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 eigenfunctions din (x) for the excited states are found by the

formula (16.12) using (2):

' (2w) - 1 d WX

(W) 2

On (X) = -

UjX -

Vn-1

Obviously, these functions have the form

,On (x) = Pn (x) e-"

-1

where Pn (x) is an nth-degree polynomial.

It can be shown that

Pn(x) =

w--x), where

is the nth Chebyshev-Hermite poly-

nomial. It is known that the system of functions H e'

-t2

is Com-

plete in L2(R). This assertion follows in general from the irreducibil-

ity of the coordinate representation and results in the preceding sec-

tion.

The function

is the density of the coordinate distribution

function in the nth state of the oscillator. It is interesting to compare

this distribution with the corresponding classical distribution. The

solution of the classical problem of an oscillator has the form

(3)

x(t) = A sin(wt + W)

where A is the amplitude of oscillation and Sp is the initial phase. The

energy of the oscillations can be computed from the formula

_ x2 w2 x

+ 2

and is equal to w2A2/2. The corresponding density of the coordinate

distribution function has the form

(4)

F(x) = 6(x - x(t)).

It is clear that a stationary state of a quantum oscillator can under

no conditions pass into a classical pure state given by the formula

(3). It is natural to assume that the limit of a quantum state will be

a classical mixed state that is a convex combination of solutions (3)

with random phases cp. For such a state the density of the coordinate

§ 17. The oscillator in the coordinate representation

Figure 5

distribution function is obtained by averaging (4):

27r

F(x) =

2-7r

S(Asin(wt + gyp) - x) dcp

(-A, A),A}o,

or, briefly,

F(x) =

O(A 2 - X2)

r A2 - x2

The graphs of the functions F(x) and k1'(x)I2 are shown in

Figure 5 for sufficiently large n under the condition that24 En

h(n + 112)w = w2A2/2. As n -+ oo, the quantum distribution will

tend to the classical distribution. The condition n --+ oo can be sat-

isfied for a given energy E if h --+ 0.

We point out an important difference between the functions F(x)

and

F(x) = 0 for IxI > A; that is, the classical particle is

24Here for convenience we write out Planck's constant h explicitly, not assuming

that h = 1

L. D. Faddeev and 0. A. YakubovskiY

always between -A and A (in the classically allowed region), while

the function 17kn(x)12 does not vanish for Jxj > A. This means that

the quantum particle can be observed with finite probability in the

classically forbidden region. For an arbitrary potential V (x) this re-

gion is determined by the condition that the total energy E is < V (x),

which corresponds to negative values of the classical kinetic energy.

§ 18. Representation of the states of a

one-dimensional particle in the sequence

space 12

According to the results in § 16, any vector W E W can be expanded

in a series of eigenvectors of the energy operator of the harmonic

oscillator:

(a*)

001

'Cl2

Cnn n

< 00.

Each vector p E 7I is uniquely determined by the coefficient se-

quence {C}, that is,

O H

If ik +-+ {B}, then

(a) = 1: BnCn

in view of the orthonormality of the system of vectors

(a*)n

On =

00-

Let us consider how the operators act in such a representation.

For this it suffices to construct the operators a and a*. Suppose that

{C} and acp {C}. We must express Cm in terms of C, .

§ 18. Representation for a one-dimensional particle 91

This can be done as follows:

(1)

acp = al: C

1Po

nlasln-1

76!

* n--1

E n G-rn

lu' 1

V(" n-- - ---1) i I P 0

vrn- -+1 Cn+1

(a*)n

00 -

The calculation used the relations [a, (a*)f] = n (a *) n and at'c = = 0,

and the last equality is obtained by the change n --+ n + 1 of the

summation index. It is clear from (1) that

(2)

Similarly,

Therefore, if a`V -> {C;;}, then

(3)

Cn = Vn-Cn-l-

The formulas (2) and (3) become especially intuitive in matrix

notation. Let us write the sequence {C} as a column:

cp H C2

Cn= Vn-+1 Cn.

E (a

!00

- Cn

/a*)ntl

* n+l

1: Vn-+l Cn

(a*)n

'LCn-1

Tn !

L. D. Faddeev and 0. A. YakubovskiI

Then the operators a and a* can be written as the matrices

(4)

vfi- 0

0 0

................

. .

. . .

. . . . .

. .

. . .

:::

...

. . . .

. .

f 0

0 ..

0 ,./2 0 ..

................

We verify that these expressions are equivalent to the relations (2)

and (3). Indeed,

0 f 0 0

...

f/Tci

0 0 f2-

0 Ci

f Ca

a`p 0

... C2

vf3-Cs

which coincides with (2); the connection between the formulas (3) and

(4) can be verified similarly. The commutation relation [a, a*] = 1 can

be verified immediately for a and a* in the representation (4). The

eigenvectors of the operator H in this representation have the form

0 0

0- of -

,3= 1

....

The vector b0 obviously satisfies the equation abo = 0.

The operators a* and a are often called the operators of creation

and annihilation of excitation, due to the fact that a* carries a state

with energy E to a state with greater energy E + w, while a carries a

state with energy E to a state with energy E -- w (the ground state

o is annihilated by a). Sometimes the so-called excitation number

operator N = a*a is introduced. In our representation it has the form

N+-+

0 0 0

...

0 0

...

0 0

...

0 0 0

...

The eigenvalues of this operator coincide with the ordinal number of

the excited state, on.

§ 18. Representation for a one-dimensional particle

Finally, we write the operators H, P, and Q in this representa-

tion:

0 0

...

(5)

H=wa*a+wH

O 2

...

2 0

0 Zw

...

............. ...

(6) P =

VW--(a - a*)

C) -

0 V-1 0 0

...

\/ _1 o

...

0 \52 0 \F3

...

v/3-- 0

.......................

It is clear from these formulas that II is represented by a diagonal

matrix; that is, this representation is an eigenrepresentation for the

Schr"finger operator of the harmonic. oscillator. Further, it is at once

obvious that P and Q are self-adjoint operators, and it can easily be

verified that they satisfy the commutation relation [Q, P] = = i.

In connection with the representation of states in the space 12,

we would like to say something about the original Heisenberg matrix

formulation of quantum mechanics. Finding the admissible values of

the energy of a system was a basic problem in the initial stage of the

development of quantum mechanics. For a system with one degree of

freedom, Heisenberg proposed the following recipe. lie considered a

classical system with Hamiltonian function H (q, p) = pl /2m + V (q) .

Self-adjoint matrices Q and P satisfying the relation [Q, P] = i were

constructed (such matrices are not uniquely determined) . The matrix

H = 1i2/2ni + V (Q) was constructed next. The last step consisted in

diagonalization of this matrix, and the eigenvalues of II were identi-

fied with the admissible values of the energy.

The formulas (6) and (7) give an example of matrices P and

Q satisfying the Heisenberg commutation relations. These matrices

fO VfY

n o

...

\A_O_

-vfl- 0 %/2- 0

V2_

0 \/2'- 0 v/3--

0 \/3- 0

a+a*

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

were chosen so that the matrix of the operator H for the oscillator is

at once diagonal. However, for an arbitrary H it is riot possible to

avoid this diagonalization step.

We see that the Heisenberg formulation essentially coincides with

the contemporary formulation of quantum mechanics if 12 is taken as

the state space.

19. Representation of the states for a

one-dimensional particle in the space V of

entire analytic functions

We consider the set of functions of a complex variable of the form

(1)

f (Z) = 1: Cn

This set V of functions becomes a Hilbert space if the scalar product

is defined by

(2)

(11,12)

ffl(z)ieI2d,L(z).

The integral is taken over the complex plane, and dp (z) = dx d y.

Let, us verify that the functions zn / n! form an orthonorrnal

basis in D. For this we compute the integral

21r

'nm =

T'n-lz1

d(z) =

fpdpfdpn+me(n_m)e_P2.

For n m we have I,nn = 0 due to the integration with respect to Sp.

For n = m,

Inn = 27r

p2n+leP2

dp =7r

the-t dt = 7rn!.

An arbitrary state p E fl with = En Cn

n, Oo can be rep-

resented by the function f (z) = En Cn LL :

cp f (z)

. The eigen

vectors On of the oscillator are represented by the basis functions

zn /

rt!.

We consider how the operators a and a* act in this representation.

Using the computations which led us to the formulas (18 2) and (18.3),

§ 20. The general case of one-dimensional motion

we can write the vectors aco and a* P in the form

n(a*)n-1

- E

* p

(a*)n+l

a;p Cn

a = 1: Cn

These vectors are represented by the functions

nzn-1

acp >Cn

f (z),

Vn-!

zn+ 1

E Cn

= Z f (z);

that is, for a and a* we have obtained the representation

a* +z.

Here are the corresponding formulas for the operators Q, P, H:

1 d

2wdz

P+-+

i v/2- \ dx - z/

HHw zd1

All the basic relations ([a, a*] = 1, [Q, P] = i, H1,z

w (n + 1/2) in) can easily be verified in this representation. The repre-

sentation constructed can turn out to be convenient if the observables

under study are polynomials in Q and P.

§ 20. The general case of one-dimensional

motion

In the preceding sections we considered two one-dimensional problems

of quantum mechanics: the problem of a free particle and the problem

of a harmonic oscillator. The free particle gives us an example of

a system with a continuous spectrum for the Schrodinger operator,

and the harmonic oscillator gives an example of a system with a

pure point spectrum. In most real physical problems the spectrum is

L. D. Faddeev and 0. A. YakubovskiI

more complicated. Let us consider the problem of the spectrum of a

Schrodinger operator

ll=-d-

+V(x)

with very general assumptions about the potential.

The forces acting on the particle usually are appreciably different

from zero in some finite region on the x-axis and tend to zero as

IxI --+ oo; therefore, it is most common to encounter potentials V(x)

that tend to constant values as x --+ ±oo. For simplicity we confine

ourselves to the case when the potential is exactly equal to constants

for x < -a and x > a. By the arbitrariness in the definition of a

potential, we can always take one of these constants equal to zero.

We consider Schr6dinger equations

(1)

0it+FO-= V(x)

under the condition that V (x) is a continuous function on the real

axis with V (x) = 0 for x < -a and V (x) = Vo for x > a.

For

definiteness we assume that Vo > 0. The graph of the potential is

pictured in Figure 6.

Figure 6

§ 20. The general case of one-dimensional motion

(2)

(3)

For x < -a and x > a the Schr"finger equation (1) simplifies:

b"+EV) =0,

x<-a,

+(E--Vo)ip =0,

x>a.

For any values of E there are two linearly independent solutions of

(1), which we denote by 01 and 02. The general solution of this

equation is

(4)

V) = C101 + C202-

In our study of the spectrum of the operator H we are interested in

either square-integrable solutions of (1) that are eigenfunctions of H

or solutions that are bounded on the whole real axis. With the help

of the latter we can describe the continuous spectrum of 11.

We now consider three cases.

1)E<0.

We rewrite the equations (2) and (3) in the form

0" -- x2ip = 0,

x < -a,

x2 = --E > 0,

x > 0;

0"-xiV)=0, x>a.

x? =--(E-Vo)>O, xl>0.

The functions

for x < -a and ell, x for x > a are linearly

independent solutions of these equations.

Therefore, an arbitrary

solution (4) of the equation (1) in the region x < -a has the form

C' e - x + C2 e"

, and in the region x > a the form C1' e - xlX + C2e xl X .

Here Ci, C2, Ci', and C2 are constants that depend linearly on Cl

and C, 2 in the formula (4). The solution i will be square integrable

under the conditions C' = 0 and C2 = = 0. Just one of these conditions

is enough for 0 to be determined up to a numerical factor. From the

condition Cl = 0, one can find the ratio of the coefficients CI and C2,

which will depend on the parameter E:

= F, (E).

Similarly, from the condition C2 = 0, we get that

ZT- = F2(E).2