(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.

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# Particle in harmonic oscillator potential

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