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

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L. D. Faddeev and 0. A. Yakubovskii

The spectrum of the operator A can be described as the support of the

distribution function of the measure. The jumps of the measure cor-

respond to the eigenvalues. If the measure is absolutely continuous,

then the spectrum is continuous.

We recall that in the finite-dimensional space Cn we defined an

eigenrepresentation for an operator A to be a representation in which

the vectors SP were given by the coefficients Spj of the expansion in an

eigenvector basis of A Suppose that the spectrum of A is simple:

Aii = ai V)i

>P =

Oi lPi ,

SPi = (W,lii).

i=1

With a vector Sp one can associate a function w(x) such that Sp(ai) =

Spi (the values of SP(x) for x

ai can be arbitrary). The distribution

function of the measure is chosen to be piecewise constant with unit

jumps at x = ai. The operator A can then be given by the formula

ASP H xSP(x),

since this expression is equivalent to the usual one:

ai;Pl

A _

SPn

a'.t 'p

Corresponding to a multiple eigenvalue ai are several eigenvec-

tors. We can introduce the function Sp(x) whose value at the point ai

is the vector with components 0ik, k = 1, 2, ... , ri.

In this case the

representation is multiple. and its iriultiplicity is equal to the multi-

plicity ri of the eigenvalue ai.

We see that in an eigenrepresentation an operator is an analogue

of a diagonal rrmatrix on C".

In an eigenrepresentation for an operator A it is easy to describe

a function f (A)

(10) I (A) p H .f (x) p(x).

§ 11. Coordinate and momentum representations

Iri particular, for the spectral function PA (A) = ©(a -- A) the formula

(10) gives

PA (A) O +-+ O(A - X) V(X) =

fcp(x),

x < A,

1.0,

x > A.

The spectrum of A will be completely described if for A we can con-

struct a spectral representation of it. A transformation carrying a

given representation into a spectral representation is called a spectral

t rarisformation.

Let us return to the coordinate representation. We see that

it really is a multiple eigenrepresentation for the three operators

Q1, Q2, Q3:

-* xi So(xl ).

The value of cp(x1) is a vector in an additional space, in this case a

function of the variables x2 and x3. The scalar product is defined by

the formula

0) =

[(xi)(dx,

that is. in the coordinate representation the measure m(x) is Lebesgue

measure, and the support of its distribution function is the whole real

axis, arid hence the spectrum of the coordinate is continuous and fills

the whole axis.

The distribution function of the coordinate Q1 in a pure state

w +-+ 1-11 has the forni

WRY (A) = (PQ1(A)'.p,) = o(xj) o(xj dxl,

from which it follows that

P(xl

, x2, x3)

(x1,x2.x3)dx2 dx3

o(x1)

(Xi) =

is the density of the distribution function of the coordinate Q1. The

expressions for the densities of the distribution functions of Q2 and

Q3 are written similarly.

It is natural to expect that I (x)12 is the

density of the distribution function common for all the coordinates;

that is, the probability of finding a particle in a region f

of three-

dimensional space is determined by the expression .f1 dx. We

L. D. Faddeev and 0. A. Yakubovskii

prove this assertion in the section devoted to systems of commuting

observables.

The momentum representation is an eigenrepresentation for the

three operators P1, P2 , P3, and I W (p) 2 is the density of the distribu-

tion function common for the three projections of the momentum.

We can now look at the uncertainty relations for the coordinates

and momenta from a new point of view. We see that these relations

are explained by a well-known property of the Fourier transforma-

tion. The more strongly the function ;p (x) is concentrated, and thus

the smaller the uncertainties A;,, Qi of the coordinates are, the more

the Fourier transform ap(p) is spread out, and thus the greater the

uncertainties 0, Pi of the momenta.

§ 12. "Eigenfunctions" of the operators Q and P

We now consider the equations for the eigenvectors of the operators

Q and P For simplicity we consider a particle with one degree of

freedom. In the coordinate representation these equations have the

form

(1) (X)

= X0 Oxo W -

(2)

It =wnW

Solving them, we gel that

(3)

'OX" (X) = 6(X - XO).

(The first formula follows at once from the property x6(x) = 0 of the

6-function, and the second is obvious. The choice of the normalization

constant will be clear from what follows.)

Although an "eigenfunction"

x,,

(x) of the coordinate operator is

a generalized function while app (x) is an ordinary function, they have

in common that they both fail to be square integrable; that is, they do

not belong to L2 The operators Q and P do not have eigenfunctions

in the usual sense of the word.

§ 12. "Eigenfunctions" of the operators Q and P

To understand how the functions cpx (x) and ypp (x) are connected

with the spectrum of the operators Q and P, we recall the meaning of

eigenvectors in C'. The problem of finding the eigenvalues and eigen-

vectors of a matrix A is connected with the problem of diagonalizing

this matrix by a similarity transformation, or in other words, with

the transformation of an operator from some initial representation to

an eigenrepresentation. A self-adjoint operator A on C' always has

a basis consisting of eigenvectors:

A i =

(pi,

0k) = Sik

We look for the eigenvectors in the initial representation, in which

I ... I10I

Gl oM

rn(i) )

-O

The components of the vector in an eigenrepresentation of A are

computed from the formula

(5)

kP(i)

k=1

We see that the matrix consisting of the numbers Uik = ;pit) con-

jugate to the, components of the eigenvectors implements a spectral

transformation:

; = E

is unitairy, since

Goi

Uik UUj =

k2)'p .7)

Vii) = Saj

k=1

k=-1

k-1

If in (5) we formally replace p by

and by gy(p). then we get

that

_ ( p)P

()fCe(x)dx.

(6)

L. D. Faddeev and 0. A. Yakubovskii

This formula is already familiar to us. We see that the function

U(p, x) = cpp(x) is the kernel of a unitary operator carrying the coor-

dinate representation into the momentum representation (the eigen-

representation for the operator P). An "eigenfunction ' of the coordi-

nate operator Q can be interpreted in the same way. The coordinate

representation is an eigenrepresentation for Q; therefore, the operator

implementing the spectral transformation must be the identity, and

,o (x) = 6(x - x0) is the kernel of the identity operator.

We see from these examples that even though the "eigenfunc-

tions" of the continuous- spectrum are not eigenelements in the usual

sense of the word, their connection with the spectral transformation

remains the same as for eigenvectors in a finite-dimensional space.

We remark that there is a construction enabling one to give so-

lutions of the equation

Acpa = A(pa

the precise meaning of eigenvectors even for the case when ;Pa does

not belong to R. To this end we introduce a broader space V D N

whose elements are linear functionals defined on some space dD C fl.

The pair of spaces

V can be constructed in such a way that each

self-adjoint operator acting in f has in V a complete system of

eigenvectors. The eigenelements of A in and not in fl are. called

generalized eigenelements. If A has a simple spectrum. then any

V' E

has an eigenelement decomposition

c(A) p,\ dA.

The Fourier transformation can be interpreted as an eigenfunc-

tion decomposition of the momentum operator :pp (x) :

(8)

') _ (27rh)

(p) ez'sI dp

The normalization constant (1/2irh)1/2 in the expression for an eigen-

function of the momentum corresponds to the normalization

§ 13. The energy, the angular momentum, and others

This formula is a consequence of the well-known integral representa-

tion for the 6-function:

(10)

ik-TdX

27ro(k).

In the last two formulas the integrals are understood in the principal

value sense. The formula (9) is an analogue of the equality

(Wi,'Pk) = sik

for eigenvectors of the discrete spectrum.

We now construct the spectral function of the momentum opera-

tor. Only here we use the letter II for the spectral function, so there

will be no confusion with the momentum operator.

In the momentum representation the operator IIp (A) is the oper-

ator of multiplication by a function:

Hp(A) P(p) = O(A - p) SP(p)

Let us pass to the coordinate representation Using the formula (11.9)

for the one-dimensional case, we get that

li a x - 1

OA -

eTp(T-y) d

) (

27Th

( p)

-- e h

dp =

V. (x) WP (y) dP.

27r h

_ y)

h _ -

The derivative of the spectral function with respect to the pa

rameter A is called the spectral density. The kernel of this operator

has the form

rip

y) = V,\ (x) V.\ (y) -

27Th

eh a(X-y)

(

)(x,

We see that this is an analogue of the projection on a one-dimensional

eigenspace.

13. The energy, the angular momentum, and

other examples of observables

We now proceed to a study of more complicated observables. Our

problem is to associate with an arbitrary classical observable f (p, q)

a quantum analogue Af of it. We would like to set Al = f (P, Q),

L. D. Faddeev and 0. A. Yakubovski

but there is no general definition of a function on noncommuting

operators. For example, it is no longer clear which of the operators

Q2P, QPQ, or PQ2 one should associate with the classical observable

q2p. However, for the most important observables such difficulties do

not arise in general, since these observables are sums of functions of

the mutually commuting components Q1, Q2, Q3 and P1, P2, P3.

We present some examples.

The kinetic energy of a particle in classical mechanics is

pi + p2 + p3

The corresponding kinetic energy operator has the form

Pl + P2 + P3

2rn

and in the coordinate representation

(1)

T,(x)

where I. = 2

+ a is the Laplace operator. In the momentum

I 2

representation the operator T is the operator of multiplication by the

corresponding function:

pi+p2+p3

27n

SA(P)

In exactly the same way it is easy to introduce the potential

energy operator V (Q 1, Q2, Q3) .

In the coordinate representation, V

is the operator of multiplication by the function

(2)

V;c(x) = V (X) SP (x),

and in the momentum representation it is the integral operator with

kernel

(9- P)

V(P q) =

27rh

V(x) e h dx.

The total energy operator (the Schrodinger operator) is defined a.5

11 = T + V.

We write the Schrodinger equation in detail

ihdV(t)

= I1 t .

(3)

dt ( }

§ 13. The energy, the angular momentum, and others

In the coordinate representation the Schr6dinger equation for a par-

ticle is the. partial differential equation

a(x, t}

Av)(x, t) + V ( X )

and in the momentum representation the integro-differential equation

a0 (P, t) P2

0(p

t) +

V (p, q) b(q, t) dq.

Along with (4) we consider the equation for the complex conju-

gate wave function:

azl) (x, t)

_ h

(5)

-ih

2 _

Oib(x, t) + V(x) P(x, 1).

Multiplying the equation (4) by 70 and the equation (5) by 0 and

subtracting one from the other, we get that

-00

ih *-+V

V))

at 01) 2m

div( grad 7P - V) grad

(6)

where

aatl + aivi = n,

- ih

(,0 grad

grad

- 2m

V)).

The equation (6) is the equation of continuity and expresses the

law of conservation of probability. The vector j is called the density

vector of the probability flow. It is clear from (6) that j has the

following meaning: fs j,, dS is the probability that a particle crosses

the surface S per unit of tirrie.

The angular velocity plays an important role in both classical and

quantum mechanics. In classical mechanics this observable is a vector

whose projections in a Cartesian coordinate system have the form

11 g2p3 -43P2

12 = g3pi - g1p3,

13=gIP2--g2pI.

L. D. Faddeev and 0. A. Yakubovski"

In quantum mechanics the operators

(7)

Lk = QlPm - QmP1

are the projections of the angular momentum. Here k, 1, in are cyclic

permutations of the numbers 1, 2, 3. The right-hand side of (7) con-

tains only products of coordinates and momentum projections with

different indices, that is, products of commuting operators. It is in-

teresting to note that the operators Lk have the same form in the

momentum and coordinate representations:

(A)

h( a al

i `C7xm

- xm

C7x(

(X)'

(i)z-

m -Pm

iP(P)

The properties of the operators Lk will be studied at length below.

Quantum mechanics can also describe systems more complicated

than just a material point, of course. For example, for a system of

N material points the state space in the coordinate representation

is the space17 L2 (R3N) of functions ;p(Xl, ... , XN) of N vector vari-

ables. The Schrodinger operator for such a system (the analogue of

the Hamiltonian functions (1.5)) has the form

N N

2mi

i+>1'j(Xi-Xj)+>Vj(X).

(9)

H=-

i=1 i<j

i=1

The physical meaning of the terms here is the same as in classical

mechanics.

We consider two more operators for a material point which turn

out to he useful for discussing the interconnection of quantum and

classical mechanics:

U(u) = e-i(U1

P1+U2 '2+u3P3)

V(V) ` e-2(v1Q1+v2Q2+V3Q3)

17 Below

we shall see that for identical particles, the state space 'H coincides with

the subspace L2 C L2 of functions with a definite symmetry

§ 13. The energy, the angular momentum, and others

where u (u x , u2 , u3) and v (vi , V2, v3) are real parameters. The opera-

tor V (v) in the coordinate representation is the operator of niultipli-

cation by a function-

V(v) V(X) = eL°XV(X)

We now explain the meaning of the operator U(u). For simplicity

we consider the one-dimensional case

U(u)

e- iuP.

Let

(u, x) = e- iu'Sp(x) Differentiating Sp(u, x) with respect to the

parameter u, we have

a;p(u, x)

- X),

&(u, x)

_ h

80(u' x)

To find the function ;p(u, x), we must solve this equation with the

initial condition

co(u, x) I,L=o = Se(x)

The unique solution obviously has the form

p(u. x) = p(x - hu)-

We see that in the. coordinate representation the operator U (u)

is the operator translating the argument of the function ,,p(x) by the

quantity -hu. In the three-dimensional case.

(12)

U(u) p(x) _ p(x - uh).

The operators U (u) and U (v) are unitary, because the operators

1'j, P2, P3 and Qi , Q2, Q3 are self-adjoint. Let us find the coxninuta-

tion relations for the operators U (u) and U (v) . In the coordinate

representation

V(V) U(U) O(X) = uh),

U(u) V(v) cp(x) = e-`°("-Dh) o(x - uh).

From these equalities it follows at once that

L°°(13)

U(u) V (v) = V (v) U(u)