8.11 Consider a two-dimensional isotropic harmonic oscillator having the Hamiltonian
\begin{equation}
H_0 = -\frac{\hbar^2}{2m} \left( \frac{\partial^2}{\partial x^2} + \frac{\partial^2}{\partial y^2} \right) + \frac{1}{2}k(x^2 + y^2)
\end{equation}
so that its energy levels are $E_n = \hbar \omega (n+1)$, with $\omega = (k/m)^{1/2}$, $n = n_x + n_y$ and $n_x, n_y = 0, 1, \dots$. Assume that a perturbation $H' = \lambda xy$ is added, where $\lambda$ is constant. Find the first-order modifications of the energy of the ground state and of the first excited state.
