(adsbygoogle = window.adsbygoogle || []).push({}); 1. The problem statement, all variables and given/known data

A particle with mass m moves in 3-dimensions in the potential [tex]V(x,y,z)=\frac{1}{2}m\omega^{2}x^{2}[/tex]. What are the allowed energy eigenvalues?

2. Relevant equations

3. The attempt at a solution

The Hamiltonian is given by [tex]H=\frac{P^{2}}{2m}+\frac{1}{2}m\omega^{2}X^{2}[/tex] where P is the momentum operator in three dimensions. Projecting this into the coordinate basis, we have [tex]\left( -\frac{\hbar^{2}}{2m}\frac{d^{2}}{dx^{2}}+\frac{1}{2}m\omega^{2}x^{2}\right)\psi = \left(E + \frac{\hbar^{2}}{2m}\left( \frac{d^{2}}{dy^{2}}+\frac{d^{2}}{dz^{2}}\right) \right)\psi [/tex]

The left side is now in the form of a simple harmonic oscillator. However, I am stuck after this point.

**Physics Forums | Science Articles, Homework Help, Discussion**

Dismiss Notice

Join Physics Forums Today!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

# Homework Help: Particle in harmonic oscillator potential

**Physics Forums | Science Articles, Homework Help, Discussion**