A particle in a 2d circle with potential

[Bokul]

Hello,

What would be the right approach to solve for a particle's wavefunction/ energy eigenvalues inside of a 2d cicrle with a potential V(r) where r is the radial distance of a particle from the center of the circle? V(r) is known and is some sort of a well potential going to infinity at R (circle's radius) and to 0 at 0.

 

[vanhees71]

I'd start to look for the right coordinates according to the given symmetries of the system. It's obvious that for your problem the best choice are plane polar coordinates. Then you write down the time-independent Schrödinger equation (i.e., the eigenvalue problem for the Hamiltonian) in these coordinates and solve for the given boundary conditions.

The good thing with polar coordinates is that the 2D Laplacian separates, i.e., you find the energy eigenfunctions through the ansatz

$$\psi(r,\phi)=R(r) \Phi(\phi).$$

 

[Bokul]

Thx, but, I guess, I asked my question too far from its main point. Here what it actually is: once I've applied the separation of variables, I will end up with a 2nd order differential equation for r with V(r) inside of it. And I don't know how to solve it. That is my main problem and, therefore, a question for you.