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

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

if a and b are elementary particles, and a many-body problem if a

and b are compound particles. We recall that the two-body problem

can be reduced to the problem of the niotion of a particle in the field

of a fixed force center by separating out the motion of the center of

inertia. This is the simplest problem of scattering theory.

In scattering by a force center, a particle can change only its di-

rection of motion due to the law of conservation of energy. In ibis case

one speaks of elastic scattering. More complex processes are possible

when compound particles collide. For example. when an electron col-

lides with a hydrogen atom, elastic scattering is possible (the state

of the atone does not change), scattering with excitation is possible

(the electron t rannsfers part, of its energy to the atorri, which then

passes into a new state), and finally, ionization of the atom by the

electron is possible Each such process is called a scattering channel.

Scatt Bring of a particle by a force center is single-channel scattering,

while scat tering of compound particles is usually nrnult ichannel scat-

tering. If, however, the colliding particles a and b are in the ground

state and the energy of their relative motion is less than the energy

of excitation, then the scattering is single channel.

The basic characteristic of the diverse scattering processes mea-

sured in experiments is their cross-section, which we define below.

Some set of possible results of scattering is called a scattering pro-

cess. The following are processes in this sense:

1) Elastic scattering into a solid angle element do constructed

about the direction n;

2) Elastic scattering at an arbitrary angle;

3) Scattering into a solid angle do with excit ation of the target

from the ith level to the kth level:

4) Scattering with ionization of the target;

5) A process consisting in scattering in general taking place. and

so on.

The probability N of some scattering process of a by b depends

on a certain quantity characterizing the accuracy of using particles

a to "shoot" a particle b.

To introduce such a characterization of

the state of an impinging particle a, we construct the plane passing

§ 36. Scattering of a one-dimensional particle

159

through the point where the scatterer b is located and perpendicular

to the momentum of a. The probability dW of a crossing an area

element dS of this plane is proportional to dS: that is, dW = I dS.

It is clear that the probability N will be proportional to the quantity

I computed at the point where the scatterer is located.``'

It now seems natural to define the cross-section a by

For the five processes listed above, the cross-sections have the

following names: 1) the differential cross-section of elastic scattering;

2) the total cross-section of elastic scattering. 3) the differential cross-

section of excitation; 4) the ionization cross-section: 5) the total cross-

section The concept of cross-section becomes especially intuitive

if we assume that there is complete deterniinisrri of the scat tering

results. In the case of such determinism. the scattering result would

be determined by the point of the transverse cross-section of the beam

through which the particle would pass in the absence of the scatterer.

To the scattering process would then correspond sonic region in the

plane of the transverse cross-section, and the cross-section would be

equal to the area of this region.

There are two problems in scattering theory: from known inter-

action potentials between the particles determine the cross-sections of

the various processes (the direct problem), and from the known cross-

section obtain information about the interaction of the particles (the

inverse problem).

In these lectures we confine ourselves to a study of the direct

problerri of scattering of a particle by a potential center, beginning

with the simplest one-dimensional case.

§ 36. Scattering of a one-dimensional particle by

a potential barrier

We structure the exposition of this problem according to the following

plan First we formulate the so-called stationary scattering problem.

35Here the scatterer b is assumed to be distant, since I characterizes the state of

a freely moving particle a

160 L. D. Faddeev and 0. A. Yakubovskj

For this we study the solutions of an equation Hb = Eik of a certain

special form. The physical meaning of such solutions will be explained

later, after we have used them to construct solutions of the nonsta-

tionary (or time-dependent) Schrodinger equation i dt = H'41

For

simplicity we employ a system of units in which h = 1 and m = 1/2.

In the coordinate representation, the Schrodinger operator has

the form

(1)

d 2

+ V W.

dX2

We take the function V (x) to be compactly supported (V(x) = 0 for

1x1 > a) and piecewise continuous.

Let us denote the regions x < -a, x > a, and -a < x < a of the

real axis by 1, 11, and 111, respectively. The scattering problem is a

problem of infinite motion of a particle. Such snot ion is possible for

E > 0, and we know that for E > 0 the spectrum of the Schrodinger

operator is continuous. From a mathematical point, of view, scat-

tering problems are problems about the continuous spectrum of the

Schrodinger operator.

The stationary Schrodinger equation with the form

(2)

--W"+V(,r)yP=k2v,

E=k2. k>0,

on the whole real axis simplifies in the regions I and II:

(3)

W' + k2b = 0.

The equation (3) has the two linearly independent solutions eikz

and a-ikx. The solutions of (2) on the whole axis can be constructed

by sewing together the solutions in the regions I III. In doing this,

we must use the conditions of continuity of the solutions and their

first derivatives at the points

-a and a. This imposes four conditions

on the six arbitrary constants appearing in the expressions for the

general solutions in the regions I III. The fifth condition for these

constants is the normalization condition Therefore, we can construct

§ 36. Scattering of a one-dimensional particle

161

linearly independent solutions of (2) with the form

(4)

V), (k, x),

eikx + A(k) e-ikx.

B(k) eikx

iP2 (k. x),

D(k) e-ikx,

a-ik2'

C(k) eika'.

in regions I and If.

For example, in constructing lpi we use the

ai bit rariness of one of the constants to get that the coefficient of

(I-

ikx is equal

to zero in the region 11. We then choose the coefficient

of eikx to be equal to 1 in the region I, thereby determining the

normalization of the function 'l

The coefficients A and B are found

from the conditions for sewing together, along with the constants m

and n. where mcpl + n'P2 is the general solution of (2) in the region

III.

It is not hard to see that the conditions for sewing together

lead to a nonhoiriogeneous linear system of equations for A, B, m,

and n with determinant that is rionzero if c1 and cp2 are linearly

independent. The solution *2 is constructed similarly. The linear

independence of 'I and /2 follows from the fact that the Wronskian

of these solutions is not zero.

Let us determine the properties of the coefficients A, B, C, , and

D. For this we note that the Wronskian W (W1 ,'p2) = ;pi p2 -'p1 Cpl of

the solutions Pi and 'p2 of (2) does not depend on x. Indeed, suppose

t hat V, and V2 satisfy (2). Then

dpi -(V-12) i -=0, Sp2-(V-k2)Sp2=0.

Multiplying the first equation by 'P2 and the second by 'p1 and sub-

tracting one from the other. we get that

-'' =0.

,F cp2}

Using this property, we can equate the Wronskian for any pair

of solutions in the regions I and H. Choosing (i1,'2), (fir,

1).

(ii'2. i'2), and ('i,1P2) as such pairs successively, and using the equal-

ities W

(e-ikx, eikx) = 2ik and Me

±ikx,efikx)

= 0, we arrive at the

361f the potential is not compactly supported but decreases sufficiently rapidly as

x1 - 0o, then these expressions for 01 and 02 in I and II should he regarded as the

asymptotics of 'i and W2 as x - tx We shall not study this case

162

L. D. Faddeev and 0. A. Yakubovsk j

following relations between the coefficients A, B. C, and D:

(5) 13=D,

(6)

JA12

1B12

= 1,

(7) 1B2 + ICI2 = 1,

(8)

At --BC-=0.

(}nor example, W (/11,1) = -2ik(1 - AA) in I and W(1 , 01)

-2ikBB in H. Equating these expressions, we get (6).)

The relations (5) (8) show that the matrix S consisting of the

coefficients A, B = D, and C, that is,

A B

B C

is symmetric and unitary. This matrix is called the scattering matrix

or simply the S-matrix. We shall see below that all the physically

interesting results can be obtained if the S-matrix is known, and

therefore the computation of its elements is the main problem in one-

dimensional scattering theory.

Let us consider the question of normalization of the functions

01 (k. x) and '02(k, x). We have the formulas

(9)

V), (ki - x)V)2( k2, X)dx = 01

ki > 01

k2 > 01

(10) i1,2 (ki , x) b1.2 (k2,

x) dx

= 27r6(ki - k2),

that is, the same relations hold as for the functions ei kr and e -

ikX

and the normalization does not depend on the formal of the potential

V (x)

The integrals in the formulas (9) and (10) are understood in

the principal value sense. Let us verify (10) for the function,01. The

remaining two relations are verified in the same way. Substitution of

i11 (k1, x) and -01 (k2, x) in (2) leads to the equalities

(11) p'(k1, x) + k20(k1, x) = V(x) '(k1, x),

(12) 011(k2, x) +k22 '(k2, x) =V (x) ib(k2, x).

(For brevity we do not write the index 1 in the notation of the so-

lution ?)1 .)

Multiplying (11) by '(k2, x) and (12) by O(kl, x) and

§ 36. Scattering of a one-dimensional particle 163

subtracting the first from the second, we get that

W (?p (k Zx), V) (k2, X))= (k1 2- k2 ? )

(/ki> ,r) V(

2 X)-

C x

this equality, we have

integrating

v 1

(13) f

Vi(ki, x) (ka, x)

= 2 2

x), G(ka

k1 - k2

-N

We are interested in the limit as N -+ oo of the integral on the

left-hand side of (13) in the sense of generalized functions. It is clear

already from (13) that this limit depends only on the form of the

solutions in the regions I and 11 (for the case of infinite potentials it

depends only on the asymptotics of the solutions as x -* ±oo).

After simple computations we get that

V (k1, x)

(k2,x) dx

-N

+ 13 k1

e`(k, -k2)N

e-z(k' - k2)1V

k _

(ki) (k2)

(

) 13 (

))

[A(k2) e-i(ki

+k2)N +A(k1) et(k1

+k2)Nj

k1 + k2

By the Rienlaun-Lebesgue theorem, the second terns tends to zero (in

the sense of generalized functions). The similar kissertion is not true

for the first term, since it is singular for k1 = k2. The singular part

of this term does not change if we replace A(k2) and B(k2) by A(k1)

and B(kl ), respectively.37 Using (6), we have

*1(kt, x) i11(k2, x) dx =

siri(k1 - k2) N + F(k1, k2. N),

-N

1 -k2

where

lim F(ki , k2, N) = 0.

Finally, using the well-known formula

lim

sin Nx

= rrb(x)

N--oc

'?For this it suffices that A (k) and B(k) be differentiable functions of k, which

can he proved

164 L. D. Faddeev and 0. A. YakubovskiT

we get that

lim

(ki, x) 01(k2. x) dx = 27rb(k1 - k2),

N-'3c -N

which coincides with (10).

§ 37. Physical meaning of the solutions V), and IP2

To clarify the physical meaning of the solutions i, and 02, we use

them to construct solutions of the nonstat ionary Schrodingcr equation

For this equation we consider the solution

(xj t)

21f

C(k) V), (k. x)

e-£k2l A

constructed from the function im (k. x). The function C(k) is assumed

to be rionzero only in a small neighborhood of the point ko. In this

case. of (x, t) has the simplest physical meaning. Moreover, we assume

that,

and then it, follows from (36.10) that,

f_:l(x,t12dx=1,

that is, the solution Pi (x, t) has the correct normalization. Using the

concentration of the function C(k) in a neighborhood of ko and the

continuity of time functions A(k) and I3(k), we can write approximate

expressions3s for the function 5P1 (x, t) in the regions I and 11:

T :

cp1(x, t)

p (x, t) + A(ko) SP- (x, t)

II.,1(x, t) ,.' B(ko)+(x, t),

where

(2)

0--- (X, t)

C k

eikzt

J°°

38These expressions can be regarded as having any desired degree of accuracy if

the interval Lk inside which C, (k) is nonzero is sufficiently small However, we cannot

pass to the limit, replacing C(k) by a 6-function, because we do not get a square-

integrahle solution of the Schrodinger equation

§ 37. Physical meaning of the solutions W1 and '02

165

The functions (pf (x, t) are normalized solutions of the Schrodinger

equation for a free particle, and were studied39 in § 15. In the same

section we constructed asymptotic expressions for these solutions as

t - ±x:

t =

C;' ± e'x + 0

(') 2t

/2 -It I

where x is a real function whose form is not important for us. From

this expression it is clear that for Itl --+ oc the functions cp j (x, t) are

nonzero only in a neighborhood of the points x = ±2k0t. Therefore,

p4 describes the state of a free particle moving from left to right with

velocity40 v = 2k0}, and :p_ describes a particle having the oppositely

directed velocity (recall that m = 1/2)

It is now easy to see what properties the solution Ppn (x, t) has as

±oc. Suppose that, t -oc. Then in the regions I and 11 we

have

y,1 (x, t) = cps (x, t),

II: P1 (x. t) = Q,

since as t -*-ocwehavecp_(x,!)=0forx<-a andpp-(x.t)=0

for x > a. Similarly, as t -- +oc

1: p (x, t) = A(kc) _ (x t),

If: ,(x,t) = B(ko)

+(x,t).

We see that long before the scattering (t --+ -oc) the probability

is 1 that the particle is to the left of the barrier and is moving toward

the barrier with velocity 2k0.

Let us compute the probabilities W1 and W11 of observing the

particle as t -, +oc in the regions I and II, respectively. We have

Wi =

= II

= I1

= iA(ko)i2.

39There is

an inessential difference in the notation Here it is more convenient

for us to take k > 0 and to write out the sign of the momentum in the exponential

etikx explicitly

Moreover, the integration in (2) can be extended to the whole real

axis, since C(k) 0 for k < 0

40More precisely, the region in which there is

a nonzero of finding the particle

moves with velocity 2k0

166

L. D. Faddeev and 0. A. Yakubovskii

t -"X

t - +3c

v= -2k0

(pl (x.t) I2

-a a x

I(.r.t)I2

v = 2k0

-a

Figure 12

It is possible to replace the region I of integration by the whole real

axis, because for t -* +o0 we have 9_ (x, t)

0 only in I. In exactly

the same way we can verify that

W11 = IThe

graphs of the function

(x, t) (2 as a function of x for t ±oo

are shown in Figure 12.

Thus, the solution yp, (x, t) of the Schrodinger equation describes a

particle that approaches the potential barrier with velocity 2ko before

the scattering and then with probability I

it is reflected from

the barrier or with probability 1112 it passes through the potential

barrier. 41

We point out that the result does not depend on the form of the

function C(k).

it is important only that the interval ok in which

41 to the one-dimensional problem the concept of cross-section loses its meaning

All information about the scattering is contained in the probabilities IA12 and Bill

§ 38. Scattering by a rectangular barrier

1 67

C(k) is nonzero be small. Physically, this requirement is understand-

able if we want to experimentally determine the dependence of, say,

the reflection coefficient J

on k: we must use particles in a state

with variance of k as small as possible (states with zero variance do

not exist). In concrete computations of the reflection and transmis-

sion coefficients IA(k)12 and IB(k)12, it is not necessary to solve the

nonstationary Schrodinger equation; it suffices to find the solution

'1(k, x) .

We remark that the solution ;02 (x, 1), which can be con-

structed from the function 2(k, x), has the same meaning, except

that the particle is approaching the barrier from the right.

Let us recall the properties of the scattering matrix S. The equal-

ity B = D leads to equality of the probabilities of passing through the

harrier in opposite directions and, as can be shown, is a consequence

of the invariance of the Schrodinger equation with respect to time

reversal. The equalities

IAl2 + 1B12 = 1,

IB12 +

IC12

= 1

express the law of conservation of probability Indeed, the normal-

ization of the solutions o 1(x, t) and c02 (x, t) does not depend on the

tune, and as t ----> oo we have

lira

l'j(x,t)l2dx = I+ I= 1.

coo

§ 38. Scattering by a rectangular barrier

We consider the concrete problem of scattering of a one-dimensional

particle by a rectangular potential barrier. Let

Vol

lxla,Vo>O,

V(x) =

lxl>a.

In this case the equation Hb = k2?k has the following form in the

regions I -III:

(1)

I and II:

1V'

+ k2 =

III:

" + a2 = 0,

where a =

k -- Vo (for definiteness we assume that a > 0 for

k2-Vo>Oand--is>Ofork2-Vo <0).