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

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Preface

book and from von Neumann's classical book, Mathematical Founda-

tions of Quantum Mechanics.

The approach to the construction of quantum mechanics adopted

in these lectures is based on the assertion that quantum and classi-

cal mechanics are different realizations of one and the same abstract

mathematical structure. The feat ures of this structure are explained

in the first few sections, which are devoted to classical mechanics

These sections are an integral part of the course and should not be

skipped over, all the more so because there is hardly any overlap of

the material in them with the material in a course of theoretical me-

chanics. As a logical conclusion of our approach to the construction

of quantum rnecha nics, we have a section devoted to the interconnec-

tion of quantum and classical mechanics and to the passage to the

limit from quantum mechanics to classical mechanics.

In the selection of the material in the sections devoted to appli-

cations of quantum mechanics we have tried to single out questions

connected with the formulation of interesting mathematical problems.

Much attention here is given to problems connected with the theory

of group representations and to mathematical questions in the theory

of scattering. In other respects the selection of material corresponds

to traditional textbooks on general questions in quwitum mechanics,

for example, the books of V. A. Fok or P. A. M. Dirar.

The authors are grateful to V. M. Bahich, who read through the

manuscript and made a number of valuable comments.

I, D. Faddeev and 0 A. Yakubovskii

Preface to the English

Edition

The history and the goals of this book are adequately described in

the original Preface (to the Russian edition) and I shall not repeat

it here. The idea to translate the book into English came from the

numerous requests of my former students, who are now spread over

the world. For a long time I kept postponing the translation because

I hoped to be able to modify the book making it more informative

However, the recent book by Leon Takhtajan, Quantum Mechanics

for Mathematicians (Graduate Studies in Mathematics, Volume 95,

American Mathematical Society, 2008), which contains most of the

material I was planning to add, made such modifications unnecessary

and I decided that the English translation can now be published.

Just when the decision to translate the book was made, my coau-

thor Oleg Yakubovskii died.

He had taught this course for more

than 30 years and was quite devoted to it. lie felt compelled to add

some physical words to my more formal exposition

The Russian

text, published in 1980, was prepared by him and can be viewed as

a combination of my original notes for the course and his experience

of teaching it. It is a great regret that he will not see the English

translation.

xii Preface to the English Edition

Leon '17akhtajan prepared a short appendix about the formalism

of classical mechanics. It should play the role of introduction for stu-

dents who did riot lake an appropriate course. which was obligatory

at St. Petersburg University

I want to acid that the idea of introducing quantum mechanics

as a deformation of classical mechanics has become quite fashionable

nowadays

Of course. whereas the term "deformation'' is not used

explicitly in the book. the idea, of deformation was a guiding principle

in the original plan for the lectures.

L. D.Faddeev

St Petersburg. November 2008

§ 1. The algebra of observables in classical

mechanics

We consider the simplest problem in classical mechanics: the problem

of the motion of a material point (a particle) with mass m in a force

field V (x), where x(x2 . x2, x3) is the radius vector of the particle. The

force acting on the particle is

F= -radV_--aV

The basic physical characteristics of the particle are. its coordi-

nates xr, x2, x3 and the projections of the velocity vector v(v1. V2, v3).

All the remaining characteristics are functions of x and v; for exam-

ple, the momentum p = mv, the angular momentum I = x x p

rnx x v, and the energy E = mv2 /2 + V (x) .

The equations of motion of a material point in the Newton form

are

{ }

mdt

ax '

It will be convenient below to use the momentum p irr place of the

velocity v as a basic variable. In the new variables the equations of

snot ion are written as follows-

follows-

dpOV dxp

( 2)

ax' dt m

Noting that

and Lv =

where H =

V (x) is the

era

C9p

I Iamilt onian function for a particle in a potential field. we arrive at

the equations in the Hamiltonian form

3 dxOH

dpOHH

ap '

It is known from a course in theoretical mechanics that a broad

class of mechanical systems, and conservative systems in particular,

2 L. D. Faddeev and 0. A. Yakubovskii

are described by the Hamiltoniann equations

all OH

(4)

q1=

pi=-- ,

i-1.2,.. n

pz qz

Here H =- 1I(g1, .. , q,1: pl ... . p,,) is the Hamiltonian function, qj

and pi are the generalized coordinates arid momenta. and n is called

the number of degrees of freedom of the system. We recall that, for a

conservative system, the Hamiltonniaxn function H coincides with the

expression for the total energy of the system in the variables qi and pi.

We write the Hamiltonian function for a system of N material points

interacting pairwise.

N 2 N N

(5)

H = 1:

+ 1: Vi.AXi - Xj) +

Vi(xi)

2mz

i=l i<j z-1

Here the Cartesian coordinates of the particles are taken as the gener-

alized coordinates q, the Yiumber of degrees of freedom of the system

is n = 3N, and Vi.? (xi -- x j) is the potential of the interaction of the

U b and jth particles. The dependence of Vi j only on the difference

xi - xj is ensured by Newton's third law. (Indeed, the force acting on

the ith particle due t o the jth particle is Fi j = -- =

_ -F ji . )

OX,

The pot exitials Vi (Xi) describe the interaction of the ith particle with

the external field The first t ernn in (5) is the kinetic energy of the

system of particles

For any mechanical system all physical characteristics are func-

tions of the generalized coordinates and momenta. We introduce the

set % of real infinitely differentiable functions f (q1, ... , qn: pI, .... p,z),

which will be called observables.1 The set % of observahles is obvi-

ously a linear space and forms a real algebra with the usual addition

and multiplication operat ions for functions. The real 2n-dixrnensional

space with elements (q1, ... , q ; pl ..... p,,) is called the phase space

and is denoted by M. Thus, the algebra of observahles in classical

mechanics is the algebra of real-valued smooth functions defined on

the phase space M.

We shall introduce in the algebra of observahles one more opera-

tion. which is connected with the evolution of the mechanical system

1 we do not discuss the question of introducing a topology in the algebra of ob-

serva,bles Fortunately, most physical questions do not depend on this topology

§ 1. The algebra of observables in classical mechanics

For simplicity the exposition to follow is conducted using the example

of a system with one degree of freedom The liamiltoriian equations

in this case have the form

(6)

OH OH

q = Op

p =

c9q

H = H(9,P)-

The Cauchy problem for the system (6) and the initial conditions

(7)

qlt=o

qo,

pl t=o = pc

has a unique solution

(8) q == q(qo, po, t) p == p(qo, po, t ).

For brevity of notat ion a point (q. p) in phase space will sometimes

he denoted by p, and the Iiamiltonian equations will he written in

the form

(9)

ft = v(p),

where v (p) is the vector field of these equations, which assigns to each

point p of phase space the vector v with components 211

Op aq

The Ilamiltonian equations generate a one-parameter commut,a

tive group of t ransforniations

Gt : M ---> M

of the phase. space into itrself,2 where Gt1z is the solution of the l{arnil-

tonian equations with the initial condition GtpIt=o = p. We have the

equalities

(10)

Gt+s = GtG8 = GSGt,

C t.

In turn, the transformations GL generate a family of transformations

Ut : 21 --a 21

of the algebra of observables into itself, where

(11)

Ut f (p) = ft (p) = f (Gtp)

In coordinates, the fuiictiori ft (q, p) is defined as follows:

ft (qO, po) = f (q (qO. po. t), p ((10, po, t))

2 V4'e assume, that the Hamiltonian equations with initial conditions (7) have a

unique solution on the whole real

axis It is easy to construct examples in which

a global solution and, correspondingly, a group of transformations C,, do not exist

These cases arc' not interesting, arid we do not consider them

L. D. Faddeev and 0. A. YakubovskiT

We find a differential equation that the function ft (q. p) satisfies.

To this end. we differentiate the identity f,-.-t(y) = f t (G sp) with

respect to the variable s and set s = 0

a fs-+

t (L) I

aft (11)

dfc(Gsµ)

s=O

= O ft (µ) ' vW =

aft ax

Oft 81I

Op aq

Thus, the function ft (q, p) satisfies the differential equation

aft

aH aft aH aft

(13)

a p

O p

and the initial condition

(14)

ft (q p) It=() = f (q, p).

The equation (13) with the initial condition (14) has a unique solu-

tion, which can be obtained by the formula (12): that is, to construct

the solutions of (13) it suffices to know the solutions of the Haniil-

tonian equations.

We can rewrite (13) in the form

(1`) dt

={H,ft}.

where {H, ft } is the Poisson bracket of the functions 11 and ft. For

arbitrary observables f and g the Poisson bracket is defined by

Of 0.9

of ag

If,

aq a

Op q

and in the case of a system with ra degrees of freedom

of ag

ff,gl = E

i=1

dpc jqi

aqi dpi

We list the basic properties of Poisson bracket s:

1) {f, g -i-

Ah) = {f, g I

-f- A { f j h I (linearity) :

2) {f,g} = - {g,f} (skew symmetry) :

3) {f, {g. h } } + {g, {h.f}} + {h. {f.g}} = 0 (Jacobi identity).

4) {f.gh} = g{ f. h.} + { f, g}h.

§ 1. The algebra of observables in classical mechanics

The. properties 1), 2), and 4) follow directly from the definition

of the Poisson brackets. The property 4) shows that the "Poisson

bracket" operation is a derivation of the algebra of observables. In-

deed, the Poisson bracket can be rewritten in the form

{f,g} =Xfg,

where X f = p -- aq dp is a first-order linear differential operator,

and the property 4) has the form

X fgh = (X1g)h + gX fh.

The property 3) can be verified by differentiation, but it can be proved

by the following argument. Each term of the double Poisson bracket

contains as a factor the second derivative of one of the functions

with respect to one of the variables; that is, the left-hand side of

3) is a linear homogeneous function of the second derivatives. On

the other hand, the second derivatives of h can appear only in the

sump {f, {g, h J} + {g, {h, f } } = = (X1Xq -- X y X f) h, but a commutator

of first-order linear differential operators is a first-order differential

operator, and hence the second derivatives of h do not appear in the

left-hand side of 3). By symmetry, the left-hand side of 3) does not

contain second derivatives at all: that is, it is equal to zero.

The Poisson bracket {f, g} provides the algebra of observables

with the structure of a real Lie algebra.3 Thus, the set of observables

has the following algebraic structure. The set 22 is:

1) a real linear space,

2) a commutative algebra with the operation f g;

3) a Lie algebra with the operation {f, g}.

The last two operations are connected by the relation

If, gh) = If, g1h + gj f, h}.

The algebra 2i of observables contains a distinguished element,

namely, the Hamiltonian function H, whose role is to describe the

3 We recall that a linear space with a binary operation satisfying the conditions

1)-3) is called a Lie algebra

6 L. D. Faddeev and 0. A. Yakubovskii

variation of observables with time:

dft

- H

,.ft}.

We show that the mapping Ut : % 2i preserves all the opera-

tions in 2i:

h = f +g--+ht=ft+9t,

h = fg-*ht=ftgt,

h={f,g} -+ht ={ft,gt};

that is, it is an automorphism of the algebra of observables. For

example, we verify the last assertion. For this it suffices to see that

the equation and the initial condition for ht is a consequence of the

equations and initial conditions for the functions ft and gt:

Oht

oft

agt

at =

, gt

+ 1f t ,

= {{H, ft}, gt} + Ift, {H, gt}} _ {H, Ift, gt}} _ {H, ht}.

Here we used the properties 2) and 4) of Poisson brackets. Further-

more,

htlt=c = {ft,gt}It=o = {f,g}.

Our assertion now follows from the uniqueness of the solution of the

equation (13) with the initial condition (14).

§ 2. States

The concept of a state of a system can be connected directly with the

conditions of an experiment. Every physical experiment reduces to

a measurement of the numerical value of an observable for the sys-

tem under definite conditions that can be called the conditions of the

experiment. It is assumed that these conditions can be reproduced

multiple times, but we do not assume in advance that the measure-

Tnent will give the same value of the observable when the experiment

is repeated. There are two possible answers to the question of how to

explain this uncertainty in the results of an experiment.

1) The number of conditions that are fixed in performing the

experiments is insufficient to uniquely determine the results of the

measurement of the observables.

If the nonuniqueness arises only

§ 2. States

for this reason, then at least in principle these conditions can be

supplemented by new conditions, that is, one can pose the experiment

more "cleanly", and then the results of all the measurements will be

uniquely determined.

2) The properties of the system are such that in repeated exper-

iments the observables can take different values independently of the

number and choice of the conditions of the experiment.

Of course if 2) holds, then insufficiency of the conditions can only

increase the nonuniqueness of the experimental results. We discuss

1) and 2) at length after we learn how to describe states in classical

and quantum mechanics.

We shall consider that the conditions of the experiment determine

the state of the system if conducting many repeated trials under these

conditions leads to probability distributions for all the observables.

In this case we speak of the measurement of an observable f for a

system in the state w. More precisely, a state w on the algebra 2t of

observables assigns to each observable f a probability distribution of

its possible values, that is, a measure on the real line R.

Let f be an observable and E a Borel set on the real line R. Then

the definition of a state w can be written as

f7 E

wt (E)

We recall the properties of a probability measure:

0 <, wf (E) <, 1 1

wf (0) = 0,

wf (R) = 1,

and if E1 f1 E2 = 0, then w f(E1 U E2) = w f(E1) + w f(E2).

Among the observables there may be some that are functionally

dependent, and hence it is necessary to impose a condition on the

probability distributions of such observables.

If an observable cp is

a function of an observable f, Sp = cp(f ), then this assertion means

that a measurement of the numerical value of f yielding a value fo is

at the same time a measurement of the observable ;p and gives for it

the numerical value cpo = p (f o) . Therefore, w f (E) and wp(f) (E) are

connected by the equality

(2) Ww(f ) (E) = wf (W-'(E)) 7