Schrodinger Equation for a central 2D potential

[EliotHijano]

Hello,
I would like to ask something about central potentials. When I am working in 3D, I haven´t got any problem solving the Schrödinger equation since I use the following change of variables:
$$-\frac{\hbar^{2}}{2m}\nabla^{2}\Psi+V(r)\Psi=E\Psi$$
$$\Psi=\frac{\chi}{r}$$
With this change of variables, I can solve and understand the equation, but when I am working in 2D( usig r and $$\varphi$$) I can't solve the equation because the laplacian is different. The concrete potential distribution I want to study is the following:
$$V(\vec{r})=0 \stackrel{if}{\rightarrow} r<a$$
$$V(\vec{r})=\infty \stackrel{if}{\rightarrow} r>a$$
And I would like to solve the equation for the first and the second energy levels of the system. I would appreciate some tips, thank you.

 

[Meir Achuz]

It's not as easy as 3D. The bound states are Bessel functions:
$$\psi=J_m(kr)\cos(m\theta)$$, with the energy determined by
$$J_m(ka)=0$$.

 

[EliotHijano]

Thank you very much clem, that will be really useful for me.