Solving using schrodinger equation techniques

[amnon_willi]

Hello,
I have problem I wish to solve, and I wonder if anyone already delt with it when solving the Schrödinger 2D equation.

say E(x,y) is a scalar field function that complies with

( \frac{d}{dx}²+\frac{d}{dy}² ) *E(x,y)+k(x,y)*E(x,y)=k1*E(x,y)


where k(x,y)={k2 for x²+y²<R² and 0 otherwise}, i.e. a tube potential.
All is known but E(x,y).
I think it can be examined as a 2D Schrödinger equation, even thow the eigenvalue k1 is known.

How can I get to start finding the solutions of this equation?
Can I expect to know how many are there? - one, two, many?

 

[StatusX]

At least in 3D, a rotationally invariant system is best dealt with in spherical coordinates, expanding in eigenfunctions of angular momentum and then solving the radial equation, which will be an ODE. In 2D, the "spherical harmonics" are just the functions e^{im \phi} where m is an integer. So try writing f_m(r,\phi) = u_m(r) e^{i m \phi}, which should give you an ordinary differential equation for u_m(r). Then the general solution will be a linear combination of the f_m, ie, you should expect a linearly independent solution for each integer m.

This is a little messier in 2D than in 3D, and the equation will have a first derivative term that doesn't appear in the ordinary Schrödinger equation. In fact, the solutions u_m(r,\phi) are probably going to be Bessel functions.

 

[per.sundqvist]

Numerically you could use pdetool within MATLAB (type: pdetool at the MATLAB prompt), where you could solve this eigen value problem in 2D (Its not that you know or could specify k1, you will get it from the numeric solution). Other better program is COMSOL multiphysics.

This problem looks like cylinder symmetry, but depend on weather k(x,y)=f(r) or not. In that case you could obtain an effective 1D eigen value problem in Psi(r), using the angular quantum quantum number m as specified in the other post here.