Computation of the scattering amplitude in the spheroidal coordinates Takuya MINE Kyoto Institute of Technology 12 October 2015 Lab Seminar at 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 can solve the Helmholtz equation by separation of variables. Today we consider the three-dimensional case, and study the harmonic analysis in the spheroidal coordinates. As an application, we try to calculate the wave functions and the scattering amplitude for the idealized Tonomura model. Takuya MINE (KIT) Spheroidal coordinate Lab Seminar 2 / 81
Review: Spherical coordinate As is well-known, the spherical coordinate in R 3 is defined as follows. x 1 = r sin θ cos ϕ, x 2 = r sin θ sin ϕ, r 0, 0 η π, π < ϕ π. x 3 = r cos θ, In the spherical coordinate, the Laplacian is written as = 1 [ sin θ ( r 2 ) + ( sin θ ) r 2 sin θ r r θ θ + 1 sin θ ] 2. ϕ 2 Takuya MINE (KIT) Spheroidal coordinate Lab Seminar 3 / 81
Review: Spherical coordinate The surface r = const. is a sphere, θ = const. a cone, ϕ = const. a half-plane. Takuya MINE (KIT) Spheroidal coordinate Lab Seminar 4 / 81
Review: Spherical coordinate The Helmholtz equation u = k 2 u is equivalent to d 2 f dr + 2 ( ) df 2 r dr + k 2 l(l + 1) f = 0, ( r 2 1 d sin θ dg ) ) + (l(l + 1) m2 sin θ dθ dθ sin 2 g = 0, θ d 2 h dϕ = 2 m2 h, where u = f (r)g(θ)h(ϕ), and l, m are the separation constants. Takuya MINE (KIT) Spheroidal coordinate Lab Seminar 5 / 81
Review: Spherical coordinate We have the solutions finite at r = 0, θ = 0, π, not diverging as r, and periodic with respect to ϕ, for l = 0, 1, 2,..., and m = 0, 1,..., l. The solutions are written as u = j l (kr)p m l (cos θ) cos(mϕ), u = j l (kr)p m l (cos θ) sin(mϕ) (m 0). Here j l (z) = π/(2z)j l+1/2 (z) is the spherical Bessel function, P m l is the associated Legendre function. The number l is called the azimuthal quantum number, and m the magnetic quantum number. The completeness of these solutions are guaranteed by the following formula. Takuya MINE (KIT) Spheroidal coordinate Lab Seminar 6 / 81
Review: Spherical coordinate 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, Then, we have x = (r sin θ cos ϕ, r sin θ sin ϕ, r cos θ), p = (k sin τ cos ψ, k sin τ sin ψ, k cos τ). e ix p = c l,m = l [ ] i l cl,mj 2 l (kr)pl m (cos θ)pl m (cos τ) cos(m(ϕ ψ)), l=0 m=0 (l m)! (2 δ 0,m )(2l + 1) (l + m)!. Takuya MINE (KIT) Spheroidal coordinate Lab Seminar 7 / 81
Review: Spherical coordinate The Fourier inversion formula is described as follows. 1 u(x) = e ix p û(p)dp (2π) 3/2 R 3 l = i l c l,m Pl m (cos θ) cos(mϕ) j l (kr)u l,m,c (k)k 2 dk l=0 m=0 + l=0 m=1 l i l c l,m Pl m (cos θ) sin(mϕ) 0 0 j l (kr)u l,m,s (k)k 2 dk, u l,m,c (k) = u l,m,s (k) = 1 (2π) 3/2 1 (2π) 3/2 π π π 0 π π π 0 c l,m P l,m (cos τ) cos(mψ)û(p) sin τdτdψ, c l,m P l,m (cos τ) sin(mψ)û(p) sin τdτdψ. Takuya MINE (KIT) Spheroidal coordinate Lab Seminar 8 / 81
Review: Phase shift and scattering amplitude Let us explain how to calculate the scattering amplitude for the Schrödinger operator H = + V (r) with a radial potential V (r) decaying sufficiently fast at. The free operator H 0 = has generalized eigenfunctions with eigenvalue k 2 (k > 0) u = j l (kr)p m l (cos θ) cos(mϕ), u = j l (kr)p m l (cos θ) sin(mϕ) (m 0), and the radial part j l (kr) has the asymptotics j l (kr) 1 ( ) kr cos (l + 1)π kr (r ). 2 Takuya MINE (KIT) Spheroidal coordinate Lab Seminar 9 / 81
Review: Phase shift and scattering amplitude If V (r) decays sufficiently fast at, we can prove that the perturbed operator H has generalized eigenfunctions with eigenvalue k 2 (k > 0) u = u l,k (r)p m l (cos θ) cos(mϕ), u = u l,k (r)p m l (cos θ) sin(mϕ) (m 0) for any l = 0, 1, 2,... and m = 0, 1,..., l, and u l,k (r) has the asymptotics u l,k (r) 1 ( ) kr cos (l + 1)π kr + δ l,k (r ), 2 where δ l,k is a real constant called the scattering phase shift. Takuya MINE (KIT) Spheroidal coordinate Lab Seminar 10 / 81
Review: Phase shift and scattering amplitude The wave operators W ± are defined by W ± u = s-lim t ± eith e ith 0 u. Notice that the solution to the Schrödinger equation i ψ(t, x) = Hψ(t, x), t ψ(0, x) = u(x) is ψ(t, x) = e ith u(x). The above definition means the solutions e ith W ± u behave like the free solutions e ith 0 u as t ±, respectively. Takuya MINE (KIT) Spheroidal coordinate Lab Seminar 11 / 81
Review: Phase shift and scattering amplitude The scattering operator S is defined by S = W +W. By the conservation of energy, the operator FSF (F is the Fourier transform) is decomposed into the direct integral of the operators S(E) (E > 0) acting on L 2 (S E ), where S E is the energy shell S E = {ξ R 3 ξ 2 = E} = { Eω ω S 2 }. Then the scattering amplitude f (k 2 ; ω, ω ) is defined by the formula (S(k 2 ) I )(ω, ω ) = ki 2π f (k 2 ; ω, ω ) (ω, ω S 2 ). The quantity f (k 2 ; ω, ω ) 2 is called the differential scattering cross section, which is proportional to the ratio of the particles with energy k 2, incident direction ω and final direction ω. Takuya MINE (KIT) Spheroidal coordinate Lab Seminar 12 / 81
Review: Phase shift and scattering amplitude Proposition 2 (Formula for the scattering amplitude) Given the scattering phase shifts δ l,k, the scattering amplitude for the whole operator is given by f (k 2 ; ω, ω ) = 1 l (e 2iδ l,k 1) c 2ik l,mp 2 l m (cos θ)pl m (cos τ) cos(m(ϕ ψ)) l=0 m=0 e 2iδ l,k 1 = (2l + 1)P l (ω ω ). 2ik l=0 The final result depends only on the angle between ω and ω, because of the spherical symmetry of the system (V is radial). Takuya MINE (KIT) Spheroidal coordinate Lab Seminar 13 / 81
Review: Phase shift and scattering amplitude In summary, (1) By the spherical coordinate, we can decompose the scattering problem in the three-dimensional space to the problem in the one-dimensional space (scattering phase shift). (2) By using the plane wave expansion formula, we can sum up the partial scattering data and get the scattering amplitude for the whole operator. This machinery also works in the case of the spheroidal coordinate. There are two spheroidal coordinates (the prolate spheroidal coordinate and the oblate spheroidal coordinate), but today we only use the oblate one. Takuya MINE (KIT) Spheroidal coordinate Lab Seminar 14 / 81
Oblate spheroidal coordinate The oblate spheroidal coordinate is defined as follows. x 1 = a cosh ξ sin η cos ϕ, x 2 = a cosh ξ sin η sin ϕ, x 3 = a sinh ξ cos η, ξ 0, 0 η π, π < ϕ π, where a is a positive constant. This definition is taken from Iwanami mathematical formulas III (there is another formulation). Takuya MINE (KIT) Spheroidal coordinate Lab Seminar 15 / 81
Oblate spheroidal coordinate The surface ξ = const. is a flattened ellipsoid, η = const. a hyperboloid of one sheet, ϕ = const. a halfplane. Takuya MINE (KIT) Spheroidal coordinate Lab Seminar 16 / 81
Laplacian in the oblate spheroidal coordinate The Laplacian is written as follows. u = 1 a 2 cosh ξ sin η(cosh 2 ξ sin 2 η) ( sin η ( cosh ξ u ) + cosh ξ ξ ξ η ) + cosh2 ξ sin 2 η 2 u. cosh ξ sin η ϕ 2 ( sin η u ) η If we put u = f (ξ)g(η)h(ϕ), the Helmholtz equation u = k 2 u is reduced to the following ordinary differential equations. Takuya MINE (KIT) Spheroidal coordinate Lab Seminar 17 / 81
Laplacian in the oblate spheroidal coordinate ( 1 d cosh ξ df ) cosh ξ dξ dξ + (a 2 k 2 cosh 2 ξ µ + m2 1 sin η d dη ( sin η dg dη d 2 h dϕ 2 = m2 h. ) cosh 2 ξ + ( a 2 k 2 sin 2 η + µ m2 sin 2 η Here m, µ are the separation constants. ) f = 0, (1) ) g = 0, (2) (3) Takuya MINE (KIT) Spheroidal coordinate Lab Seminar 18 / 81
Laplacian in the oblate spheroidal coordinate 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,...). Takuya MINE (KIT) Spheroidal coordinate Lab Seminar 19 / 81
Laplacian in the oblate spheroidal coordinate We put λ = µ a 2 k 2, c = iak. By the change of variable z = i sinh ξ in (1), and w = cos η in (2), we have ( d (1 z 2 ) df ) ) + (λ c 2 z 2 m2 f = 0, (4) dz d dw dz ( (1 w 2 ) dg dw 1 z ) 2 + (λ c 2 w 2 m2 1 w 2 ) g = 0. (5) Thus the two equations (4) and (5) are equivalent as equations for complex variables. Especially when c = 0, these equations become the associated Legendre differential equation, which is obtained from the equation for g(θ) in the spherical coordinate by the change of variable w = cos θ. Takuya MINE (KIT) Spheroidal coordinate Lab Seminar 20 / 81
Angular spheroidal wave function By the requirement (a), we need the solutions to (5) finite at w = ±1. For fixed m = 0, 1, 2,..., there are at most countable values of λ s for which the equation (5) has a non-trivial solution finite at w = ±1. We denote such values by λ ml (l = m, m + 1, m + 2,...), and corresponding solutions by S ml (c, w). We call S ml (c, w) the angular spheroidal wave function (of the first kind). When c = 0, S ml (0, w) coincides with the associated Legendre function P m l (w). S ml (c, w) is normalized as 1 1 S ml (c, w) 2 dw = 2 (l + m)! 2l + 1 (l m)!. Takuya MINE (KIT) Spheroidal coordinate Lab Seminar 21 / 81
Radial spheroidal wave function Notice that S ml (c, z) is also a solution to the radial equation (4) with the same parameter λ = λ ml. We introduce another solution R (1) ml (c, z), which is a constant multiple of S ml(c, z), and behaves like R (1) 1 ml (c, z) cz cos ( cz l + 1 2 π Notice that z i corresponds the limit ξ, and ) as z i. (6) 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. Takuya MINE (KIT) Spheroidal coordinate Lab Seminar 22 / 81
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 Pl m (z)). Then, there exists a non-trivial solution to (4) which has the opposite parity to that of R (1) (5) (j) ml (c, z). We denote such solution by R ml (c, z) (R ml (c, z) for j = 2, 3, 4 are already defined in the handbook of Abramowitz and Stegun). We normalize R (5) ml (c, z) so that there exists a constant δm l,k such that ( R (5) 1 ml (c, z) cz cos cz l + 1 ) 2 π + δm l,k as z i. (7) Takuya MINE (KIT) Spheroidal coordinate Lab Seminar 23 / 81
Matching condition Let us consider the requirement (c). Notice that in the spheroidal coordinate x 1 = a cosh ξ sin η cos ϕ, x 2 = a cosh ξ sin η sin ϕ, x 3 = a sinh ξ cos η, the two coordinates (ξ, η, ϕ) and ( ξ, π η, ϕ) give the same point x. 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 and g have the same parity with respect to z and w, respectively. Takuya MINE (KIT) Spheroidal coordinate Lab Seminar 24 / 81
The generalized eigenfunctions for In summary, we obtain generalized eigenfunctions for with eigenvalue k 2 u = S ml ( iak, i sinh ξ)r (1) ml ( iak, cos η) cos mϕ, u = S ml ( iak, i sinh ξ)r (1) ml ( iak, cos η) sin mϕ, 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. Takuya MINE (KIT) Spheroidal coordinate Lab Seminar 25 / 81
Level surfaces of a Dirichlet eigenfunction Takuya MINE (KIT) Spheroidal coordinate Lab Seminar 26 / 81
Level surfaces of a Dirichlet eigenfunction Takuya MINE (KIT) Spheroidal coordinate Lab Seminar 27 / 81
Level surfaces of a Dirichlet eigenfunction Takuya MINE (KIT) Spheroidal coordinate Lab Seminar 28 / 81
Level surfaces of a Dirichlet eigenfunction Takuya MINE (KIT) Spheroidal coordinate Lab Seminar 29 / 81
Level surfaces of a Dirichlet eigenfunction Takuya MINE (KIT) Spheroidal coordinate Lab Seminar 30 / 81
Idealized Tonomura model We consider a ring in R 3, R = {x R 3 x = a, x 3 = 0} in which a quantized magnetic flux is enclosed. This is an idealized model to the experiment by Tonomura et al. Actually, we can obtain the explicit generalized eigenfunctions for this model, by using the oblate spheroidal coordinate. Takuya MINE (KIT) Spheroidal coordinate Lab Seminar 31 / 81
Idealized Tonomura model The corresponding Hamiltonian is H = ( ) 2 1 i A on R 3, where A C (R 3 \ R; R 3 ) is the magnetic vector potential satisfying A = 0 in R 3 \ R, ( A) n ds = A dl = π D for any small disc D pierced by the ring R, where n is the unit normal vector on D (the direction of n is appropriately fixed). Actually we can take A satisfying the above conditions and the support A is bounded in R 3 (we assume this condition below). D Takuya MINE (KIT) Spheroidal coordinate Lab Seminar 32 / 81
Idealized Tonomura model Take x 0 R 3 sufficiently large, and consider a phase function defined by the line integral ( x ) Φ(x) = exp i A dl. x 0 The function Φ is two-valued, since exp ( i ) A dl = e iπ = 1 for any small disc D pierced by R. Moreover, ( ) ( ) 1 1 Φ i Φ 1 u = i A u. D Takuya MINE (KIT) Spheroidal coordinate Lab Seminar 33 / 81
Idealized Tonomura model Thus we have the intertwining relation We put v = Φ 1 u. Then, Hu = Φ( )Φ 1 u. Hu = k 2 u v = k 2 v, which is the Helmholtz equation. But v is a two-valued function in the sense that v(x) changes the sign when x moves along an edge of a small disc pierced by the ring R. Takuya MINE (KIT) Spheroidal coordinate Lab Seminar 34 / 81
Idealized Tonomura model We need the solution for v = k 2 v, 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 small disc 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. Takuya MINE (KIT) Spheroidal coordinate Lab Seminar 35 / 81
Idealized Tonomura model In summary, H has generalized eigenfunctions with energy k 2 u = Φ S ml ( iak, i sinh ξ)r (5) ml ( iak, cos η) cos mϕ, u = Φ S ml ( iak, i sinh ξ)r (5) ml ( iak, cos η) sin mϕ, m = 0, 1, 2,..., l = m, m + 1, m + 2,.... As an application, we give some level surfaces for an eigenfunction of H in a flattened ellipsoid with Dirichlet boundary conditions. Takuya MINE (KIT) Spheroidal coordinate Lab Seminar 36 / 81
Level surfaces of a Dirichlet eigenfunction for H Takuya MINE (KIT) Spheroidal coordinate Lab Seminar 37 / 81
Level surfaces of a Dirichlet eigenfunction for H Takuya MINE (KIT) Spheroidal coordinate Lab Seminar 38 / 81
Level surfaces of a Dirichlet eigenfunction for H Takuya MINE (KIT) Spheroidal coordinate Lab Seminar 39 / 81
Level surfaces of a Dirichlet eigenfunction for H Takuya MINE (KIT) Spheroidal coordinate Lab Seminar 40 / 81
Level surfaces of a Dirichlet eigenfunction for H Takuya MINE (KIT) Spheroidal coordinate Lab Seminar 41 / 81
Plane wave expansion formula In order to develop the scattering theory, we use the plane wave expansion formula in spheroidal coordinate, which is given in the Flammer s book. Proposition 3 We use the oblate spheroidal coordinate in x-space and the spherical coordinate in p-space, that is, x 1 = a cosh ξ sin η cos ϕ, x 2 = a cosh ξ sin η sin ϕ, x 3 = a sinh ξ cos η, p 1 = k sin τ cos ψ, p 2 = k sin τ sin ψ, p 3 = k cos τ. Takuya MINE (KIT) Spheroidal coordinate Lab Seminar 42 / 81
Plane wave expansion formula Proposition (continued) Then, we have e ix p = m=0 l=m i l cl,mr 2 (1) ml ( iak, i sinh ξ) S ml ( iak, cos η)s ml ( iak, cos τ) cos m(ϕ ψ). Here, the normalization constants c l,m is the one in the Rayleigh s formula. Notice that R (1) ml ( iak, i sinh ξ) and j l(kr) has the same asymptotics as ξ (r ). Takuya MINE (KIT) Spheroidal coordinate Lab Seminar 43 / 81
Scattering theory in the spheroidal coordinate Remind that the phase shifts δ m l,k R (1) ml ( iak, i sinh ξ) 1 ( kr cos kr R (5) ml ( iak, i sinh ξ) 1 ( kr cos kr are introduced as follows. (l + 1)π 2 (l + 1)π 2 ), + δ m l,k as r. In this case, δl,k m depends on both l and m (in the case of radial V, it depends only on l). Then the scattering amplitude is calculated as follows. ), Takuya MINE (KIT) Spheroidal coordinate Lab Seminar 44 / 81
Scattering theory in the spheroidal coordinate Theorem 1 We introduce the spherical coordinate (τ, ψ) in S 2 as ω = (sin τ cos ψ, sin τ sin ψ, cos τ), ω = (sin τ cos ψ, sin τ sin ψ, cos τ ). Then, the scattering amplitude with energy k 2 for the pair H and H 0 = is f (k 2 ; ω, ω ) = 1 (e 2iδm l,k 1)c 2 2ik l,m m=0 l=m S ml ( iak, cos τ)s ml ( iak, cos τ ) cos(m(ϕ ψ)). Takuya MINE (KIT) Spheroidal coordinate Lab Seminar 45 / 81
Numerical calculation of the scattering wave The plane wave expansion formula is also useful in the calculation of the plane wave scattered by the Tonomura ring. The incident plane wave with momentum p in the perturbed system is described as W e ix p = Φ m=0 l=m i l cl,me 2 iδm (5) l,k R ml ( iak, i sinh ξ) S ml ( iak, cos η)s ml ( iak, cos τ)cos(m(ϕ ψ)), where Φ is the (two-valued) gauge function used in the construction of the vector potential A. Takuya MINE (KIT) Spheroidal coordinate Lab Seminar 46 / 81
Numerical calculation of the scattering wave There is ambiguity of the choice of the gauge function Φ. For simplicity, we take Φ = 1, then the wave function satisfies the boundary condition u(x 1, x 2, +0) = u(x 1, x 2, 0), u (x 1, x 2, +0) = u (x 1, x 2, 0) x 3 x 3 for x 2 1 + x 2 2 < a 2. Thus the wave function might have discontinuity on the disc enclosed by the ring R. In the real experiment, we can observe only the square of the absolute value of the wave function, as the hitting probability of the scattered particles. So, this is not essential. Takuya MINE (KIT) Spheroidal coordinate Lab Seminar 47 / 81
Numerical calculation of the scattering wave According to the numerical calculation, the phase shift δl,k m decays very rapidly as l. If δl,k m is very small, then we have e iδm l,k R (5) ml (1) ( iak, i sinh ξ) R ( iak, i sinh ξ). Taking the difference with the usual plane wave expansion, we have W e ix p e ix p + δ m l,k : not small i l c 2 l,m ml ( ) e iδm (5) l,k R ml R (1) ml ( iak, i sinh ξ) S ml ( iak, cos η)s ml ( iak, cos τ)cos(m(ϕ ψ)). This approximation can greatly unburden the numerical calculation. Takuya MINE (KIT) Spheroidal coordinate Lab Seminar 48 / 81
Numerical calculation of the scattering wave For the calculation of the phase shift δl,k m, we need another spheroidal function R (2) ml (c, z), which has the asymptotics Then we have R (2) 1 ml (c, z) cz sin ( cz l + 1 ) 2 π as z i. R (5) ml (c, z) = cos(δm l,k)r (1) ml (c, z) sin(δm l,k)r (2) ml (c, z), ( ) arctan R (1) (2) δl,k m ml (c, 0)/R ml = (c, 0) (l m: even), ( ) arctan (R (1) ml ) (c, 0)/(R (2) ml ) (c, 0) (l m: odd). Takuya MINE (KIT) Spheroidal coordinate Lab Seminar 49 / 81
Numerical calculation of the scattering wave Fortunately, Wolfram Mathematica knows how to calculate R (1) ml and R (2) ml. We give here the tables of eiδm l,k 1 for several l and m, in the case a = 1 and k = 1. l\m 0 1 2 3 0 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 iδm l,k 1 for a = 1 and k = 1. Thus, taking only the term for (l, m) = (0, 0) is not so bad approximation. Takuya MINE (KIT) Spheroidal coordinate Lab Seminar 50 / 81
Numerical calculation of the scattering wave We assume a = 1 and the incident direction is x 1 -direction (p = (1, 0, 0)), so k = 1, τ = π/2, ψ = 0. The approximation of the incident plane wave is ( ) W e ix 1 e ix 1 + c0,0 2 e iδ0 (5) 0,1 R 00 R (1) 00 ( i, i sinh ξ) S 00 ( i, cos η)s 00 ( i, 0). From the next page, we shall exhibit the time propagation of the incident plane wave, by plotting the imaginary part of the wave function on xy-plane and xz-plane. Takuya MINE (KIT) Spheroidal coordinate Lab Seminar 51 / 81
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
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 53 / 81
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 54 / 81
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 55 / 81
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 56 / 81
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 57 / 81
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 58 / 81
Time propagation of the horizontal plane wave Figure: The imaginary part of the incident plane wave on xz-plane. Takuya MINE (KIT) Spheroidal coordinate Lab Seminar 59 / 81
Time propagation of the horizontal plane wave Figure: The imaginary part of the incident plane wave on xz-plane. Takuya MINE (KIT) Spheroidal coordinate Lab Seminar 60 / 81
Time propagation of the horizontal plane wave Figure: The imaginary part of the incident plane wave on xz-plane. Takuya MINE (KIT) Spheroidal coordinate Lab Seminar 61 / 81
Time propagation of the horizontal plane wave Figure: The imaginary part of the incident plane wave on xz-plane. Takuya MINE (KIT) Spheroidal coordinate Lab Seminar 62 / 81
Time propagation of the horizontal plane wave Figure: The imaginary part of the incident plane wave on xz-plane. Takuya MINE (KIT) Spheroidal coordinate Lab Seminar 63 / 81
Time propagation of the horizontal plane wave Figure: The imaginary part of the incident plane wave on xz-plane. Takuya MINE (KIT) Spheroidal coordinate Lab Seminar 64 / 81
Time propagation of the horizontal plane wave Figure: The imaginary part of the incident plane wave on xz-plane. Takuya MINE (KIT) Spheroidal coordinate Lab Seminar 65 / 81
Numerical calculation of the scattering wave Next we assume a = 1 and the incident direction is x 3 -direction (p = (0, 0, 1)), so k = 1, τ = 0, ψ = 0. The approximation of the incident plane wave is ( ) W e ix 3 e ix 3 + c0,0 2 e iδ0 (5) 0,1 R 00 R (1) 00 ( i, i sinh ξ) S 00 ( i, cos η)s 00 ( i, 1). From the next page, we shall again exhibit the time propagation of the incident plane wave, by plotting the imaginary part of the wave function on xy-plane and xz-plane. Takuya MINE (KIT) Spheroidal coordinate Lab Seminar 66 / 81
Time propagation of the vertical plane wave Figure: The imaginary part of the incident plane wave on xy-plane. Takuya MINE (KIT) Spheroidal coordinate Lab Seminar 67 / 81
Time propagation of the vertical plane wave Figure: The imaginary part of the incident plane wave on xy-plane. Takuya MINE (KIT) Spheroidal coordinate Lab Seminar 68 / 81
Time propagation of the vertical plane wave Figure: The imaginary part of the incident plane wave on xy-plane. Takuya MINE (KIT) Spheroidal coordinate Lab Seminar 69 / 81
Time propagation of the vertical plane wave Figure: The imaginary part of the incident plane wave on xy-plane. Takuya MINE (KIT) Spheroidal coordinate Lab Seminar 70 / 81
Time propagation of the vertical plane wave Figure: The imaginary part of the incident plane wave on xy-plane. Takuya MINE (KIT) Spheroidal coordinate Lab Seminar 71 / 81
Time propagation of the vertical plane wave Figure: The imaginary part of the incident plane wave on xy-plane. Takuya MINE (KIT) Spheroidal coordinate Lab Seminar 72 / 81
Time propagation of the vertical plane wave Figure: The imaginary part of the incident plane wave on xy-plane. Takuya MINE (KIT) Spheroidal coordinate Lab Seminar 73 / 81
Time propagation of the vertical plane wave Figure: The imaginary part of the incident plane wave on xz-plane. Takuya MINE (KIT) Spheroidal coordinate Lab Seminar 74 / 81
Time propagation of the vertical plane wave Figure: The imaginary part of the incident plane wave on xz-plane. Takuya MINE (KIT) Spheroidal coordinate Lab Seminar 75 / 81
Time propagation of the vertical plane wave Figure: The imaginary part of the incident plane wave on xz-plane. Takuya MINE (KIT) Spheroidal coordinate Lab Seminar 76 / 81
Time propagation of the vertical plane wave Figure: The imaginary part of the incident plane wave on xz-plane. Takuya MINE (KIT) Spheroidal coordinate Lab Seminar 77 / 81
Time propagation of the vertical plane wave Figure: The imaginary part of the incident plane wave on xz-plane. Takuya MINE (KIT) Spheroidal coordinate Lab Seminar 78 / 81
Time propagation of the vertical plane wave Figure: The imaginary part of the incident plane wave on xz-plane. Takuya MINE (KIT) Spheroidal coordinate Lab Seminar 79 / 81
Time propagation of the vertical plane wave Figure: The imaginary part of the incident plane wave on xz-plane. Takuya MINE (KIT) Spheroidal coordinate Lab Seminar 80 / 81
Remaining problems (1) What we exhibited here are the imaginary part of the wave functions, 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. Takuya MINE (KIT) Spheroidal coordinate Lab Seminar 81 / 81