Computation of the scattering amplitude in the spheroidal coordinates … · 2015-10-12 · Computation of the scattering amplitude in the spheroidal coordinates Takuya MINE Kyoto

Computation of the scattering amplitude in the

spheroidal coordinates

Takuya MINE

Kyoto Institute of Technology

12 October 2015 Lab Seminarat Kochi University of Technology

Takuya MINE (KIT) Spheroidal coordinate Lab Seminar 1 / 81

Orthogonal curvilinear coordinates

There are several orthogonal curvilinear coordinates in which we cansolve the Helmholtz equation by separation of variables.

Today we consider the three-dimensional case, and study theharmonic analysis in the spheroidal coordinates. As an application,we try to calculate the wave functions and the scattering amplitudefor the idealized Tonomura model.


Review: Spherical coordinate

As is well-known, the spherical coordinate in R3 is defined as follows.x1 = r sin θ cosϕ,

x2 = r sin θ sinϕ,

x3 = r cos θ,

r ≥ 0, 0 ≤ η ≤ π, −π < ϕ ≤ π.

In the spherical coordinate, the Laplacian is written as

∆ =1

r 2 sin θ

[sin θ

∂

∂r

(r 2∂

∂r

)+

∂

∂θ

(sin θ

∂

∂θ

)+

1

sin θ

∂2

∂ϕ2

].



The surface r = const.is a sphere, θ = const.a cone, ϕ = const. ahalf-plane.



The Helmholtz equation −∆u = k2u is equivalent to

d2f

dr 2+

2

r

df

dr+

(k2 − ℓ(ℓ+ 1)

r 2

)f = 0,

1

sin θ

d

dθ

(sin θ

dg

dθ

)+

(ℓ(ℓ+ 1)− m2

sin2 θ

)g = 0,

d2h

dϕ2= −m2h,

where u = f (r)g(θ)h(ϕ), and ℓ,m are the separation constants.



We have the solutions finite at r = 0, θ = 0, π, not diverging asr → ∞, and periodic with respect to ϕ, for ℓ = 0, 1, 2, . . ., andm = 0, 1, . . . , ℓ. The solutions are written as

u = jℓ(kr)Pmℓ (cos θ) cos(mϕ),

u = jℓ(kr)Pmℓ (cos θ) sin(mϕ) (m = 0).

Here jℓ(z) =√π/(2z)Jℓ+1/2(z) is the spherical Bessel function, Pm

ℓ

is the associated Legendre function. The number ℓ is called theazimuthal quantum number, and m the magnetic quantum number.The completeness of these solutions are guaranteed by the followingformula.



Proposition 1 (Rayleigh’s plane wave expansion formula)

Let (r , θ, ϕ) and (k , τ, ψ) are the spherical coordinates for x and p,respectively. That is,

x = (r sin θ cosϕ, r sin θ sinϕ, r cos θ),

p = (k sin τ cosψ, k sin τ sinψ, k cos τ).

Then, we have

e ix ·p =∞∑ℓ=0

ℓ∑m=0

[i ℓc2ℓ,mjℓ(kr)P

mℓ (cos θ)P

mℓ (cos τ) cos(m(ϕ− ψ))

],

cℓ,m =

√(2− δ0,m)(2ℓ+ 1)

(ℓ−m)!

(ℓ+m)!.



The Fourier inversion formula is described as follows.

u(x) =1

(2π)3/2

∫R3

e ix ·pu(p)dp

=∞∑ℓ=0

ℓ∑m=0

i ℓcℓ,mPmℓ (cos θ) cos(mϕ)

∫ ∞

0

jℓ(kr)uℓ,m,c(k)k2dk

+∞∑ℓ=0

ℓ∑m=1

i ℓcℓ,mPmℓ (cos θ) sin(mϕ)

∫ ∞

0

jℓ(kr)uℓ,m,s(k)k2dk ,

uℓ,m,c(k) =1

(2π)3/2

∫ π

−π

∫ π

0

cl ,mPℓ,m(cos τ) cos(mψ)u(p) sin τdτdψ,

uℓ,m,s(k) =1

(2π)3/2

∫ π

−π

∫ π

0

cl ,mPℓ,m(cos τ) sin(mψ)u(p) sin τdτdψ.


Review: Phase shift and scattering amplitude

Let us explain how to calculate the scattering amplitude for theSchrodinger operator H = −∆+ V (r) with a radial potential V (r)decaying sufficiently fast at ∞. The free operator H0 = −∆ hasgeneralized eigenfunctions with eigenvalue k2 (k > 0)

u = jℓ(kr)Pmℓ (cos θ) cos(mϕ),

u = jℓ(kr)Pmℓ (cos θ) sin(mϕ) (m = 0),

and the radial part jℓ(kr) has the asymptotics

jℓ(kr) ∼1

krcos

(kr − (ℓ+ 1)π

2

)(r → ∞).



If V (r) decays sufficiently fast at ∞, we can prove that the perturbedoperator H has generalized eigenfunctions with eigenvalue k2 (k > 0)

u = uℓ,k(r)Pmℓ (cos θ) cos(mϕ),

u = uℓ,k(r)Pmℓ (cos θ) sin(mϕ) (m = 0)

for any ℓ = 0, 1, 2, . . . and m = 0, 1, . . . , ℓ, and uℓ,k(r) has theasymptotics

uℓ,k(r) ∼1

krcos

(kr − (ℓ+ 1)π

2+ δℓ,k

)(r → ∞),

where δℓ,k is a real constant called the scattering phase shift.



The wave operators W± are defined by

W±u = s-limt→±∞

e itHe−itH0u.

Notice that the solution to the Schrodinger equation

i∂

∂tψ(t, x) = Hψ(t, x), ψ(0, x) = u(x)

is ψ(t, x) = e−itHu(x). The above definition means the solutionse−itHW±u behave like the free solutions e−itH0u as t → ±∞,respectively.



The scattering operator S is defined by S = W ∗+W−. By the

conservation of energy, the operator FSF∗ (F is the Fouriertransform) is decomposed into the direct integral of the operatorsS(E ) (E > 0) acting on L2(SE ), where SE is the energy shell

SE = {ξ ∈ R3 | |ξ|2 = E} = {√Eω | ω ∈ S2}.

Then the scattering amplitude f (k2;ω, ω′) is defined by the formula

(S(k2)− I )(ω, ω′) =ki

2πf (k2;ω, ω′) (ω, ω′ ∈ S2).

The quantity |f (k2;ω, ω′)|2 is called the differential scattering crosssection, which is proportional to the ratio of the particles with energyk2, incident direction ω′ and final direction ω.



Proposition 2 (Formula for the scattering amplitude)

Given the scattering phase shifts δℓ,k , the scattering amplitude for thewhole operator is given by

f (k2;ω, ω′)

=1

2ik

∞∑ℓ=0

(e2iδℓ,k − 1)ℓ∑

m=0

c2ℓ,mPmℓ (cos θ)P

mℓ (cos τ) cos(m(ϕ− ψ))

=∞∑ℓ=0

e2iδℓ,k − 1

2ik(2ℓ+ 1)Pℓ(ω · ω′).

The final result depends only on the angle between ω and ω′, becauseof the spherical symmetry of the system (V is radial).



In summary,

(1) By the spherical coordinate, we can decompose the scatteringproblem in the three-dimensional space to the problem in theone-dimensional space (scattering phase shift).

(2) By using the plane wave expansion formula, we can sum up thepartial scattering data and get the scattering amplitude for thewhole operator.

This machinery also works in the case of the spheroidal coordinate.There are two spheroidal coordinates (the prolate spheroidalcoordinate and the oblate spheroidal coordinate), but today we onlyuse the oblate one.


Oblate spheroidal coordinate

The oblate spheroidal coordinate is defined as follows.x1 = a cosh ξ sin η cosϕ,

x2 = a cosh ξ sin η sinϕ,

x3 = a sinh ξ cos η,

ξ ≥ 0, 0 ≤ η ≤ π, −π < ϕ ≤ π,

where a is a positive constant. This definition is taken from ‘Iwanamimathematical formulas III’ (there is another formulation).


Oblate spheroidal coordinate

The surface ξ = const.is a flattened ellipsoid,η = const. a hyper-boloid of one sheet,ϕ = const. a half-plane.


Laplacian in the oblate spheroidal coordinate

The Laplacian is written as follows.

∆u =1

a2 cosh ξ sin η(cosh2 ξ − sin2 η)

·

(sin η

∂

∂ξ

(cosh ξ

∂u

∂ξ

)+ cosh ξ

∂

∂η

(sin η

∂u

∂η

)

+cosh2 ξ − sin2 η

cosh ξ sin η

∂2u

∂ϕ2

).

If we put u = f (ξ)g(η)h(ϕ), the Helmholtz equation −∆u = k2u isreduced to the following ordinary differential equations.



1

cosh ξ

d

dξ

(cosh ξ

df

dξ

)+

(a2k2 cosh2 ξ − µ+

m2

cosh2 ξ

)f = 0, (1)

1

sin η

d

dη

(sin η

dg

dη

)+

(−a2k2 sin2 η + µ− m2

sin2 η

)g = 0, (2)

d2h

dϕ2= −m2h. (3)

Here m, µ are the separation constants.



We require

(a) g(η) is finite at η = 0, π,

(b) h(ϕ) has period 2π, and

(c) u is single-valued with respect to the original coordinate x .

By (b) and (3), we have

h(ϕ) = cos(mϕ) (m = 0, 1, 2, . . .),

h(ϕ) = sin(mϕ) (m = 1, 2, . . .).



We put λ = µ− a2k2, c = −iak . By the change of variablez = i sinh ξ in (1), and w = cos η in (2), we have

d

dz

((1− z2)

df

dz

)+

(λ− c2z2 − m2

1− z2

)f = 0, (4)

d

dw

((1− w 2)

dg

dw

)+

(λ− c2w 2 − m2

1− w 2

)g = 0. (5)

Thus the two equations (4) and (5) are equivalent as equations forcomplex variables. Especially when c = 0, these equations becomethe associated Legendre differential equation, which is obtained fromthe equation for g(θ) in the spherical coordinate by the change ofvariable w = cos θ.


Angular spheroidal wave function

By the requirement (a), we need the solutions to (5) finite atw = ±1. For fixed m = 0, 1, 2, . . ., there are at most countablevalues of λ’s for which the equation (5) has a non-trivial solutionfinite at w = ±1. We denote such values by

λmℓ (ℓ = m,m + 1,m + 2, . . .),

and corresponding solutions by Smℓ(c ,w). We call Smℓ(c ,w) theangular spheroidal wave function (of the first kind) . When c = 0,Smℓ(0,w) coincides with the associated Legendre function Pm

ℓ (w).Smℓ(c ,w) is normalized as∫ 1

−1

|Smℓ(c ,w)|2dw =2

2ℓ+ 1

(ℓ+m)!

(ℓ−m)!.


Radial spheroidal wave function

Notice that Smℓ(c , z) is also a solution to the radial equation (4) withthe same parameter λ = λmℓ. We introduce another solutionR

(1)mℓ (c , z), which is a constant multiple of Smℓ(c , z), and behaves like

R(1)mℓ (c , z) ∼

1

czcos

(cz − ℓ+ 1

2π

)as z → i∞. (6)

Notice that z → i∞ corresponds the limit ξ → ∞, and

cz = −iak · i sinh ξ = k · a sinh ξ ∼ kr (ξ → ∞).

Thus (6) means R(1)mℓ (c , z) behaves like usual spherical Bessel

function at infinity. We call R(1)mℓ (c , z) the radial spheroidal wave

function of the first kind.


Radial spheroidal wave function

Notice that the left hand side of the equation (4) preserves the parity

of f . Actually, R(1)mℓ (c , z) is an even function if ℓ−m is even, and an

odd function if ℓ−m is odd (the same as Pmℓ (z)). Then, there exists

a non-trivial solution to (4) which has the opposite parity to that of

R(1)mℓ (c , z). We denote such solution by R

(5)mℓ (c , z) (R

(j)mℓ(c , z) for

j = 2, 3, 4 are already defined in the handbook of Abramowitz andStegun). We normalize R

(5)mℓ (c , z) so that there exists a constant δmℓ,k

such that

R(5)mℓ (c , z) ∼

1

czcos

(cz − ℓ+ 1

2π + δmℓ,k

)as z → i∞. (7)


Matching condition

Let us consider the requirement (c). Notice that in the spheroidalcoordinate

x1 = a cosh ξ sin η cosϕ,



the two coordinates (ξ, η, ϕ) and (−ξ, π − η, ϕ) give the same pointx . Thus (c) implies

f (ξ)g(η) = f (−ξ)g(π − η).

Since z = i sinh ξ and w = cos η, the above condition requires f andg have the same parity with respect to z and w , respectively.


The generalized eigenfunctions for −∆

In summary, we obtain generalized eigenfunctions for −∆ witheigenvalue k2

u = Smℓ(−iak , i sinh ξ)R(1)mℓ (−iak , cos η) cosmϕ,

u = Smℓ(−iak , i sinh ξ)R(1)mℓ (−iak , cos η) sinmϕ,

m = 0, 1, 2, . . . ,

ℓ = m,m + 1,m + 2, . . . .

As an application, we give some level surfaces of an eigenfunction for−∆ in a flattened ellipsoid with the Dirichlet boundary conditions.


Level surfaces of a Dirichlet eigenfunction










Idealized Tonomura model

We consider a ring in R3,

R = {x ∈ R3 | |x | = a, x3 = 0}

in which a quantized magnetic fluxis enclosed. This is an idealizedmodel to the experiment by Tono-mura et al. Actually, we can obtainthe explicit generalized eigenfunc-tions for this model, by using theoblate spheroidal coordinate.



The corresponding Hamiltonian is

H =

(1

i∇ − A

)2

on R3,

where A ∈ C∞(R3 \ R ;R3) is the magnetic vector potential satisfying

∇× A = 0 in R3 \ R ,∫D

(∇× A) · n dS =

∫∂D

A · dℓ = π

for any small disc D pierced by the ring R , where n is the unit normalvector on D (the direction of n is appropriately fixed). Actually wecan take A satisfying the above conditions and the support A isbounded in R3 (we assume this condition below).



Take x0 ∈ R3 sufficiently large, and consider a phase function definedby the line integral

Φ(x) = exp

(i

∫ x

x0

A · dℓ).

The function Φ is two-valued, since

exp

(i

∫∂D

A · dℓ)

= e iπ = −1

for any small disc D pierced by R . Moreover,

Φ

(1

i∇)Φ−1u =

(1

i∇− A

)u.



Thus we have the intertwining relation

Hu = Φ(−∆)Φ−1u.

We put v = Φ−1u. Then,

Hu = k2u ⇔ −∆v = k2v ,

which is the Helmholtz equation. But v is a two-valued function inthe sense that v(x) changes the sign when x moves along an edge ofa small disc pierced by the ring R .



We need the solution for −∆v = k2v , by putting v = f (ξ)g(η)h(ϕ).The equations for f , g , h are the same as before, but we require

(a) g(η) is finite at η = 0, π,

(b) h(ϕ) has period 2π, and

(c)’ v(x) changes the sign when x moves along an edge of a smalldisc pierced by the ring R .

The requirement (c)’ is equivalent to the condition

f (ξ)g(η) = −f (−ξ)g(π − η).

Thus f and g have opposite parities with respect to z and w ,respectively.



In summary, H has generalized eigenfunctions with energy k2

u = Φ · Smℓ(−iak , i sinh ξ)R(5)mℓ (−iak , cos η) cosmϕ,

u = Φ · Smℓ(−iak , i sinh ξ)R(5)mℓ (−iak , cos η) sinmϕ,

m = 0, 1, 2, . . . ,

ℓ = m,m + 1,m + 2, . . . .

As an application, we give some level surfaces for an eigenfunction ofH in a flattened ellipsoid with Dirichlet boundary conditions.


Level surfaces of a Dirichlet eigenfunction for H










Plane wave expansion formula

In order to develop the scattering theory, we use the plane waveexpansion formula in spheroidal coordinate, which is given in theFlammer’s book.

Proposition 3We use the oblate spheroidal coordinate in x-space and the sphericalcoordinate in p-space, that is,

x1 = a cosh ξ sin η cosϕ,



p1 = k sin τ cosψ,

p2 = k sin τ sinψ,

p3 = k cos τ.


Plane wave expansion formula

Proposition (continued)

Then, we have

e ix ·p =∞∑

m=0

∞∑ℓ=m

i ℓc2ℓ,mR(1)mℓ (−iak , i sinh ξ)

·Smℓ(−iak , cos η)Smℓ(−iak , cos τ) · cosm(ϕ− ψ).

Here, the normalization constants cℓ,m is the one in the Rayleigh’sformula.

Notice that R(1)mℓ (−iak , i sinh ξ) and jℓ(kr) has the same asymptotics

as ξ → ∞ (r → ∞).


Scattering theory in the spheroidal coordinate

Remind that the phase shifts δmℓ,k are introduced as follows.

R(1)mℓ (−iak , i sinh ξ) ∼ 1

krcos

(kr − (ℓ+ 1)π

2

),

R(5)mℓ (−iak , i sinh ξ) ∼ 1

krcos

(kr − (ℓ+ 1)π

2+ δmℓ,k

),

as r → ∞. In this case, δmℓ,k depends on both ℓ and m (in the case ofradial V , it depends only on ℓ). Then the scattering amplitude iscalculated as follows.


Scattering theory in the spheroidal coordinate

Theorem 1We introduce the spherical coordinate (τ, ψ) in S2 as

ω = (sin τ cosψ, sin τ sinψ, cos τ),

ω′ = (sin τ ′ cosψ′, sin τ ′ sinψ′, cos τ ′).

Then, the scattering amplitude with energy k2 for the pair H andH0 = −∆ is

f (k2;ω, ω′)

=1

2ik

∞∑m=0

∞∑ℓ=m

(e2iδmℓ,k − 1)c2ℓ,m

·Smℓ(−iak , cos τ)Smℓ(−iak , cos τ ′) cos(m(ϕ− ψ)).


Numerical calculation of the scattering wave

The plane wave expansion formula is also useful in the calculation ofthe plane wave scattered by the Tonomura ring. The incident planewave with momentum p in the perturbed system is described as

W−eix ·p = Φ ·

∞∑m=0

∞∑ℓ=m

i ℓc2ℓ,meiδmℓ,kR

(5)mℓ (−iak , i sinh ξ)

·Smℓ(−iak , cos η)Smℓ(−iak , cos τ)cos(m(ϕ− ψ)),

where Φ is the (two-valued) gauge function used in the constructionof the vector potential A.



There is ambiguity of the choice of the gauge function Φ. Forsimplicity, we take Φ = 1, then the wave function satisfies theboundary condition

u(x1, x2,+0) = −u(x1, x2,−0),

∂u

∂x3(x1, x2,+0) = − ∂u

∂x3(x1, x2,−0)

for x21 + x22 < a2. Thus the wave function might have discontinuityon the disc enclosed by the ring R . In the real experiment, we canobserve only the square of the absolute value of the wave function, asthe hitting probability of the scattered particles. So, this is notessential.



According to the numerical calculation, the phase shift δmℓ,k decaysvery rapidly as ℓ→ ∞. If δmℓ,k is very small, then we have

e iδmℓ,kR

(5)mℓ (−iak , i sinh ξ) ∼ R

(1)mℓ (−iak , i sinh ξ).

Taking the difference with the usual plane wave expansion, we have

W−eix ·p

∼ e ix ·p +∑

δmℓ,k : not small

i ℓc2ℓ,m

(e iδ

mℓ,kR

(5)mℓ − R

(1)mℓ

)(−iak , i sinh ξ)

·Smℓ(−iak , cos η)Smℓ(−iak , cos τ)cos(m(ϕ− ψ)).

This approximation can greatly unburden the numerical calculation.



For the calculation of the phase shift δmℓ,k , we need another spheroidal

function R(2)mℓ (c , z), which has the asymptotics

R(2)mℓ (c , z) ∼

1

czsin

(cz − ℓ+ 1

2π

)as z → i∞.

Then we have

R(5)mℓ (c , z) = cos(δmℓ,k)R

(1)mℓ (c , z)− sin(δmℓ,k)R

(2)mℓ (c , z),

δmℓ,k =

arctan(R

(1)mℓ (c , 0)/R

(2)mℓ (c , 0)

)(l −m: even),

arctan((R

(1)mℓ )

′(c , 0)/(R(2)mℓ )

′(c , 0))

(l −m: odd).



Fortunately, Wolfram Mathematica knows how to calculate R(1)mℓ and

R(2)mℓ . We give here the tables of |e iδmℓ,k − 1| for several l and m, in the

case a = 1 and k = 1.

l\m 0 1 2 30 0.610919 - - -1 0.079627 0.105036 - -2 0.001266 0.001833 0.006314 -3 0.000009 0.000012 0.000029 0.000163

Table: |e iδmℓ,k − 1| for a = 1 and k = 1.

Thus, taking only the term for (ℓ,m) = (0, 0) is not so badapproximation.



We assume a = 1 and the incident direction is x1-direction(p = (1, 0, 0)), so k = 1, τ = π/2, ψ = 0. The approximation of theincident plane wave is

W−eix1 ∼ e ix1 + c20,0

(e iδ

00,1R

(5)00 − R

(1)00

)(−i , i sinh ξ)

·S00(−i , cos η)S00(−i , 0).

From the next page, we shall exhibit the time propagation of theincident plane wave, by plotting the imaginary part of the wavefunction on xy -plane and xz-plane.


Time propagation of the horizontal plane wave

Figure: The imaginary part of the incident plane wave on xy -plane.Takuya MINE (KIT) Spheroidal coordinate Lab Seminar 52 / 81














Figure: The imaginary part of the incident plane wave on xz-plane.Takuya MINE (KIT) Spheroidal coordinate Lab Seminar 59 / 81














Next we assume a = 1 and the incident direction is x3-direction(p = (0, 0, 1)), so k = 1, τ = 0, ψ = 0. The approximation of theincident plane wave is

W−eix3 ∼ e ix3 + c20,0

(e iδ

00,1R

(5)00 − R

(1)00

)(−i , i sinh ξ)

·S00(−i , cos η)S00(−i , 1).

From the next page, we shall again exhibit the time propagation ofthe incident plane wave, by plotting the imaginary part of the wavefunction on xy -plane and xz-plane.


Time propagation of the vertical plane wave




























Remaining problems

(1) What we exhibited here are the imaginary part of the wavefunctions, not the interference pattern in the real experiment.How should we take the physical parameters?

(2) We need more technique for the numerical calculation.Sometimes results by Mathematica are unreliable...

Figure: The graph of (R(2)11 )′(−i , ix) by Mathematica?

Formulas in Flammer’s book might help us.


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Computation of the scattering amplitude in the spheroidal coordinates … · 2015-10-12 · Computation of the scattering amplitude in the spheroidal coordinates Takuya MINE Kyoto