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

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L. D. Faddeev and 0. A. Yakubovskii

Vectors e1, ... , e, form an orthonormall ° basis in C7t if

(2)

(ei, e j) = bi J,

where b;J is the Kronecker delta symbol.

The expansion of an arbitrary vector

in the vectors of the basis

el, ... , e,z has the form

(3) --

i =

rl'he vect or is uniquely determined by the numbers 1it ...

1. .. , ').

If one chooses a basis, then one thereby chooses a concrete realization

for vectors, or one specifies a representation. Let e1, ... , e,, be a basis.

Then the vectors

-° 1] Uikek,

k=1

i=1.2,...,n,

also form a basis if the matrix U = {Ujk} is invertible and

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Uik1

= Uki.

A matrix whose elements satisfy the equality (5) is said to be unitary.

The passage from one basis to another is a unitary transformation.

If a representation has been chosen, then the scalar product is

given by the formula

(6)

ini

i=1

In the given basis, operators are represented by matrices:

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A H {A2k},

Aik = (Aek, ei).

Indeed, if 17 = then

n n

rIi = e) =

(Akekei)

= >

k (Aek, ei) = E AikG-

k=1

1oIn what follows, the word orthonormal will often be omitted, since we do not

consider other bases

§ 5. A finite-dimensional model of quantum mechanics

An operator A* is called the adjoint of A if

(8)

17) =

(A*

, n)

for any pair of vectors

and q. Obviously. A i = Aki. An operator

is said to be self-adjoint if A* = A.

For a self-adjoint operator,

Ails = Aki .

It follows immediately from the definition of the adjoint

of an operator that

(AB)* = B*A*,

(cA)*

= aA*,

where rt is a complex number.

We now construct a realization of the algebra of observables. Let,

21 be the set of self-acljoint operators on C7t. In what follows, self-

adjoint operators will often be called observables. The operations

of addition and multiplication by a number are defined on the set,

of operators in the usual way. If A E 2i, B E %, and A E R, then

(A+B) E % and AA E %, because (A+B)* = A+B and (AA)* = AA.

It is natural to regard t hese operations as operations of addition of

observables and multiplication by a number.

Our next problem is to find out how to construct functions of

observables.

It can be assumed here that the usual definition of a

function of an operator works. We can justify this assumption after

we have learned how to construct probability distributions for ob-

servables in quantum mechanics. Then we shall be able to verify the

formula w f(A) (E) = wA(f -1(E)), which is equivalent to the general

definition of a function of an observable as given in the preceding

section.

We recall that there are several equivalent definitions of a function

of an operator. If f (x) has a power series expansion

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f (X) _ E

Cnxn

n=0

valid on the whole real axis, then f (A) is defined by the formula

f (A)

c,A

n=0

L. D. Faddeev and 0. A. Yakubovskii

A second definition uses the existence for self-adjoint operators of an

cigenvector basis-ii

Ajpi = aiSpi,

i = 1.... , n.

Here the Spi are cigenvectors with (pj, ;p3) = bi j, and the ai are eigen-

values of the operator A. To define a linear operator f (A) it suffices

to define the result of the action of f (A) on the vectors of a basis By

definition,

(12)

f (A) pj = f (ai) oj-

In an eigenvector basis the matrix A is diagonal with the eigenval-

ues on the diagonal, that is, A13 = aibij.

In the same representation,

[1(A)]13 = f (ai) bid . We remark that a self-ad j oint operator corre-

sponds to a real function, that is, f (A) E %.

The operation A o B is defined by the formula (4.0), which for

self-adjoint operators has the form

(A + B)2 - (A - B)2 _ AB + BA

(13)

A o B =

4 2

The self-adjointness of the operator A o B is obvious.

It remains for us to construct a Lie operation. To this end, we

consider the commutator [A, B] = AB - BA of operators A and B.

The operation [A, B] has the following properties:

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1) [A, B) = - [Bj A],

[A+.AB,C] _ [A, C] + A[B, C'],

3) [A,BoC]=[A,B]oC+Bo[A,CJ,

4) [A, [B, C]] + [B, [C, A]] + [C, [A. B]] = 0.

All these properties can be verified directly. We remark that the

property 3) is valid also for the nonsynarietrized product. Indeed,

[A, BC] = ABC - BCA + BAC - BAC = [A, B]C + B [A, C].

We see that the commutator has the properties of a Lie operation, but

[A, B] is not self-adjoint, that is, [A, B] V Z. However, the expression

"We re( all that the eigenvalues of a self-adjoi nt operator are real and that eigen-

vectors corresponding to different eigenvalues are orthogonal

if an eigenvalue has

multiplicity r, then there are r linearly independent eigenvectors corresponding to it,

and they can always be chosen to be orthonornial

§ 5. A finite-dimensional model of quantum mechanics

(i/h) [A, B], which differs from the conirnutator only by the imagi-

nary factor i/h, does satisfy all the requirements. We remark that

algebras % constructed with different constants h are not isomorphic.

The choice of h can be made only after comparing the theory with

experiment. Such a comparison shows that h coincides with Planck's

constant. In what follows we use the notation

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JAI B}h

=i-[A,

and call {A, B } h the quantum Poisson bracket.

It is interesting to note that in classical mechanics we could have

replaced the definition

{f,g}-8fdg

afag

ap dq

aq Op

of the Poisson bracket by the definition

{fig} = o

8f dg

Ofa

ap dq aq ap

where a is a real constant. It is not hard to see, however, that in

classical mechanics this new definition of the Poisson bracket leads

to an algebra that is isomorphic to the original algebra. Indeed, the

change of variables p = a j p', q = sgn a

f a j q' returns us to the old

definition.

An important role in quantum mechanics is played by the trace

Tr A of an operator. By definition,

TrA = >Ajj -- (Ae,e).

i=1

We recall the basic properties of this operation. The trace does not

depend on the choice of basis. In particular, if we take an eigenvector

basis for an operator A, then

IA'A=Eai;

i=x

that is, the trace is the sum of the eigenvalues.

If aZ is a multiple

eigenvalue, then it appears as a term in the sure as many times as its

multiplicity.

L. D. Faddeev and 0. A. YakubovskiT

The trace of the product of two operators does not depend on the

order of the factors:

Tr A B = Tr BA.

In the case of several fact ors, cyclic permutations can be taken under

the sign of Tr.

Tr ABC = '1'r BCA.

The trace of a self-adjoins matrix is a real number, since its eigenval-

ues are real.

§ 6. States in quantum mechanics

In this section we show how states are given in quantum mechanics.

We recall that we kept the definition in § 2 of a state. There we also

showed that specifying the probability distributions is equivalent to

specifying the mean values for all observables. The arguments that

led us to this result remain in force also in quantum mechanics since

t hey did not use a concrete realization of the algebra of observables of

classical mechanics. To specify a state means to specify on the algebra

% of observables a functional (w I A) with the following properties:

rii

1) (1\A+Bjw)=1\(Ajw)+(Bjw),

(421w)>O

(CIw)=C,

(AIw)=(AIw).

We have already discussed the property 1) . The property 2) expresses

the fact that for the observable A2, which has the sense of being non-

negative, the mean value cannot be a negative number. The property

3) asserts that the mean value of the constant observable C in any

state coincides with this constant. Finally, the property 4) says that

the mean values are real. Thus, a state in quantum mechanics is a

positive linear functional on the algebra 2 of self-adjoint operators.

The general form of such a functional is

(2)

(A I w) = Tr MA,

§ 6. States in quantum mechanics

where M is an operator on C' satisfying the conditions

1) M*=M,

(3)

Tr M = 1.

The operator M is called the density matrix, and in quantum me-

chanics it plays the same role as the distribution function p(q, p) does

in classical mechanics.

We show that the properties of the density matrix follow from

the properties formulated above of the averaging functional (A I U)).

Indeed, since the functional (A I w) is real,

n n

AM =

Ails Mki =

A*i Mik

i,k=1 i,k=1

1: Aki

*k = Tr AM* = Tr AM.

i,k=1

Setting X = A, + i A2, where A 1 and A2 are self-adj oint operators,

we get from the last equality that

TXM=TrXM*,

where X is an arbitrary operator on Cn. The property 1) follows at

once from the arbitrariness of the operator X.

We now use the positivity of the functional (A I u)):

TrA2M0.

Let A = P,7, where 1-11 is the operator of projection on a normalized

vector r, (I I q I I = 1) :

Pn =(C'9)

To compute the trace it is convenient to use any basis in which the

first basis vector is,-r/ (e l = Ti, e2,

... , (2,L) , so that

Tr Pn M = Tr P,,M = (Mrf, 77) > 0.

We see that the positivity of the density matrix is a necessary con-

dition for the positivity of the functional (A I w). The sufficiency is

L. D. Faddeev and 0. A. Yakubovskii

verified as follows,.

n n

Tr A2M = Tr AMA =

E(AMAei,

ei) = E(MAei, Aei) > 0,

i=1

since each term under the summation sign is nonnegative

Finally, the normalization condition Tr M = 1 of the density

matrix follows at once from the property 3) of the functional.

Thus, we have showed that states in quantum mechanics are de-

scribed by positive self-adjoint operators with trace 1. (Recall that a

state in classical mechanics is given by a nonnegative function p(q, p)

normalized by the condition f p(q, p) dq dp = 1.)

Any operator M with the properties (3) describes some state of

the system; that is, there is a one-to-one correspondence between the

set of states and the set of density matrices M.

wHM.

If Ml and M2 are operators with the properties (3), then it is obvious

that a convex combination

M=aM1+(1-a)M2,

0<a<1,

of them also has these properties and hence corresponds to some state

w=awl+(1--a)w2.

We see that the set of states in quantum mechanics is convex.

All the requirements of the density matrix are satisfied by a one-

dimensional projection P (11011 = 1). Indeed,

1) (P1 ,rl)_(i1,

P7l)

=(b, )=1.3) TrPO

At the end of the section we show that the state P. cannot be

decomposed into a convex combination of different states; that is, it

is a pure state

(Recall that in classical mechanics a pure state has

a distribution function of the form p(q. p) = b(q - go)S(p - pc).) We

note that a pure state is determined by the specification of a vector

', but there is not a one-to-one correspondence between pure states

and vectors, because if 0' differs from 0 by a numerical factor with

§ 6. States in quantum mechanics

modulus 1, then P,,' = P

Thus. corresponding to a pure state is a

class of vectors with norm 1 that differs from each other by a factor

(-iQwith aER.

The vector i' is usually called a state vector, and the space in

which the self-adjoint operators (observables) act is called the state

space. For the present we take the space C" as the state space.

We show that any st ate in quantum mechanics is a convex com-

bination of pure states. To this end, we observe that M, like any

operator, has an eigenvector basis:

self-elf

Mcpi =

Let us expand an arbitrary vector in this basis and act on it by the

operator M:

- (

i=1

Pi v, Vii) Oi = > pi Poi .

i-I i=1

Since

is arbitrary,

(4)

M =

EIuij SOi

i=1

All the numbers lui are nonnegative since they are eigenvalues of a

positive operator. Moreover.

j=j /,ti = Tr M = 1, and hence the

state M is indeed a convex combination of the pure states P,,i . For

the mean value of an observable in a pure state w

P, , the forrrlula

(2) takes the form

(5)

(AIw) = (A),

I1'1I = 1.

For the case of a mixed state w M

pi Pit , we have

(6)

(A I w) =

1:,ui

i-1

This formula shows that. as in classical mechanics, the assertion that

a system is in a mixed state M = E i

r 1-i PS,, is equivalent to the

assertion that the system is in the pure state P., with probabilityi

fori=1,...,n.

L. D. Faddeev and 0. A. YakubovskiT

In concluding this section, we show that the state described by the

density matrix M = P is pure. We have to show that the equality

(7)

'V =aM1+(1-o)M2, 0<a <1,

implies that M1 = M2 = Pv.

For the proof we use the fact that for a positive operator A and

any vectors ;p and

(8) I

(A)(Ap).

It follows from (8) that AW = 0 if (Acp, gyp) = 0. (The inequality (8) is

a condition for the positivity of a Hermitian form: (A(ap + 60). app +

bib) = (Ap,p)a+ (Acp, V))ab + (Aip.p)ab+ (AiP,ib)bb > 0,where a

and b are complex numbers.)

Let RI be the subspace orthogonal to the vector ib. Then P.0 = 0

if V E f l .

Using (7) and the positivity of the operators Ml and M2,

we have

0 < a (Ml p, P) < a (Ml 0, p) + (1 -- a) (M2 gyp, Sp) =

that is, M1 p = 0. Since M1 is self-adjoint, we get for any vector

that

OERJJ

hence, M1 = Q V), and in particular, Mxib = C,, Here C#) is a

constant depending on the vector An arbitrary vector

can be

represented in the form

ipExz,

hence, Ml = CJP

that is, All = C,,, Pg,. Finally, we get from the

condition Tr M1 = 1 that C. b = 1, and thus M1 = Pp and M2 = P,.

The converse can also he proved If a state is pure, then there

exists a vector 0 E Cn with I I V 1 I = 1 such that M = Per,

§ 7. Heisenberg uncertainty relations

In this section we show how the uncertainty relations mentioned in

4 follow from the mathematical apparatus of quantum mechanics,

and we give them a precise formulation.

§ 7. Heisenberg uncertainty relations 37

Let us consider the variance of two observables A and B in a state

w. We recall that the variance of an observable is defined by

A2 A

= (w I (A - Aavg) 2) _ (w I A2) - (w I

A)2'

where Aavg = (w I A) . Let Ow A be denoted by Ow A .

The uncertainty relations assert that for any state w

A, A B >

w IIt

suffices to prove the uncertainty relations for pure states. For

mixed states they will then follow from the inequality (2.17), which

holds also in quantum Ynechanics, since its derivation does not depend

on the realization of the algebra of observables.

We begin with the. obvious inequality

((A + jaB),O, (A+iaB)iP) > 0,

11,011=17

a E R.

Expanding the left-hand side, we get that

(A2b,i') + a2(B2 b, s) + ia((AB --- BA) ,') > 0.

Using the definition (5.15) of the quantum Poisson bracket and the

formula (6.5) for the mean value of an observable in a pure state

we rewrite. the last inequality in the form

(A21w)+a2(B21w)-ah({A,B}hlw) > 0

for any a E R, and hence

(A21w)(B21w)>h ({A,B}hIw)2.

The uncertainty relations (1) follow immediately from this inequality

if we make the substitutions A A - Aavg, B B - and take

into account that {A, B}h = ((A - Aavg), (B - Bg) }h.

The uncertainty relations show that the variances of two observ-

ables can be simultaneously zero only if the mean value of the Poisson

bracket is zero. For commuting observables, { A, B}h = 0, and the

right-hand side is zero for all states. In this case the uncertainty rela-

tions do not impose any restrictions, and such observables are simul-

twieously measurable. Below we shall see that there really is a theo-

retically possible way of simultaneously measuring such observables

The strongest restrictions are imposed on the pairs of observables