Consider the 3-dimensional oscillator with Hamiltonian

$$H=-\frac{\hbar^{2}}{2 m} \nabla^{2}+\frac{m \omega^{2}}{2}\left(x^{2}+y^{2}+4 z^{2}\right)$$

Find the ground state energy and the spacing between energy levels. Find the degeneracies of the lowest three energy levels.

[You may assume that the energy levels of the 1-dimensional harmonic oscillator with Hamiltonian

$$H_{0}=-\frac{\hbar^{2}}{2 m} \frac{d^{2}}{d x^{2}}+\frac{m \omega^{2}}{2} x^{2}$$

$\left.\operatorname{are}\left(n+\frac{1}{2}\right) \hbar \omega, n=0,1,2, \ldots .\right]$