- #1

raul_l

- 105

- 0

## Homework Statement

Hello. I'd like to solve this: [tex] -\frac{\hbar^2}{2m}\nabla^2 \Psi(r,\theta,\phi) -U(r) \Psi(r,\theta,\phi) = E\Psi(r,\theta,\phi) [/tex]

## Homework Equations

## The Attempt at a Solution

I can separate the variables, but that's about it.

[tex] \frac{1}{R(r)} \frac{d}{dr}(r^2 \frac{d}{dr}R(r))+\frac{2mEr^2}{\hbar^2} + \frac{2m\gamma r}{\hbar^2} = C_r [/tex]

[tex] \frac{1}{F(\phi)} \frac{d^2}{d\phi^2} F(\phi) = C_\phi [/tex]

[tex] -C_r-\frac{sin\theta}{P(\theta)} \frac{d}{d\theta} (sin\theta \frac{d}{d\theta} P(\theta))=C_\phi [/tex]

The answer should be something like this but I don't know how to get there.

[tex] \psi_{nlm}(r,\vartheta,\varphi)=\sqrt{{\left(\frac{2}{n a_0} \right)}^3\frac{(n-l-1)!}{2n(n+1)!}}e^{-\rho /2} \rho^{1} L_{n-l-1}^{2l+1}(\rho)\cdot Y_{lm}(\vartheta, \varphi) [/tex]

If somebody could offer me any idea on how to proceed that would be great. I'm not even sure of how to choose the righ boundary conditions.