Discuss the quantum mechanics of the one-dimensional harmonic oscillator using creation and annihilation operators, showing how the energy levels are calculated.

A quantum mechanical system consists of two interacting harmonic oscillators and has the Hamiltonian

$$H=\frac{1}{2} \hat{p}_{1}^{2}+\frac{1}{2} \hat{x}_{1}^{2}+\frac{1}{2} \hat{p}_{2}^{2}+\frac{1}{2} \hat{x}_{2}^{2}+\lambda \hat{x}_{1} \hat{x}_{2}$$

For $\lambda=0$, what are the degeneracies of the three lowest energy levels? For $\lambda \neq 0$ compute, to lowest non-trivial order in perturbation theory, the energies of the ground state and first excited state.

[Standard results for perturbation theory may be stated without proof.]