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

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

We construct a solution 11(k x). In the regions I -III this solu-

tion has the form

(2)

1. eikx +

Ae-ika-

(3) If:

Beik' .

III: meickx + ne z`Yx .

The coefficients A, B. m. n are found from the conditions that Sb1

and its derivative be continuous at the points a and -a

e-ika +

APika

In(,-ina +

rtei(tia

(e-ika

- Aeika) _

a(rne-iara

neictia).

rne? +

ne,,-icva = Beika

ce(mezCYa - rte - 2`xa) =

kBeika

We wi ite the expression for the coefficient B:

-1

f(a + k)2

2i(k-(x)a

(a -

k)2

4ak

4ok

It is not hard to verify that B -* 0 under any of the conditions

1) k-0,

2) Vooc,

3) a - oc, k2 <Vo.

On the other hand. B -* I under one of the conditions

4) k -+oc, 5) V0--+ 0.

6) a-40.

We see that in the limit cases 1) 5) the results obtained on the basis

of quant um mechanics coincide with the classical results.42 Figure 13

shows the graph of the function I B(k)12. From this graph it is clear

that for certain finite values of k the probability of transmission is

lB2l = 1. It is interesting to note that the equations (1) together

with the conditions (2) and (3) describe the passage of light waves

through transparent plates. Here I B 12 and J 2 are proportional to

the intensities of the transmitted and reflected waves. In the case

I B 1 2 = 1, lAl2 = 0 the reflected wave is absent. This phenomenon is

used for coated optics.

42 According to classical mechanics a particle passes through the harrier with

probability I when k2 > V0 and is reflected with probability 1 when k2 < V0

§ 39. Scattering by a potential center

IB(k)12

------------ ---

Figure 13

§ 39. Scattering by a potential center

169

For the problem of scattering by a potential center, the Schrodinger

operator has the form

(1)

H_-0+V(x).

(We again assume that in = 1/2 and It = 1.) In what follows we

usually assume that the potential V (x) is a compactly supported

function (V(x) = 0 for lxi > a). The exposition is according to the

same scheme as in the one-dimensional case, that is, we first consider

some solutions of the stationary Schrodinger equation, and then we

explain their physical meaning with the help of the nonstationary

Schrcidinger equation

Our first problem is to formulate for the solut ion of the equation

-AO(x) + V (x) i' (x) = k2/'(x)

an asymptotic (as r - oo) condition that corresponds to the physi-

cal picture of scattering and which is an analogue of the conditions

(36 4) for the one-dimensional problem. It is natural to expect that

one terni will correspond asymptotically to the particle impinging on

the scattering center in a definite direction, while the second terra

will correspond to the scattered particle, which can have different

directions of motion after the scattering and which goes away from

the center.

The analogues of the functions eik.c and a-ik, in the

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

three-dimensional case are the functions eik' /r and e-ikr/r. There-

fore, it is reasonable to assume that the terms (e - i kr /r) b (n + w) and

(ei kr /r) S(k, n, w) correspond to the particle before and after scat-

t eying. We use t he following notation: k is the momentum of the

impinging particle. w = k/k. n = x/r, S(n - no) is the s-function on

the unit sphere, defined by the equality

f(n)h(n--no)dn = f(no),

dn=sin0dOdV;.

and S(k, n, w) is a certain function which will be shown to contain all

information about t he scattering process and which must be found in

solving the problem We shall see t hat S(k, n, w) is the kernel of a

certain unitary operator S, which is called the scattering operator

We come to the following statement of the problem of scattering

by a force center: it is required to find for the equation

(2) -A(x, k) + V(x) i(x. k)

k2V)(x, k)

a solution which has the asymptotic behavior

e- ikr ,ikr i

(3)

V1 (x, k) =

S(n + w)

S(k, n, w) + o

asr -moo.

This formulation of the problem caxi be justified only with the

help of the nonstationary formalism of scattering theory, and this

will be done in the next section. The question of the existence of

a solution of (2) with the condition (3) will also be discussed later.

However, this question has a simple answer for the case V (x) = 0.

We show that the function

x k) =

k eikx

ei kr nw

vc (

2iri 2iri

which is obviously a solution of (2) for V (x) = 0. also satisfies the

condition (3). and we find the form of the function S(k. n, w) for this

case.

We are looking for the asynnptotics of the function ip0 (x, k) in the

class of generalized functions, and therefore we must find an asymp-

totic expression for the integral

I= f(n)

(n),Oo(rn,J

§ 39. Scattering by a potential center

171

as r -> oc, where f (n) is a smoot h function. Using the explicit form

(x, k) and introducing a spherical coordinate system wit Ii polar

axis directed along the vector w, we have

27H

j27r

I = d f (Cob B.

) eikr cos a sin 9 dO

dr1 f (rl,

V) eikr,.

2ir2

Integrating by parts, we get that

2-/r

d,^ -- f

(rl, ) eik"I

27ri ilr

r I l

21ri

f27

djp

di f' (77, V)

eikr7j.

ikr

I ntegradrig by parts again, we easily see that the second term has

order 0(1/r 2), and hence

d(p (eikl f(w) _

eikr f(-w))

+ 0

f r2

e-ikr

eiki

f(-w) -

f (w} + n

Thus, we have shown that

iki

eikr

(4)

V)o(x. k) = -

o(n + w) -

b(n - w) + 0

Comparing (4) with (3), we find the function S for a free particcle:

SO(k, n. w) = 6(n - w).

Using (4), we can rewrite the asymptotic condition (3) in the

forma

ikr

(i).

(5) (x. k) =

eik+ f(k,

n. )+ o

2 ri r

where the function f (k, n, (A;) is connected with the function S(k. n. W)

by the relation

(6) S(k, n, w =bn - w+knw.

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

The function ,f (k, n. w) is called the scattering amplitude. Obviously,

f (k, n, w) = 0 for V (x) = 0. We shall see that the scattering cross-

section is most simply expressed in terms of this function.

Instead of the asymptotic condition (5). one often

uses

eikr

(7)

(x, k) - eik7 nw + f(k. n.

-+0 -

(r)

which leads to a more convenient normalization of the function zit(x, k).

Let us now consider the properties of the function S (k, n, W). The

following auxiliary assertion turns out to be convenient for the study

of these properties.

Suppose that the functions '1 (x) and IP2 (x) satisfy the equation

(2) and the asymptotic conditions

ikr

e2kr

. ,

(x) = A, (n)

+ B1(n)

V)2(x) = A2(n)

1,r

0 (r)

as r --+ coo, which can be differential end once with respect to r. Then

(8)

fAi(n)B2(n)dn=

A2 (n) B1(n) do

fS2

This assertion is easily proved with the help of Green's formula

- ' dX

(9)

i) 1

j(11b2

We should choose the domain of integration sl to be the ball of radius

R and take the limit as R - oo, obtaining

e- ikr ,ikr

c'i (x) = /'(x, k) = -b(n + w) -

-S(k, n. w) + o

j'2(x)-= '(x,k')=

e- ikr

,ikr

b(n+w') -

S(k.n,w')+o - ,

I,, r

-ikr

eikr

1 (x,k} - --

S(k.n.w/)+

bt(n+w')+o ,

where k' = kw'. Applying the formula (8) to the functions vI and

'V2, we get that

S(k, --w, w') = S(k, - w', w),

§ 39. Scattering by a potential center

173

or. replacing w by -w,

(10)

S(k, w, w') = S(k. -w', -w).

This formula is an analogue of the equality B = D for the one-

dimensional scattering problem and expresses the fact that the values

of S for the direct (w' w) and time-reversed (-w ---+ --w') collision

processes coincide. It can be shown that this property (just like the

symmetry of the S-matrix in the one-dimensional case) is a conse-

quence of the invariance of the Schrodinger equation with respect to

time reversal.

Further, applying the formula (8) to the functions ipx and 1/12, we

get that

(11) f S(k, n, w') S(k, n, w) do = 8(w - w').

If we consider the function S (k, n, w) as the kernel of an integral

operator

S;p(w) =

S(k, w, w') ;p(w') dw'

acting in L2 (S2), then (11) can be rewritten in the form

SE`S = I

In view of (10) it follows from (11) that

SS* = I.

The last two relations imply that the operator S is unitary.

We see that for the problem of scattering by a potential cen-

ter the operator S has the same properties as the S-matrix for the

one-dimensional scattering problem. Since S is unitary, we have the

important relation

(k,n,w)I2dn= 4k Imf(k7 w,),

(12)

152 If

174

L. D. Faddeev and 0. A. Yakubovskii

which is called the optical theorem. Indeed, using (6) and (11), we

get that

S z

LS(n - w') -

21r

(k' n, [(n - w) +

f (k, n, w)J do

= 8(w - w') + 2 [f (k, w , w) - f (k, w, w')]

2 r

+ k2J f(k,n,w)f(k,n,w')dn=S(w-w').

4'rs2

Setting w = w', we immediately arrive at the formula (12).

In § 42 we shall see that the integral on the left-hand side of (12)

coincides with the total cross-section a for scattering of a particle by

a potential center. Therefore, (12) can be rewritten as

4rr

cr=

Tinf(k,w,w).

This formula connects the total scattering cross-section with the imag-

inary part of the zero-angle scattering amplitude.

By using the unitarity of the operator S, it is easy to show that

the functions

(x, k) satisfying the asymptotic condition (7) have the

normalization

(13)k(x,k) (x, k') dx = (27r)sb(k - k'):

IR3

that is, they are normalized just like the free-particle solutions

eakx.

To verify (13) we multiply the equalities

o1(x, k) + k2L(x, k) = V(x) O(x, k).

o(x, k') + k'2 7P (x, k') = V(x) iP(x, k')

by 0 (x, k') and i (x, k), respectively, subtract the first from the sec-

ond, and integrate over the ball of radius R. Then

y)(x, k) (x. k') dx

S2R

k12

[(x, k

xk' "'' xk' O xk dx.

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

175

With the help of Green's formula, this equality can be rewritten in

the form

O(x, kb(x,k')dx

fb(x,

k2 - k'a

j(ii(x,k)k')

igr 49r

where SR is the sphere of radius R. The formula (13) is obtained from

the last relation by passing to the limit as R -+ oo, and the integral

on the right-hand side is computed with the help of the asymptotic

expression for O(x, k). We omit the corresponding computations,

since they repeat literally those which led us to the formulas (36.9)

and (36.10).

§ 40. Motion of wave packets in a central force

field

With the help of the function O(x, k), we construct for the nonsta

tionary Schrodinger equation

iab(x't)

= H x t

a solution of the form

(x,t)

1 2

C(k) V)(x, k)

e-tk2t

C21r) .R,3

If the function C(k) satisfies the condition

(2)

1R3

then O(x, t) has the correct normalization

J3 Iib(x,t)I2dx = 1

in view of (39.]3).

176

L. D. Faddeev and D. A. YakubovskiI

As in the

one-dimensional case, it is reasonable to consider a

solution with the

function C (k) concentrated in a small neighbor-

hood of the point ko = kowo. By the Riemann

Lebesgue theorem as

(x, t) 2 dx

0 for any finite

- oo, we have V

)(x, t) --+ 0, and f

region Q. Therefore, the function P (x, t) describes an infinite motion

of a particle. We are interested in the behavior of this solution as

It I

--+ oo and r --- oo, and we can replace ' (x, k) in the formula (1)

by its asymptotic expression

tai /e-ikT

eikr \ /1\

I r f .

(x, k) _

w) - r S(k, n, w)) +o

Asr ---+oo,wehave

2 00

7ra

x t

kdk dwC k w

o s2

f -ikr

ikr

(-5(n+') _

S(k, n, w) a-ik2t

= V11 (X, t) + 02 (X, 0,

where '1(x, t) and2 (x, t) correspond to the converging wave e-ikr/r

and the diverging wave eikr /r in the asymptotic..

The function 01(x, t) can be rewritten in the form

(3)

x t=

kC(k. -n)

e-i1CTe-ik

t dk

r 2i -o o

(we consider integration over the whole real axis, setting C(k, n) =

0 for k < 0). Computing the integral (3) by the stationary phase

Ynethod, we get that

(4)

(X, t)

\-2t,-n) etx +0

\ItJ2/

here x, as always, denotes a real function that is not of interest to us.

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

177

To compute 12 (x, Owe use the formula (39.6) for the function S.

02(x,t) = - Z

11)0

kdkJ

r 2 x SZ

(5)

[S(n-

f k n, w

eikre- ik2t

k A

V"2--,-7 r

_o c

Here we have introduced the notation C, (k) = fC(k, w) dw, and,

taking into account that C(k, w) is S-shaped, we have replaced the

function f (k, n, w) by its value at the point w = wo. The integral in

(5) can be computed by the stationary phase method:

1p2(x, t) _

x [Gry\2t'nl+47rtC,

2t/

(2t'n'w0)Je`x,+0(

ItI2/

x I Qk, n) + :Cl (k) .f (k, n, wo) Ct(kr-k2t)

Finally, since C1 (k) is S-shaped (it is concentrated in a neighborhood

of ko), we get that

(6)

V;2 (Xi 0 =

x [+

\ 2t /

(ko, n, WO)

e'xj .+-p I

\P /

It is clear from (4) and (6) that 01 contributes to 0 (x, t) only as

t - -oc, and 02 only as t --> +oo. For the density Ii/'(x, t)2 of the

coordinate distribution function, we have

r l 2

(7)

1b(x,t)I2 ='

81t13

2t'

-n/ 1

t --> -oo,