u) V (v) e- � uv e -i uP e - i vQ , and formally using the Baker-Campbell-Hausdorff formula, we get S ( u, v) = e- i( uP+vQ) . (3.4) =

=

Thus the Weyl transform W(f)

=

1 - r (u , v) e-i(uP+vQ) �u dnv (27r )n }JR2n f

can be considered as a non-commutative Fourier transform valued generalization of the "ordinary" Fourier transform ] (p , q)

=

-

an operator­

r f(u, v) e- i(up+vq) dnu dnv . -1-)n }JR2n (27r

Remark. The Stone-von Neumann theorem is a very strong result. In par­

ticular, it implies that creation and annihilation operators for n degrees of freedom, 1 1 a = (Q + tP ) , a * -/2ii (Q - iP ) , -/2ii .

=

124

2. Basic Principles of Quantum Mechanics

where P and Q are the corresponding generators 24 satisfying Heisenberg commutation relations, are unitarily equivalent to the corresponding oper­ ators in the Schrodinger representation. Thus there always exists a ground state - a vector 7/Jo E £, annihilated by the operators a = (a 1 , . . . , an ) · The corresponding statement no longer holds for quantum systems with in­ finitely many degrees of freedom, described by quantum field theory, where one needs to postulate the existence of the ground state ( the "physical vac­ uum" ) . The Weyl transform in the S chrodinger representation £ = £2 ( IRn , dn q ) can be described explicitly as a bounded operator with an integral kernel. Namely,

( 3.5)

(S(u, v ) 7jJ ) ( q ) = /� uv - ivq 7/J ( q - fiu ) , 7/J E L 2 (1Rn , �q ) , and for 1/J1 , 1/J2 E L2 (1Rn , dnq) and f E L 1 ( IR2n) n L2 (IR2n) we have

1

}'R2n j(u , v ) ( S(u, v ) 7/J1 , 7/J2 )dnu �v

( W (f) 7/J 1 , 1/J2 ) = - f ( 27r ) n

=

1 n r r e % uv -ivqf( u, v ) 7fJl (q - fiu ) 7/J (q)dn q) dn u dn v 2 ( 27r ) }JR2n ( }JRn

= in (in K (q, q') 7/Jl (q')�q') 7/J2 (q)dn q,

where

( 3.6) The interchange of the order of integrations is justified by the Fubini theo­ rem. Thus W ( f) is an integral operator with the integral kernel K (q, q' ) , ( W (f) 7J; ) (q)

=

}JRn{ K(q , q') 7/J (q' ) dnq',

7jJ

E

L 2 (1Rn , dn q ) .

By the Plancherel theorem,

r l f(u , vWdn u dnv, }JRr 2n I K (q , q'Wdn q dn q' = }JR2n

so that W (f) is a Hilbert-Schmidt operator on £. The linear subspace L1 (JR2n ) n L2 (JR2n ) is dense in L2 ( IR2n) and the Weyl transform extends to the isometry L2 (IR2n) Y2 , where Y2 is the Hilbert space of Hilbert­ Schmidt operators on £ = L2 (1Rn, dnq) . Since every Hilbert-Schmidt oper­ ator on £ is a bounded operator with the integral kernel in L2 ( IR2n ) , the mapping W is onto. Thus we have proved the following result.

W

(Qt, ... ,Qn)-

:

--->

2 4 Here it is convenient to lower the indices

by

the Euclidean metric on

JRn

and use Q

=

3. Weyl relations

125

Lemma 3 . 1 . The Weyl transform W defines the isomorphism L 2 ( JR.2n ) :::: Y2 .

Problem 3 . 1 . Let P = (P1 , . , Pn) and Q = (Q1 , . . . , Qn ) be self-adjoint op­ erators25 on .Yt' satisfying Heisenberg commutation relations (2.4) on some dense domain D in .Yt', and such that the symmetric operator H = I:: �=l (Pf + Q D , defined on D , is essentially self-adjoint . Also suppose that every bounded operator which commutes with P and Q is a multiple of the identity operator in .Yt'. Prove that P and Q are unitarily equivalent to momenta and coordinate operators in the Schrodinger representation. ( Hint: By the Stone-von Neumann theorem, it is sufficient to show that P and Q define an integrable representation of the Heisen­ berg algebra. One can also prove this result directly by using commutation relations between the operators H, a * , a and the arguments in Section 2 . 6 . This would give another proof of the Stone-von Neumann theorem. ) .

.

Problem 3 . 2 . Prove formula (3.4) . 3.2. Invariant formulation. Here we present a coordinate-free descrip­

tion of the Schrodinger representation. Let (V, w ) be a finite-dimensional symplectic vector space, dim V = 2n. Recall that (see Section 2 . 1 ) the Heisenberg algebra g = g (V ) is a one-dimensional central extension of the abelian Lie algebra V by the skew-symmetric bilinear form w. As a vector space, g = V EB JR.e, with the Lie bracket [u + ac, v + ,Be] = w ( u, v ) e, u, v E V, a, ,8 E R

Every Lagrangian subspace f of V defines an abelian subalgebra f EB JR.e of g. A Lagrangian subspace f' is complementary to f in V if f EB f' = V. A choice of the complementary Lagrangian subspace f' to £ establishes the isomorphism gj (f EB JR.c ) :::: f' .

The Heisenberg group G = G(V) is a connected and simply-connected Lie group with the Lie algebra g. By the exponential map, G :::: V EB JR.e as a manifold, with the group law exp(v 1 + a1 e) exp(v2 + a 2 c ) = exp(v 1 + v2 + (a1 + a 2 + � w (v1 , v2 )) c ) , where v1 , v2 E V, a1 , a 2 E R A choice of the symplectic basis for V es­ tablishes the isomorphism G(V) :::: Hn with the matrix Heisenberg group defined in Section 2 . 1 . The volume form d2n v 1\ de, where d2 n v E A 2n V* is a volume form on V and V * is the dual vector space to V, defines a hi-invariant Haar measure on G. Every Lagrangian subspace f defines an 25See previous footnote.

2.

126

Basic Principles of Quantum Mechanics

abelian subgroup L = exp ( R EB JR c) of G, and a choice of the complementary Lagrangian subspace R' establishes the isomorphism

GjL c::: e' . ( 3.7 ) The isomorphism A2n V* ::: Anf* 1\ Ant• gives rise to the volume form dnv' on R' and defines a measure dg on the homogeneous space G / L. The measure dg is invariant under the left G-action and does not depend on the choice of R' . For a given n E JR, the function X L C, i x ( exp ( v + ac) ) e na, v E V, a E JR, defines a one-dimensional unitary character of L, :

=

--t

Definition. The Schrodinger representation Se of the Heisenberg group

G ( V ) , associated with a Lagrangian subspace R of V, is a representation of G induced by the one-dimensional representation X of the corresponding abelian Lie group L , Se lndt X · =

By definition of the induced representation, the Hilbert space .Ytf_ of the representation Se consists of measurable functions f G C satisfying the property f (g l ) = x (z) - 1 f ( g) , g E G, l E L, and such that :

I I J II 2 =

r

I J ( g) l 2 dg

____,

la;L Corresponding unitary operators Se (g) , g E G, are defined by the left trans­ lations, ( Se ( g) J ) ( g' ) = f ( g-1 g ' ) , f E .Ytf_ , g E G. In particular, ( Se ( exp ac) J ) (g) J ( exp ( - ac)g) = J (g exp ( -ac) ) ei !ia J (g) , so that Se ( exp a c) = e i na I, where I is the identity operator on .Ytf.. For every Lagrangian subspace R of V, the representation Se is unitarily equivalent to the Schrodinger representation. This can be explicitly shown as follows. By ( 3.7 ) , a choice of a complementary Lagrangian subspace R' gives rise to the unique decomposition g exp v ' exp ( v + ac) , and for f E .Ytf_ we get ' J (g) = f ( exp v ' exp ( v + ac) ) = e - i !ia f ( exp v ' ) , v E R, v E R' , a E R =

< oo .

=

=

3.

127

Weyl relations

Thus every f E ft£ is completely determined by its restriction to exp P' � P' , and the mapping %'e : ft£ L2 ( P', dnv') , defined by %'e (f) (v') = f(exp *v') , v' E f' , is the isomorphism of Hilbert spaces. For the corresponding representation S e = %i!Se%'e-I in L2 (P', dnv') we have ( S e(exp v ) if! ) (v ' ) = e iw(v ,v')if! ( v') , v E f, v ' E £' , ( S e (exp u' ) if! ) (v' ) 'lf! ( v ' - liu ') , u' , v' E f' . Let e i , . . . , en , !I , . . . , fn be the symplectic basis for V such that f = R. e i EB EB .lRe n and .e' .IR JI EB · · · EB .lR fn , and let (p, q) = (p i , . . . , Pn , q i , . . . , qn ) be the corresponding coordinates in V ( see Section 2.6 in Chapter 1 ) . Then L2 (.e', dn v ' ) � L2 (.1Rn , dn q) and Se( exp uX) = U(u) and S e (exp v Y) = V (v ) , where uX = L�=I u k fk , v Y L�= I Vk e k , and U(u ) = e-iuP, V (v ) e- i vQ are, respectively, n-parameter groups of unitary operators correspond­ ing to momenta and coordinate operators P and Q. The mapping * h - the new associative product on L 1 ( .IR2n ) introduced in the last section - can also be described in invariant terms. Let Bn_ = G jrn_ be the quotient group, where rn_ = { exp e�n c) E G I n E Z } is a discrete central subgroup in G, and let db = d2nv A d*a be the left-invariant Haar measure on Bn_, where d*a is the normalized Haar measure on the circle R./ 2;z. The Banach space L I (Bn_, db) has an algebra structure with respect to the convolution: --+

=

·

·

=

·

=

(
{

}B,.