Energy levels from the Schrodinger eqn

[Brewer]

Homework Statement

I'm just wondering how you get the energy levels from the Schrödinger equation. I've got the equation in the form H(psi) = E(psi) with all the H expanded out, I just don't know how to calculate the energy levels from it. Its probably something I did once know how to do, but for the time being it has slipped my mind. Any help would be appreciated.

In the example I'm looking at (and following to an extent) its a particle in a uniform magnetic field that's perpendicular to the motion of the particle, but with no potential amd he suddenly says that its clearly a harmonic oscillator and that the energy levels are obvious the normal value for such.

In our question we have a potential that I have dealt with (succesfully I think) and I've followed up until this point that I mentioned. But I can't see how he made the step that he did. I'm pretty sure that this is just a standard method, but I can provide the specific examples if required for this (my tex isn't great, so I'm trying to avoid it where possible!).

Thanks for any help

Homework Equations

The Attempt at a Solution

 

[lzkelley]

To find the energy levels you need to know the potential - your teacher is saying that the potential energy goes like V=kx^2 -> but i don't understand how he's coming to that conclusion either.