(i) Let $a$ and $a^{\dagger}$ be the annihilation and creation operators, respectively, for a simple harmonic oscillator whose Hamiltonian is

$$H_{0}=\omega\left(a^{\dagger} a+\frac{1}{2}\right),$$

with $\left[a, a^{\dagger}\right]=1$. Explain how the set of eigenstates $\{|n\rangle: n=0,1,2, \ldots\}$ of $H_{0}$ is obtained and deduce the corresponding eigenvalues. Show that

(ii) Consider a system whose unperturbed Hamiltonian is

$$H_{0}=\left(a^{\dagger} a+\frac{1}{2}\right)+2\left(b^{\dagger} b+\frac{1}{2}\right)$$

where $\left[a, a^{\dagger}\right]=1, \quad\left[b, b^{\dagger}\right]=1$ and all other commutators are zero. Find the degeneracies of the eigenvalues of $H_{0}$ with energies $E_{0}=\frac{3}{2}, \frac{5}{2}, \frac{7}{2}, \frac{9}{2}$ and $\frac{11}{2}$.

The system is perturbed so that it is now described by the Hamiltonian

$$H=H_{0}+\lambda H^{\prime},$$

where $H^{\prime}=\left(a^{\dagger}\right)^{2} b+a^{2} b^{\dagger}$. Using degenerate perturbation theory, calculate to $O(\lambda)$ the energies of the eigenstates associated with the level $E_{0}=\frac{9}{2}$.

Write down the eigenstates, to $O(\lambda)$, associated with these perturbed energies. By explicit evaluation show that they are in fact exact eigenstates of $H$ with these energies as eigenvalues.

$$\begin{aligned} & a|0\rangle=0 \\ & a|n\rangle=\sqrt{n}|n-1\rangle, \quad n \geqslant 1, \\ & a^{\dagger}|n\rangle=\sqrt{n+1}|n+1\rangle, \quad n \geqslant 0 . \end{aligned}$$