# Numerical solution of Schrödinger equation

I want to solve the following differential equation numerically. The geometry of the problem is as shown below. Electron 1 is located on the inner ring of radius $R_1$ and electron 2 is located on the outer ring of radius $R_2$ and are vertically separated by a distance $d$ given by

$d^2(\phi_1,\phi_2;R_1,R_2,\alpha)=[R_2 \cos \phi_2-R_1 \cos \phi_1]^2+[R_2 \cos \alpha-R_2+R_2 \cos\alpha \sin\phi_2-R_1 \sin \phi_1]^2+R_2^2 \sin^2 \alpha [1+\sin \phi_2]^2$

1/Subscript[R, 1]^2 D[ψ[Subscript[ϕ, 1], Subscript[ϕ,2]], {Subscript[ϕ, 1], 2}] -
1/Subscript[R, 2]^2 D[ψ[Subscript[ϕ, 1], Subscript[ϕ,2]], {Subscript[ϕ, 2], 2}] +
1/d ψ[Subscript[ϕ, 1], Subscript[ϕ, 2]] =
e ψ[Subscript[ϕ, 1], Subscript[ϕ, 2]]


where $\phi_i\in[0,2\pi)$ with periodic boundary conditions $\Psi(0,\phi_2)=\Psi(2\pi,\phi_2)$,$\Psi(\phi_1,0)=\Psi(\phi_1,2\pi)$

• ...and you've already looked at the docs for NDSolve[]? – J. M.'s discontentment Oct 29 '15 at 15:38
• @J.M I'm trying it! – user0322 Oct 29 '15 at 15:40
• You are trying to solve the time-independent Schrodinger equation and it is an eigenvalue problem. One way is to write the equation in the difference form and diagonalize the Hamiltonian. Here is an example for 1D version: mathematica.stackexchange.com/q/32293/1364 I think NDSolve maybe more useful for time-dependent Schrodinger equation, which is a PDE problem. – xslittlegrass Oct 29 '15 at 16:02
• Please edit your question to specify d. If you wish a numerical solution, also specify R1 and R2. Finally, please provide your equation in Mathematica format as well. Because this appears to be an eigenvalue problem, there is no need to specify the normalization of Psi. Thanks. – bbgodfrey Oct 30 '15 at 16:43
• You may want to look at NDEigensystem which can solve eigensystem numerically for a differential operator. I have added a 1D example in the same link. – xslittlegrass Oct 30 '15 at 22:34

You have not specified the region of interest. But suppose it is that region between the two rings (say of radius 3 & 5 ), with appropriate units taken into account, we examine the first eigenfunction using

 {vals, funs} =      NDEigensystem[{-Laplacian[u[x, y], {x, y}],
DirichletCondition[u[x, y] == 0., x == 0]}, u[x, y],
Element[{x, y}, RegionSymmetricDifference[Disk[{0, 0}, 5],
Disk[{0, 0}, 3]]], 1]

Plot3D[funs, Element[{x, y}, RegionSymmetricDifference[Disk[{0, 0}, 5],
Disk[{0, 0}, 3]]], PlotRange -> All, PlotLabel -> vals,
PlotTheme -> "Scientific"]


Visualize the eigenfunction below. If this is not what you are after, then amend the first part using {x, y} \[Element] Disk[]

• I am using V8 when I try to run it shows me error message. – user0322 Oct 31 '15 at 10:14
• No error for 10.3, probably the newer version has upgraded features – thils Oct 31 '15 at 10:30
• I got the result {{0.0427379}, {InterpolatingFunction[{{-5., 5.}, {-5., 5.}}, <>][x,y]}} but takes too much time for the plot. – user0322 Oct 31 '15 at 16:11
• That's gd. Computation time is a bit long here as well. – thils Oct 31 '15 at 22:23
• It's strange I couldn't see the plot you posted, still running. – user0322 Nov 1 '15 at 13:26