Solution
= Solution
The unperturbed ground state is $|0,0\rangle$ with energy $\hbar\omega$. Since
$$
H_{\rm int}|0,0\rangle
=\frac{\lambda\hbar}{2\omega}|1,1\rangle,
$$
the <first-order energy correction> vanishes and <first-order nondegenerate perturbation theory> gives
$$
\boxed{|\Omega\rangle=|0,0\rangle-\frac{\lambda}{4\omega^2}|1,1\rangle+O(\lambda^2),
\qquad
E_\Omega=\hbar\omega+O(\lambda^2).}
$$