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ELASTIC SCATTERING OF A QUANTUM PARTICLEBY A CENTRAL POTENTIAL

Pupyshev V.V.

JINR, BLTP

June 28, 2012

Pupyshev V.V. (JINR) Elastic scattering June 28, 2012 1 / 16
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1. INTRODUCTION

By definition,

p1 is a quantum particle with mass m;

R3 is the three dimensional coordinate space in which this particle moves;

S3is the cartesian coordinate system in this space.

In this system r = (r, , ), k and E = (k)2/(2m) are the radius- and wavevectors and the total energy of the particle p1;

V(r) is the central (spherically-symmetric) potential acting on the particle p1and obeying the following condition

b

a

|V(r)|r dr < , a, b : 0 a< b . (1)

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The wave function (r; k) describing the elastic scattering of the particle satisfies thethree dimensional Schrodinger equation

22m

r + E V(r)

(r, k) = 0 (2)

and the scattering boundary condition

(r; k) exp(kr) + f()exp(kr)

r, r . (3)

The function is represented as series

(r; k) =1

r

=0

u(r; k) P(cos ) , (4)

where P, = 0, 1, 2 . . ., are the Legendre polynomials, and is the angle between thevectors r and k.

By substitution (4) equation (2) is reduced to the infinite set ( = 0, 1, 2, . . .) ofuncoupled one dimensional Schrodinger equations for the radial components u.

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The Schrodinger equations for the radial components u with given reads as

2r + k2 ( + 1)

r2 V(r)

u(r; k) = 0 , r 0 . (5)

Two solutions of this equation are physically interesting:

the regular solution u+ with asymptotics

u+

(r; k) = O(+1) , kr 0 ; u+ (r; k) sin [ /2 + (k) ] , ,

(6)

and irregular solution u having the asymptotics

u (r; k) = O() , 0 ; u (r; k) cos [ /2 + (k) ] , . (7)

Here (k) is the phase-shift generated by the potential V(r).

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2. STATEMENT OF THE PROBLEM

The main aims of our work is

to develop the linear version of variable phase approach

and propose the methods allowing one derive

the explicit asymptotics of both solutions u+ and u

in the four physically interesting limits.

They are:

the limit of small values of the argument r,

the limit of large values of this argument,

the limit of large integer angular momentum ,

and the limit of small positive energy E = (k)2/(2m) 0+ .

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3. METHODS

The suggested methods are based on the linear version of the variable phase

approach to the potential scattering.

3.1 The linear version of the variable phase approach

In this version

first, the Riccati-Bessel functions j() and n() are used as the solutions of free radialSchrodinger equation (5);

second, two amplitude functions c and s obeying the Lagrange identity

j() rc(r; k) n() rs(r; k) = 0 kr r 0 , (8)

are introduced

and, finally, the wave-function u in represented as

u (r; k) = N(k) U(r; k) , U(r; k) = c(r; k)j() s(r; k) n() , (9)

where N(k) is the searched normalization factor.

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Then by substitution (9) the Schrodinger equation (5) is reduced to the system of the

first-order differential equations for the amplitude functions c and s

rc(r; k) = k1 V(r) [j() s(r; k) n() n() ] n() , (10)

rs(r; k) = k1 V(r) [j() s(r; k) n() n() ] j()

and these functions are obeyed the following boundary conditions at the point r = 0

c(r; k) = 1 , s(r; k) = 0 , r = 0 , (11)

for regular function u+ and

c(r; k) = 0 , s(r; k) = 1 , r = 0 . (12)

for irregular wave function u .

And, finally, the phase-shift (k) and the norm factor N(k) are defined as the followinglimits at large argument r:

(k) = limr

arctg [ s(r; k)/c(r; k) ] , N(k) = limr

[ c(r; k)2 + s(r; k)2 ]1/2 .

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3.2 Method for construction of the coordinate asymptotics

In the suggested method

first, the differential equations for the amplitude functions are rewritten in the

integral form;

second, the two obtained uncoupled integral equations are iterated;

and, finally, the principle of contracting mapping is used to prove

the uniform convergence of the iterations under well-defined conditions.

As a result the following three theorems were proven.

One has to point out that in these theorems the exponential function

B(b, r; k) expk1 r

b

V(t)j(kt) n(kt) dt (13)is included in the asymptotics and the residual term is defined by the integral

v(b, r)

2

2 + 1

rb

|V(t)|t dt. (14)

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Theorem 1: asymptotics at small argument r.

Let and k are fixed v2(0, r) and b is a root of the inequality

v(0, b)

2

2 + 1

b0

|V(t)|t dt < 1 . (15)

Then in the limit r/

b

0 the regular wave function u+

has the asymptotics

u+

(r; k) = N(k) { [ 1 + (r; k) + O() ] j() [ (r; k) + O())n() ] } , (16)

where

(r; k) k2 r

0

n2(kt) B2(0, t; k) dt

t

0

j2 (kz) B2(0, z; k) dz, (17)

(r; k) k1r

0

j2 (kt) B2(0, t; k) dt.

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Theorem 2: asymptotics at large argument r.

Let and k are fixed v2 (r, ) and b is a root of the inequality

v(b, )

2

2 + 1

b

|V(t)|t dt < 1 . (18)

Then in the limit r/b the regular wave function u+ has the asymptotics

u+

(r; k) = N(k) { [ (r; k) + O() ] j() [ (r; k) + O() ] n() } , (19)

where

(r; k) c(; k) s(; k) k1 r

n2

(kt) B2

(, t; k) dt, (20)

(r; k) s(; k) + c(; k) k1r

j2 (kt) B2(, t; k) dt.

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Theorem 3: asymptotics at large angular momentum .

Let k > 0, v2L (0, ) and L is such that the following inequality holds

vL(0, )

2

2L + 1

0

|V(t)|t dt < 1 . (21)

Then in the limit /L the regular wave function u+ has the asymptotics

u+

(r; k) = N(k) { [ 1 + (r; k) + O() ] j() + [ (r; k) + O() ] n() } ,

where and are the functions from Theorem 1.

The analogues of the Theorems 1-3 are proven for the irregular wave function u .

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3.3 Method for construction of low-energy asymptotics

The proposed method is based on the differential equations for the amplitude

functions (10) and the well-known expansions of the Riccati-Bessel functions

j(kr) = (kr)+1

n=0

an k2n r2n , n(kr) = (kr)

n=0

bn k2n r2n

where are an and bn the numerical coefficients.

The amplitude functions are searched as series

c(r; k) =

n=0

k2n cn(r) , s(r; k) = (k)2+1

n=0

k2n sn(r) .

As result for the regular wave function one obtains the expansion

u+(r; k) = k+1 N(k)

n=0

k2n Un(r) , Un(r)

p+q=n

[ apcq(r) + bpsq(r) ] .

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In this expansion the functions cn and sn are defined the recurrence chain of the

energy-independent equations.

The first system of this chain reads as

rc0(r) =V(r)

2 + 1

rc0 + r

2s0

, (22)

rs0(r) =

V(r)

2 + 1 r

2+1 rc0 + r

2

s0

.

The second system has a form

rc1(r) =V(r)

2 + 1 rc1 + r

2s1 +r2

2 1

2

2 + 3

rc0 + r2s0 , (23)

rs1(r) = V(r)

2 + 1r2

+1

rc1 + r

2s1 r2

2 + 3

rc0

2

2 1r2s0

.

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The next system is

rc2(r) = V(r)2 + 1

rc2 + r2s2 + r

2

2 1

2

2 + 3rc1 + r2s1

+

+r4

(2 1)(2 3)

6

(2 + 3)(2 + 5)rc0 +

1

2 1r2s0

,

rs2(r) = V(r)

2 + 1r2

+1 rc2 + r2s2 r2

2 + 3 rc1 2

2 1r2s1 + (24)

+r4

(2 + 3)(2 + 5)

+ 2

2 + 3rc0 +

6

(2 1)(2 3)r2s0

.

At the point r = 0 the boundary conditions are

c0(r) = 1 , s0(r) = 0 ; cn(r), sn = 0 n = 1, 2, . . .

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The obtained representation of the irregular wave function u is the series

u(r; k) = k N(k)

n=

0

k2n Un(r) , Un(r)

p+q=n[ apcq(r) + bpsq(r) ]

in which the functions cn and sn satisfy the systems (22)(24) and the boundary

conditions

c0(r) = 0 , s0(r) = 1 ; cn(r), sn = 0 n = 1, 2, . . .

The next result is the proof of the following theorem.Theorem 4.

Let energy tends to zero.

If for any n = 0, 1, 2, . . . the potential V satisfies the condition

limr

rnV(r) = 0 .

then the obtained expansions for regular and irregular wave-functions converge

uniformly in respect to the argument r and angular momentum .In the opposite case, when the potential V is the long-range one, these series

converge uniformly if r < .

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4. Summary

The linear version of the variable phase approach to the elastic scattering of a quantum

particle by a central potential is supplemented by the methods for construction of thecoordinate and low-energy asymptotics of radial regular and irregular wave-functions.

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