Faddeev L.D., Yakubovskii O.A. Lectures on Quantum Mechanics for Mathematics Students

Подождите немного. Документ загружается.

L. D. Faddeev and 0. A. YakubovskiI

H are known, then it is easy to construct a solution of the Cauchy

problem

do(t)

= H

V(t)}

V/ (t) I t=O = 0,

for the Schrodinger equation. For this it suffices to expand the vector

zG in the eigenvector basis of the operator H,

E ciWoi

C2 ... (tb, 7'Z)

i-1

and to use the formula (9). Then we get the solution of the Cauchy

problem in the form

(13)(t} ---

hit

2=1

This formula is usually called the formula for the expansion of a so-

lution of the Schrodinger equation in stationary states.

In the conclusion of this section we obtain the time-energy un-

certainty relation. Setting B = H in (7.1), we have

w .

1({ AH}h(14)

w w

Recalling that t = {H, A } h in the Heisenberg picture and using the

obvious equality dt = dc I w), we rewrite the relation (14) as

A, A

H>h

dAavg

We introduce the time interval &, t A = Ow A / I

dALvg

. In the time

&,t A the mean value of the observable Aa,,,g is displaced by the

"width" &,A of the distribution. Therefore, for the state w and

the observable A, OW t A is the characteristic time during which the

distribution function WAN can change noticeably.

It follows from

the inequality

&WtA&H > h/2

that the set of values of A,1 A for all possible observables A in the

state w is bounded from below.

§ 10. Quantum mechanics of real systems 49

Setting At = infA OWtA arid denoting &,H by A E, we get that

for any

state w.

(15)

AE At > h/2.

This is the time-energy uncertainty relation. The physical meaning

of this relation differs essentially from the meaning of the uncertainty

relations (7.1). In (7.1), &,A and A , B are the uncertainties in the

values of the observables A and B in the state w at, one and the same

niorrient of time. In (15), Al is the uncertainty of the energy, and

this does not depend on the time, and At characterizes the time over

which the distribution of at least one of the observables can change

noticeably. The smaller the time At, the more the state w differs from

a stationary state. For strongly nonstationary states, At is small and

the uncertainty DE of the energy must be sufficiently large. On the

other hand, if of = 0, then At = oo.

In this case the state is

stationary.

§ 10. Quantum mechanics of real systems. The

Heisenberg commutation relations

We have already mentioned that the finite-dimensional model con-

structed is too poor to correspond to reality and that it can be im-

proved by taking the state space to be a complex Hilbert space fl

and the observables to be self-adjoint operators acting in this space.

One can show that the basic formula for the mean value of an

observable A in a state w keeps its form

(1)

(A I w) = Tr AM,

where M is a positive self-adjoins operator acting in fl with trace 1.

For any self-adjoint operator there is a spectral function PA (A),

that is, a family of projections with the following properties:

1) PA (A) < PA (li) for A < p, that is, PA (A) PA (li) = PA (A) ;

2) PA (A) is right-continuous, that is, lim;,_.., A+c PA (1u) = PA (A) ;

3) PA(-oo) = lima-,_0CPA (A) = 0 and PA (oo) = I;

4) PA (A) B = B PA (A) if B is any bounded operator commuting

with A.

L. D. Faddeev and 0. A. Yakubovskii

A vector p belongs to the domain of the operator A (cp E D(A))

and then

(2)

jA2d(PA(A),)

< oc,

AdPA(A) o.

A function f (A) of the operator A is defined by the formula

(3)

f (A) p = f-

f (A) dPA(A) p-

The domain D (f (A)) of this operator is the set of elements :p such

that

oc.

The spectrum of a self-adjoint operator is a closed subset of the

real axis and consists of all the points of growth of the spectral func-

tion PA (A) . The jumps of this function correspond to the eigenvalues

of the operator A, and PA (A + 0) - PA (A -- 0) is the projection on the

eigenspace corresponding to the eigenvalue A. The eigenvalues form

the point spectrum. If the eigenvectors form a complete system, then

the operator has a pure point spectrum. In the general case the space

can be split into a direct sum of orthogonal and A-invariant subspaces

7x and H2 such that A acting in the first has a pure point spectrum

and acting in the second does not have eigenvectors. The spectrum

of the operator acting in the subspace f2 is said to be continuous.

It follows from the basic formula (1) that in the general case the

distribution function of A in the state w H M has the form

(4) WA (A) = Tr PA (A) M,

while for pure states M = Pp,

11,011 = 1, we have

(5)

WA(A) = (PA(A),O,

Unlike in the finite-dimensional model, the function WA (A) is not nec-

essarily a step function. The set of admissible values of an observable

A coincides with the set of points of growth of the function WA (A) for

all possible states w. Therefore, we can assert that the set of possible

results of measurements of A coincides with its spectrum. We see

§ 10. Quantum mechanics of real systems

that the theory makes it possible to describe observables with both a

discrete and a continuous set of values.

Our problem now is to give rules for choosing the state spaces

and to learn how to construct the basic observables for real physical

systems. Here we shall describe quantum systems having a classi-

cal analogue. The problem is posed as follows.

Suppose that we

have a classical system, that is, we are given its phase space and

llamiltonian function. We must find a quantum system, that is, con-

struct a state space and a Schrodinger operator, in such a way that

a one-to-one relation f A f is established between the classical

observables (functions on the phase space) and quantum observables

(operators acting in the state space). Furthermore, the Hamiltonian

function must correspond to the Schr"finger operator. This one-to-

one relation certainly cannot be an isomorphism fg F-- / -+ A f o A9,

f f, g } +- / -* {A, A9 J h (it is therefore that quantum mechanics dif-

fers from classical mechanics), but it must become an isomorphism as

h -+ 0 (this ensures that quantum mechanics approaches classical me-

chanics in the limit). The quantum observables A f usually have the

same names as the classical observables f. We remark that we must

not exclude the possibility of the existence of quantum systems that

do not have simple classical analogues. For such systems there can be

observables that do not correspond to any function of the generalized

coordinates and momenta.

The correspondence rules and the approach to the classical me-

chanics limit will be described at length in § 14. For the present we

establish a correspondence only between the most important observ-

ables, and we show how to construct the state space for the simplest

systems.

Let us first consider a material point. Its phase space is six-

dimensional, and a point in it is determined by specifying three Caxte-

sian coordinates q1, q2, q3 and three momentum projections p1, p2, p3.

It is not hard to compute the classical Poisson brackets for any pairs

of these observables:

{q,q}=0,

{pi,pj} =0,

{pi,gj}--Sip, i,j=1,2,3.

L. D. Faddeev and 0. A. YakubovskiT

For a particle in quantum mechanics we introduce six observables Q,

Q2- Q3- P1, P2. PP.3 for which the quantum Poisson brackets have the

same values:

(6)

{Q1.Qj}h = 0,

{P, P3 }h = o,

{IQj}h =- Ihj3.

where I is the identity operator. These observables will be called the

coordinates and momentum projections.

Below we shall justify this association of operators with coordi-

nates and momenta by the following circumstances. After studying

the observables determined by the relations (6), we shall see that they

have many of the properties of classical coordinates and xrionienta.

For example, for free particles the projections of the momentum are

integrals of motion, and the mean values of the coordinates depend

linearly on the time (uniform rectilinear motion).

Further. the correspondence qi H "L}i, pi E--3 PZ, i = 1, 2, 3, is

fundamental for us for the construction of a general correspondence

f (q, p) H Af . The quantum observables defined in this way, including

the coordinates and momenta, turn into the corresponding classical

observables in the limit.14 Finally, the correspondence rules f ; Al

make. the theory completely concrete, and its results can be verified

in experiments. It is the agreement of theory with experiment that

should be regarded as the definitive justification of the assumption

(6) and of all of quantum mechanics.

The conditions (6) are called the Heisenberg quantization condi-

tions. Together with (7.1) they at once imply the Heisenberg uncer-

tainty relations for the coordinates and momenta. We have already

discussed these relations.

The formulas (6) are often written for the commutators:

(7)

[Q7, QkJ = O, [PjiPk1=O,

[Q3, Pk]= ihSjk.

These relations are called the Heisenberg commutation relations.

14The exact meaning of this assertion is discussed in § 14

§ 10. Quantum mechanics of real systems

We now state without proof the remarkable von Neumanii-Stone

theorer1m.15

For the system of relations (7) there exists a unique irreducible

representation by operators actin( in a Hilbert space (unique tip to a

unitary transformation).

We recall what an irreducible representation is. One usually uses

two equivalent definitions:

1) A representation of the relations (7) is said to be irreducible

if there does not, exist an operator other than a multiple C1 of the

identity that commutes with all the operators Qi and Pi.

2) A representation is said to be irreducible if there does not exist

a nontrivial subspace Ho of x that is invariant under all the operators

Q1 and P.

We verify the equivalence of these definitions. If there is a non-

trivial subspace Ho invariant under Qi and I i, then the projection

PI-j0 onto this subspace commutes with Qi and Pi, and obviously

PH() CL

If there is an operator A CI commuting with all the Qi and Pi,

then A* and hence also A + A* commute with Qi and Pi. Therefore,

we can assume from the very beginning that A is self-adjoint. The

self-adjoint operator A has a spectral function PA (A) which for some A

is different from zero and CI. The operator PA (A) commutes with Qi

and Pi, and thus the subspace Ho onto which it projects is invariant

with respect to Qi and Pi.

It follows from the von Neumann-Stone theorem that if we find

some representation of the Heisenberg commutation relations and

prove that it is irreducible, then all other irreducible representations

will differ from it by a unitary transformation.

We know that the physical interpretation of the theory is based

on the formula (A I w) = RAM for the mean value. The right-hand

side of this formula does not change under a unitary transformation

15The formulation given is not precise, since there

are subtleties connected with

the unboundedness of the operators P and Q that we are not going to discuss Sim-

ilar difficulties do not arise if instead of the operators P and Q themselves we con-

sider bounded functions of these operators We present a precise formulation of the

von Neumann-Stone theorem in § 13

L. D. Faddeev and 0. A. Yakubovskii

of all the operators. Therefore, the physical results of the theory do

not depend on which of the unitarily equivalent representations we

choose.

§ 11. Coordinate and momentum

representations

We describe the so-called coordinate representation most often used

in quantum mechanics. It is sometimes called the Schrodinger repre-

sentation. As the state space for a material point in this representa-

tion, we choose the space L2 (R3) of square-integrable complex-valued

functions p (x) = Sp(x 1

, X2, X3). The scalar product is defined by

(1)

((P1, 02) =

1(X) So2 (x) dX.

The operators Qj and Pj, = 1, 2, 3, are introduced by the for-

rnulas

Qj c0 (X) = xj fi(x),

(2)

h (9

Pj O(x)

O(x)

8xj

These are unbounded operators, and this corresponds to the phys-

ical meaning of the observables, since the numerical values of the

coordinates and momenta are unbounded. The operators Q., and P

themselves and all their positive integer powers are defined on the do-

main D formed by the infinitely differentiable functions that decrease

faster than any power. Obviously, D is dense in W. It can be shown

that the operators Qj and P3 have unique self-adjoint extensions upon

closure. It is easy to verify that these operators are symmetric. For

example, for P1 and any functions b, ;p E D we get by integrating by

parts that

(Pi 5P,

} =

V)(x)

11)0

dx2 dx3.

Ax)'P(X)

R.3

ox,

«,

f0C

h 8ip(x) h ft (x)

dx so(x)

dxi

dx So(x)

i ax,

- p,

Pi h)

§ 11. Coordinate and momentum representations

Let us verify the commutation relations (10.7). The first two of

them are obvious, and the third follows from the equalities

PkQ3Se(x)

i vex

(xjfi(x))

bjk'P(x) + xj

SP(X),

2 i(Xk

SP(X) = x

3 i

S O

Thus, we have indeed constructed a representation of the relations

(10.7) by linear operators. The irreducibility of this representation

will be proved later

We present one more example of a representation (the momentum

representation). In this representation the state space 7-1 is also the

complex space L`2 (R3) with elements co(p), and the operators Q.7 and

PJ , j = 1, 2, 3, are defined as follows:

Qj P(p) = i

5P-j

(3)

SP(P),

P(p) _ pj SP(P)-

The commutation relations (10.7) are verified in the same way

as for the coordinate representation, and the proof of irreducibility is

t he same. By the von Neumann Stone theorem, these representations

are unitarily equivalent; that is, there is a unitary transformation 16

Se(x)

- (p)

such that the operators defined by (3) are transformed into the oper-

ators (2). It is not hard to see that the Fourier transformation

(4)

x =

SP()

(27rh

cP(P) P

(5)

SP(p) =

2i h

e- hp"cp(x) dx

is such a transformation

The unitarity of the Fourier transformation follows from the Par-

seval equality

fR3 Isa(X)I2 ax = 1R3

dp.

16We intentionally use one and the same symbol to denote the different functions

.P(x) and cp(p), since both these functions represent one and the same state p

L. D. Faddeev and 0. A. Yakubovskii

to the expression (4), we get thatApplying the operator h

OX j

h a

i a:r.j

27rh

-1

f' h pX

(P) dP =

27Th

ppp(P) dP

We see that tinder the unitary transformation W, the operator

is transformed into the operator of multiplication by the variable

z {`fix J

Pi. It can he shown similarly that

xjp(x)

----p ih

SA(P)

Thus. we have proved the unitary equivalence of the coordinate and

the momentum representations.

We write the formulas for the transformation of integral operators

in passing from the coordinate to the momentum representation:

A(X,Y) P(Y) dY,

A4p(X) =

AiP(P) =

where

1 2

(27rh

iRS e

h P dx

()3fR9

e' 1iP(9) dxdYdq

P q) F(9) d4,

fR3 A(

(6) A(p, q) = 27Th

_ pXA(x, y) ,dx

dy.

fR6

If an operator in the coordinate representation is the operator of

multiplication by the function f (x), then it can be regarded as the

integral operator with kernel

A(x, y) = f (x) J(x - y).

In the momentum representation the kernel of such an operator de-

pends only on the difference of the variables and has the form

(7)

A(P,q)=

()3ff(x)e1_cix.

§ 11. Coordinate and momentum representations

The transformation formulas from the momentum representation

to the coordinate representation differ by the sign in the exponent:

(8)

A(X,Y)= (27rh)

f1R;6

hpA(Aq)e

"`"'dPdq,

and if A(p, q) = F(p)8(p - q), then

(9)

A(x, y) = (27rh ) fR

F(p) eT p(X-r) dp.

We now explain the physical meaning of the functions

(x) and

,o(p), which are usually called wave functions. We begin with the

wave function in the coordinate representation yp(x). In this repre-

sentation the operators Q (i = 1, 2; 3) are the operators of multipli-

cation by the variables x1; that is, the coordinate representation is an

eigenrepresentation for the operators Q j. To see this we make a little

mathematical digression.

To give a function representation of the Hilbert space 7-1 means

to specify a one-to-one correspondence between vectors in this space

and functions of a real variable, p

;p(x), and to specify a measure

on the real axis such that

(Sp, *) =

f(x)Ydrn(x).

The values of ;p(x) can be complex numbers or functions of other

variables, that is, they are elements of some additional space, and

(x)'(x) then donotes the scalar product in this additional space

In the first case the representation is said to be simple, and in the

second to be multiple. Two functions p(x) are regarded as the same

if they differ only on a set of m-measure zero.

A representation is called a spectral representation for an opera-

tor A if the action of this operator reduces to multiplication by some

function f (x).

representation is called a direct representation or eigenrepre-

sentation for an operator A if f (x) = x:

xo(x)