Numerical solution of Schrödinger equation

Question

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 $\Psi(0,\phi_2)=\Psi(2\pi,\phi_2)$,$\Psi(\phi_1,0)=\Psi(\phi_1,2\pi)$

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: http://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
Your kinetic energy term is probably wrong (not positive definite). You're also using notation like E and = that isn't syntactically correct. — Jens, Oct 31 '15 at 22:26
@Jens, The electronic Hamiltonian of the system is H=T+u^-1 where T= T1 + T2 — user0322, Nov 02 '15 at 18:53

score 1 · Answer 1 · answered Oct 31 '15 at 00:35

1

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[]

answered Oct 31 '15 at 00:35

thils

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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 01 '15 at 13:26
Generating the plot was no problem. My code is just a rough guess based on your question, is this what you are after? – thils Nov 01 '15 at 22:00
U using version 10.3? – thils Nov 01 '15 at 22:20
Yes, I'm using 10.3 – user0322 Nov 02 '15 at 08:14
there's a considerable difference the first eigenfunction with this paper http://arxiv.org/abs/1301.0649 – user0322 Nov 02 '15 at 19:22
Not sure how to respond......as my naïve code differs hugely from the problem specified in the paper....my response is only a guess – thils Nov 09 '15 at 22:12

Numerical solution of Schrödinger equation

1 Answers1