A three-dimensional oscillator has Hamiltonian

$$H=\frac{1}{2 m}\left(\hat{p}_{1}^{2}+\hat{p}_{2}^{2}+\hat{p}_{3}^{2}\right)+\frac{1}{2} m \omega^{2}\left(\alpha^{2} \hat{x}_{1}^{2}+\beta^{2} \hat{x}_{2}^{2}+\gamma^{2} \hat{x}_{3}^{2}\right),$$

where the constants $m, \omega, \alpha, \beta, \gamma$ are real and positive. Assuming a unique ground state, construct the general normalised eigenstate of $H$ and give a formula for its energy eigenvalue. [You may quote without proof results for a one-dimensional harmonic oscillator of mass $m$ and frequency $\omega$ that follow from writing $\hat{x}=(\hbar / 2 m \omega)^{1 / 2}\left(a+a^{\dagger}\right)$ and $\left.\hat{p}=(\hbar m \omega / 2)^{1 / 2} i\left(a^{\dagger}-a\right) .\right]$

List all states in the four lowest energy levels of $H$ in the cases:

(i) $\alpha<\beta<\gamma<2 \alpha$;

(ii) $\alpha=\beta$ and $\gamma=\alpha+\epsilon$, where $0<\epsilon \ll \alpha$.

Now consider $H$ with $\alpha=\beta=\gamma=1$ subject to a perturbation

$$\lambda m \omega^{2}\left(\hat{x}_{1} \hat{x}_{2}+\hat{x}_{2} \hat{x}_{3}+\hat{x}_{3} \hat{x}_{1}\right),$$

where $\lambda$ is small. Compute the changes in energies for the ground state and the states at the first excited level of the original Hamiltonian, working to the leading order at which nonzero corrections occur. [You may quote without proof results from perturbation theory.]

Explain briefly why some energy levels of the perturbed Hamiltonian will be exactly degenerate. [Hint: Compare with (ii) above.]