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

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178

L. D. Faddeev and 0. A. Yakubovskj

(g)

I/'(x,t)I2 -

81t13

1IC(2tr,n)

ImCl ()c(j-,n)f(ko,n,wo)

+ 41r2

I Cl

\2tl

If(ko,n,)o)I2],

t --> +oo.

It follows from (7) and (8) that the asymptotic expressions obtained

for

(x, t) have the correct normalization as t --+ ±oo. (This follows

trivially for the case t - -oo and is a consequence of (39.12) for the

case t -' +oo.)

Recalling that C(k, w) is nonzero only in a small neighborhood

of the point k0w0 and Cl (k) is nonzero only in a small neighborhood

of k0, we see that as t -* -oo the density IV) (x, t)12 of the coordinate

distribution function is nonzero in a neighborhood of the point r =

- 2 k0 t, n = -WO. As t --+ +oo the density

1,0 (x, t) 12 is

nonzero interior

to a thin spherical shell of radius r = 2k0 t. The angular probability

distribution can be obtained43 by integrating (8) with respect to the

variable r with the weight r2 .

It is clear that the first two terms in

(8) contribute to this distribution only in directions close to WO. The

angular distribution with respect to all the remaining directions is

proportional to If (ko, n,

wo)12.

We can now easily see how the motion takes place for a particle in

the state described by the function(x, t). Long before the scattering

(t --+ -oo) the particle approaches the scattering center with velocity

2k0, moving in the direction w0. After the scattering (t - +00) it

moves away from the scattering center with the same velocity, and

it can be observed at any point of the spherical shell of radius r

2kot with angular probability distribution depending on C(k) and

f (k0, n, wo). In Figure 14 we have shaded the regions in which the

probability of observing the particle is large as t --p ±oo. The regions

43 We do not write out the exact formulas for the angular probability distribution,

since it is essentially dependent on the form of the function C(k) and is therefore not

a convenient characteristic of the scattering process (the function C(k) corresponding

to a concrete scattering experiment is never known) The cross-section is a suitable

characteristic As we shall see, the cross-section turns out to be insensitive to the form

of C(k) it is only important that this function be concentrated in a small neighborhood

of ko Physically, this requirement means that the momentum of the impinging particle

must be almost specified

§ 40. Motion of wave packets in a central force field

179

t -+ --oo

r = -- 2k0C

v = 2kow0

Figure 14

w{)

in which this probability is nonzero even in the absence of a scattering

center is crosshatched.

The three terms in the formula (8) admit the following interpre-

tation. The integral Wl over the whole space of the sum of the first

two terms is the probability that the particle goes past the force cen-

ter without scattering. This probability is less than 1 because of the

second term. The integral W2 of the third term is the probability of

scattering. We have already noted that the asymptotic expression (6)

for 4 (x, t) has the correct normalization, and hence

W1+W2=1.

We see that the solution i&(x, t) of the Schrodinger equation con-

structed with the help of the function O(x, k) correctly describes the

physical picture of the scattering. This justifies the choice of the

asymptotic condition for ip(x, k) .

We mention some more features of the solution iP(x, t). It is not

hard to see that as t -p -oo this function has the same asymptotics

as the solution

(x, t)

C(k)

ei(kx-k2t)

I /

scattering

center

/ /

/ I

\\\

\v=2k()

t -4 +00

180

L. D. Faddeev and 0. A. Yakubovskii

of the Schrodinger equation for a free particle. (Here C(k) is the same

function as in the integral (1).) Indeed, the diverging waves eikr /r

do not contribute to the asymptotics as t --+ -oo. and the coefficients

of e -ikr/r in the asymptotic expression for the functions ekx and

7'(x, k) coincide.

We show that, also for t -* +oc the solution V )(x, t) tends asymp-

tot ically to some solution of the Schrodinger equation for a free par-

ticle. Using the fact that for t

+oc the converging waves

a-ikl/r

do not contribute to the asymptotics, we get that

V/ (Xj t=

(1)2

C k x k

e-ik2t

dk d W C w)

27ri

S(k, n, w)

eikr

_ik2t

Joy k2

27ri

S k.

2kT

w', W) C(k,w) S(w' - n)

e_ik2t

_ - I

2 00

k2 dk I

2'ri

C(k, w') S(w'

- n)

\2r/

e-ik2t

O(k) ei(kx-k2t) dk = AX, t).

2ir

Here

(9) C(k) = O(k,

') =

S(k,w,w') C(k,w') dw'.

We see that the solution i'(x, t) of the Schrodinger equation tends

asymptotically to the solution yp(x. t) for a free particle as t - oo. The

function C(k), which determines the final state of the free motion, is

obtained from the function C(k) giving the initial state as a result of

the action of the operator S. The unitarity of S ensures the correct

normalization of

(x, t), since

fR3

IO(k)I2dk= 1

in view of the unitarily.

41. The integral equation of scattering theory

181

41. The integral equation of scattering theory

Most approaches to the construction of the solutions L(x, k) and the

scattering amplitude f (k. n, w) are based on the integral equation

ikjx-yj

(1) b(x, k) = euI c

47r

V(y) V) (y, k) dy,

R 3

Ix-yl

which is often called the Lippmann-Schwinger equation.

We verify that a solution of this equation satisfies the equation

(39.2) and the asymptotic condition (39.7).41 Indeed, using the for-

mulas

(O + kz) e:t« = 01

(Q + k 2 )

eikr/r

= -4a8(x)-

we get that

a(X - y) V(y) V)(y, k) dy = V(x) Vi(x, k).

(o + k 2) O(X, k) = o + f

We verify the asymptotic condition for the case of a compactly

supported potential (V(x) = 0 for r > a). We have

1 1

, r = lxl,

13'! < a,

lx - yl

(r2

s /

Ix -yI=r

1-2xY+Y

=r-ny+Ol

1r)

r2 r2

eikIX-YI =

i(kr-kny)

r7'1 ,

Therefore, for the function O(x, k) we get that

,O(X, k)

= etc -

r f

0(y, k) dY + 0 (-k)

Comparing the last formula with (39.7), we see that the solution of

the integral equation has the correct asymptotics, and moreover, we

get that

V) (x, kw) dx.

(2)

.f (k, n, w) = -4

440f course, it is also possible to verify the converse assertion

182

L. D. Faddeev and 0. A. Yakubovskii

The formula (2), in which the scattering amplitude is expressed in

terms of the solution of (1), often turns out to be useful for the

approximate determination of f (k, n, w).

One of the approximation methods of scattering theory is based

on the use of a series of iterations of the equation (42.1).

(3)

(x, x) = E V)

(n) (x,

k),

n=0

where

(°) (x, k) = eikx,

eik x-yl

(x, k) = _

V(Y) V)

(n)

(yj k) dy.

47r

frR,,J 1X-Y1

The series (3) is called the Born series, and substitution of (3) in

(2) gives the Born series for the scattering amplitude. The Born

series for the problem of scattering by a potential center has been

thoroughly studied. It is known, for example, that it converges under

the condition that

dy < 47r.

InXXR4 IV(Y)

Ix-yl

It is also known that under this condition on the potential the

operator H does not have a discrete spectrum. If a discrete spectrum

is present, then there are values of k for which the series (3) does not

converge. At the same time, for sufficiently large k it does converge

for a very broad class of potentials.

The simplest approximation for the scattering amplitude is ob-

tained if in place of O(x, k) we substitute 0(°) (x, k) = eikx:

k n, w = - 1

e-iknxv

eikwx

dx.

This approximation is called the Born approximation. The precise

assertion is that for large k

f(k,n,w) - f (k,n,w) = o(1)

uniformly with respect to n and w.

§ 42. Derivation of a formula for the cross-section 183

The integral equation is used for mathematical investigation of

the scattering problem. For a suitable choice of function space, this

equation can be reduced to an equation of the second kind

+AV)

with a compact operator. Therefore, the Fredholm theorems hold for

the equation (1). For a broad class of potentials it has been shown

that the corresponding homogeneous equation can have solutions only

for imaginary values of the parameter k (k9, = ix7z, x, > 0). By re-

peating the calculations carried out in the beginning of this section,

it is easy to see that the solution V,, (x) of the homogeneous equation

satisfies the Schrodinger equation [-+V(x)]i/'(x) = -x/'(x) and

has the asymptotic expression 0,, (x) '=-' f (n) e - xn T /r, where f (n) is

a function defined on the unit sphere. This means that the solutions

of the homogeneous equation are eigenfunctions of the discrete spec-

trum of H. The completeness of the collection {fin (x),

(x, k) I of

eigenfunctions of H was proved with the help of the integral equation

(1)

§ 42. Derivation of a formula for the

cross-section

The main characteristic of the process of scattering of a particle by

a potential center is the differential cross-section. In accordance with

the general definition of the cross-section, the differential cross-section

is defined by the formula

dor =

where dN is the probability of observing the particle after the scat-

tering (t -+ oo) in the solid angle element do about some direction n,

and I is the probability of a free particle crossing a unit area element

oriented perpendicular to the motion of the particle.

This proba-

bility is defined at the point where the scattering center is located.

We stress that dN is a characteristic of a particle in the field of the

scattering center, while I is a characteristic of a free particle.

Little is known about the state w of a free particle in the beam.

Indeed, it is known that the particle has momentum approximately

1 84

L. D. Faddeev and 0. A. Yakubovskii

equal to ko = kowo, and the variance

(Ak1)avg+(Ok2)avg+

(k3)vg of the momentum is known; moreover, for the scattering

experiments to admit a simple interpretation, one tries to use beams

of particles with a small variance of the. momentum. Very little is

usually known about the coordinates of a particle in the beam. But

nevertheless it is known that at some moment of time (which can be

taken to be t = 0), the particle is in a macroscopic region between the

accelerator and the t arget. The transverse size of this region can be

identified with the diameter of the beam. We remark that according

to the Heisenberg uncertainty relation, the existence of such a region

imposes lower bounds on the variance of t he momentum.

In deriving the formula for the differential cross-section, we make

two assumptions about the smallness of the square root ok of the

variance of the momentum:

1) Ok<<ko,

2) I

- f (k'. n, w') I

< 1f(k.n')1 if Ikw -- k'w'l < Ak.

In principle these requirements can always be satisfied if f (k, n, w)

is a continuous function of k = kw (the continuity of f can be proved

for a broad class of potentials). These conditions usually hold in prac-

tice. However, there are cases when a small change in the variable k

leads to a sharp change in the function f (k, n, w) (the case of narrow

resonance). In the final analysis the quantity ok is determined by the

construction of the accelerator and in practice cannot be made arbi-

trarily small. Therefore, in some experiments the condition 2) may

not hold. In this case one cannot use the formula obtained below for

the cross-section.

We show that under the conditions 1) and 2) the differential cross-

section depends on the state of the impinging particle only in terms

of k0, and that

do- = I

The most general state for a free particle is a mixed state, given by

a density matrix Mo (t) which we represent as a convex combination

of pure states:

Mo(t) = EasPPSt ), Ics = 1.

§ 42. Derivation of a formula for the cross-section

185

The pure state Pp, (t) is determined by a wave function Vs (x, t) with

the form

X, 0 =

(-_)2f

Cs k

ei(x-k2t)

dk.

Ps (

27t

It can be asserted that an appreciable contribution is given only by

the pure states in which the momentum variance does not exceed the

momentum variance in the state M0 (t) .

Therefore, we assume that

all the functions C., (k) are nonzero only in a small neighborhood of

the point k0 = k0w0, and that the diameter of this neighborhood does

not exceed Ak. We do not have any information about the concrete

form of the function45 Cs (k) nor about the weights as, and we do not

make any assumptions about them.

We first compute the probability I. Let us choose a coordinate

system with origin at the force center and with x3-axis directed along

the vector w0. For a pure state Spa (t) it is easiest to compute the

probability I(s) by using the density vector

(.s) (X,

t) = i (;p. O S03

- S0.14 V SPs }

of the probability flow, and then

{s}

33s)

(x,t)

Computing

(s)

ass

J3 (x,t)IX

= i

x.=0

dt.

1 dk dk/

i(k 2 -k2)t

3 3

(k3

3) Cs (k} C's (k) e

1!2 s

45 Corresponding to different functions C, (k) are both different forms and different

arrangements of wave packets in the configuration space at some moment of time For

example. suppose that some function CY (k) gives a wave packet whose c enter of gravity

(x(t)

moves along the x3-axis directed along the vector w0 and passes through the

scattering center Then corresponding to the function C1 (k) - eik"1 C4 (k) is the packet

shifted by the vector u1 in the configuration space (ezk" is the shift operator, written

in the momentum repres('ntation) As a classical analogue (though not completely

accurate) of the state p. (x, t) we can take the state of a particle moving on a rectilinear

trajectory that passes through the scattering center Then corresponding to the state

SP1(x, t) is a trajectory passing by the scattering center at a distance p equal to the

projection of the vector ul on the plane of the (rocs-section of the beam In classical

scattering theory the distance p by which the particle would go by the center if there

were no interaction is called the aiming parameter

186

L. D. Faddeev and 0. A. Yakubovskil

we get that

ei(

2)t

(v)

= 1

A dk'

°°

C (k')

(2ir)3

(k3

(k)

1-00

(2ir)2

A JIF3

' (k3 + IC3) Cs(k) CS(k')

k2)

(27r)2 fR

the 3

Cs(k)

(27r)2 f A I A' Cs(k) CY(k') d(k'

- k).

The computations use the equality

(2)

6(k2

k'2)

= 2k(6(k -

k') + b(k + k'))

and the condition Ok K ko, which enables us to replace (k3 + k3)/2k

by 1. The second term in (2) dog not contribute to the integral,

because k > 0 and k' > 0 in the region of integration.

It is obvious that for a mixed state Mg(t)

I -

as (21r)2

J, A f

dk' C.,(k) Gs(k') 8(k - k').

Let us now compute dN. In the computation we must use the

density matrix M (t) describing a state which long before the scat-

tering (t -+ -oo) tends asymptotically to Mo(t). The operator M(t)

can be written in the form

M(t)

as (t),

where the pure states Pw ,(t) are determined by the wave functions

- 1

-ik2t

(3)

03(t)

21r

3(k),O(x k) e

dk C'

The density of the coordinate distribution function in a pure state

PP,

(t) is f'3 (x, t )12 , and for the probability d N(-) of observing the

particle as t --+ oo in a solid angle element do in the state P(t) we

get

(4) dN(3) = lien do r2 dr IV)5(x, t)12.

t-'+00

§ 42. Derivation of a formula for the cross-section

We know that as t --+ oc the. particle goes to infinity, and therefore

when we substitute (3) into (4) we can replace b(x. k) by its asymp-

totic expression, obtaining

dN(s) = urn do

r2 dr

t-+oc

1 2

2-7r

CK (k)

ikr

eikx +

f (k, n, w)

e-ik2t

187

The function (2,7r) -312 fR3 dk C(k) eikx-- ik2 t is a wave packet for

a free particle and is nonzero only in that, part of the configuration

space where there is a finite probability of observing an unscattered

particle of the beam, that is, inside a narrow cone constructed about

the direction wo. For all the remaining directions, only the diverging

wave gives a contribution, and for such directions

dN(q) = line

-dn

t- (27r) 3

f Cs (k) f (k, n, w)

eikr-ik2t

Using the condition 2), we can replace the function f (k. n, w) un-

der the integral sign by its value at the point k = ko, w = wo

and take it out from under the integral sign. Further, the integral

JR3 Cs

(k)eu1'2t dk becomes

a one-dimensional wave packet after

integration with respect to the angle variable of the vector k, and

as a function of r it is nonzero only for large positive r as t --+ oc.

Therefore, the integration with respect to r in (5) can be extended

over the whole real axis. Then

d N (`s) = limn

t-y+oc (2ir)3

- 00

A Al 8(k) C,(k')

ei(k-k')re -i(k2-k'2)t

;

= do If (ko, n, wo)I2

()2JR

C. (k) Cs(k') 6(k - k').