Scattering 1

Quantum Mechanics
                      K. Konishi, G. Paffuti

Check in Mathematica the asymptotic form of the scattering solutions.

NOTE 1: To avoid interference with other notebooks it is recommended to quit and start again the Kernel.

Global variables and initialization

Off[General::spell, General::spell1];
ClearAll["Global`*"];

Theory and notations

Spherical Bessel functions

Spherical Bessel (and Hankel) functions are defined by

(Subscript[j, L])[x] = (Sqrt[(π/(2x))]) (Subscript[J, L+(1/2)])[x];   (Subscript[η, L])[x] = (Sqrt[(π/(2x))]) (Subscript[Y, L+(1/2)])[x];  (Subscript[h, L]^((1,2)))[x] = (Subscript[j, L])[x]± i (Subscript[η, L])[x].

In Mathematica (version 6) these functions are defined explicitly with SphericalBesselJ, SphericalBesselY, SphericalBesselH1 etc.

In the following we will use the shortand notation

sJ = SphericalBesselJ ; sY = SphericalBesselY ; sH = SphericalHankelH1;

All these functions satisfy the radial Schrödinger equation for free particles

(1/(r^2))(d/dr)(r^2)(d/dr) (Subscript[R, L]) + (k^2) (Subscript[R, L])- ((L(L+1))/(r^2))(Subscript[R, L]) = 0 ;                      (k^2)= ((2 m E)/(ℏ^2)) .

sJ are the only regular at the origin.  The small and large r behavior is

sJ[L,x](Underscript[→, x→0]) (((2^(-1-L)) (Sqrt[π]) (x^L))/(Gamma[(3/2)+L])) ;  sY[L,x](Underscript[→, x→0]) -(((2^L)  Gamma[(1/2)+L](x^(-1-L)) )/(Sqrt[π])) ;

sJ[L,x](Underscript[→, x→∞]) (1/x) Sin[x-L (π/2)];   sY[L,x](Underscript[→, x→∞]) - (1/x) Cos[x-L (π/2)];  sH[L,x](Underscript[→, x→∞]) (1/x)(1/i)(e^(i(x-((Lπ)/2)))) .

The plane wave solution (regular at the origin) admits an expansion in terms of eigenstates of L and of energy of the form:

Exp[i k z] = (∑_(L=0)^∞)(i^L) (2 L+1) LegendreP[L,Cos[θ]] sJ[L,k r].

Check in Mathematica

sJ = SphericalBesselJ ; sY = SphericalBesselY ; sH = SphericalHankelH1;

Here we verify the radial equation

FullSimplify[((1/(r^2))(∂_r)((r^2)(∂_r)#1[L,k r])+ (k^2) #1[L,k r] -(( L(L+1))/(r^2))#1[L,k r]== 0)&/@{sJ,sY,sH}]/.{k-> 1}

The short distances expansion is

Simplify[Normal[Series[#1[L,x],{x,0,1}]]&/@{sJ,sY},Assumptions-> L∈ Integers]

The asymptotic expansion

Simplify[Normal[Series[#1[L,x],{x,∞,0},Assumptions-> x>0]]&/@{sJ,sY},Assumptions-> L∈ Integers]

The leading term coincides with ().

For the series expansion we limit ourselves to check the first maxL orders

Block[{k=1,maxL = 10},
Simplify[Normal[Series[Sum[(I^L) (2 L+1) LegendreP[L,Cos[θ]] sJ[L,k r],{L,0,maxL}],{r,0,10}]]] == 
Simplify[Normal[Series[Exp[I k r Cos[θ]],{r,0,maxL}]]]]

General radial solution

A generic regular radial solution of the radial Schrödinger equation will be named (Subscript[R, L])[k, r]:

(1/(r^2))(d/dr)(r^2)(d/dr) (Subscript[R, L]) + ((k^2) - ((2m)/(ℏ^2))U[r])(Subscript[R, L])- ((L(L+1))/(r^2))(Subscript[R, L]) = 0 .

For potentials decaying sufficiently rapidly at infinity the asymptotic form of R is  (up to a normalization)

R → ((2 Sin[k r-((π )/2)L+δ[L]])/r)

In fact the reduced radial solutions are a linear superposition of the free solutions Sin[k r] and Cos[k r]. In the following we will take the normalization () for the elementary solutions. The phases (Subscript[δ, L]) are the scattering phases and are 0 for a free particle, as is easily seen from () and the asymptotic expansion (). As combination of sJ and sY we can define the asymptotic (elementary) solution as

Ras[L,k,r]= 2k (Cos[δ[λ]] sJ[λ,k r] - Sin[δ[λ]] sY[λ,k r]).

This combination reproduces asymptotically (), at is easly seen using (). In this normalization

RasFree[L,k,r]= 2k  sJ[λ,k r].

The general regular solution, invariant under rotations around z axis, will have the form, at fixed k:

ψ[r,θ]= (∑_(L=0)^∞) (Subscript[A, L]) LegendreP[L,θ] R[L,k,r].

All  solutions have this form (we have just performed an expansion in terms of angular momentum eigenfunctions). We look for solutions corresponding to a scattering, this means that asymptotically the solution must behaves as a superposition of the incident plane wave plus an outgoing spherical wave:

ψ → Exp[i k z] + ((f[θ])/r)Exp[i k r].

f[θ] is the scattering amplitude and the differential cros section is ( dΩ = Sin[θ] dθ dφ is the infinitesimal solid angle):

dσ = |f[θ]((|)^2) dΩ.

The form () of the solution we look for impies that the only incoming spherical waves are contained in the incident plane wave. This fix the form of the coefficients (Subscript[A, L])in the general solution ().

In the free solution we have, see eq.():

(∑_(L=0)^∞)(i^L) (2 L+1) (Subscript[P, L])[Cos[θ]] sJ[L,k r]= (∑_(L=0)^∞)(i^L) (2 L+1) (Subscript[P, L])[Cos[θ]] (1/2)((Subscript[h, L]^((1)))[k r] + (Subscript[h, L]^((2)))[k r])

In the exact solution asimptotically

(∑_(L=0)^∞)(Subscript[A, L]) (Subscript[P, L])[Cos[θ]] Ras[L,k r]= (∑_(L=0)^∞)(Subscript[A, L]) (Subscript[P, L])[Cos[θ]] 2k (Cos[(Subscript[δ, L])]sJ[L,k r]- Sin[(Subscript[δ, L])] sY[L,kr])=
                             (∑_(L=0)^∞) (1/2)(Subscript[P, L])[Cos[θ]]( 2k (Subscript[A, L]))(Cos[(Subscript[δ, L])]((Subscript[h, L]^((1)))+ (Subscript[h, L]^((2)))) - (1/i)((Subscript[h, L]^((1)))- (Subscript[h, L]^((2)))))

The coefficient of the convergent wave (Subscript[h, L]^((2))) must be equal, then

2k (Subscript[A, L]) (e^(-i (Subscript[δ, L])))= (i^L)(2L+1)   ⇒    (Subscript[A, L])= ((2L+1)/(2k))(i^L) (e^(i (Subscript[δ, L])))

By the definition of the asymptotic form

(Subscript[ψ, as]) = Exp[i k z]+ (1/r)(∑_(L=0)^∞)f[L,k] LegendreP[L,Cos[θ]]Exp[i k r]≡ Exp[i k z]+ (1/r) f[θ]Exp[i k r]

one has

f[L,k] = ((2L+1)/(2k i))( (e^(2i (Subscript[δ, L])))-1); f[θ] =(1/(2i k)) (∑_(L=0)^∞)(2L+1)( (e^(2i (Subscript[δ, L])))-1) (Subscript[P, L])[Cos[θ]]

Using

∫dΩ (Subscript[P, M])[Cos[θ](Subscript[P, L])[Cos[θ]] = ((4π)/(2L+1))(Subscript[δ, LM])

one easily gets

σ = (π/(k^2))(∑_L)(2L+1)| (e^(2i (Subscript[δ, L])))-1((|)^2) = ((4π)/(k^2))(∑_L)(2L+1)((Sin[(Subscript[δ, L])])^2).

The scattering phases δ are to be fixed by imposing the regularity condition at the origin, or the boundary condition for an "excluded volume" problem.

Check in Mathematica

Let us check our algebra in Mathematica

A[L_]:= ((2L+1)/(2k))(I^L) (E^(I δ[L]));

The coefficient of the L - th Legendre polynomial in the asymptotic solution and in the free solutions are (Subscript[A, L])Ras and (i^L)

(Subscript[A, L]) Ras[L,k,r];   (i^L)(2L+1)sJ[L,k r] ;

respectively. In the difference

Ras[L_,k_,r_]:= 2k (Cos[δ[L]] sJ[L,k r] - Sin[δ[L]]sY[L,k r]);
coeffFree[L_,k_,r_]:= (I^L)(2L+1)  sJ[L,k r] ;

Simplify[A[L] Ras[L,k,r]-coeffFree[L,k,r]]

it apperas, correctly, only the combination J+iY (the Hankel h1 function) which describes an outgoing wave.

The above coefficient is

coeff = (1/2) (I^L) (-1+(E^(2 I δ[L])));

Substituting for (Subscript[h, L]^((1)))its asymptotic expansion one get

h1As = (1/(k r))(1/I)(((-I))^L) Exp[I k r];

Simplify[coeff*h1As,Assumptions -> L∈Integers]

which gives the correct expression for f[L, k].