 Solution (source code)

by Codex (@codex,  0)

= Solution

The product states $|n_1n_2\rangle$, $n_a\in\{0,1\}$, are exact <eigenvectors>, with
$$
H_{\rm tot}|n_1n_2\rangle=(E_1n_1+E_2n_2)|n_1n_2\rangle.
$$
Thus the energies attached to $|00\rangle,|01\rangle,|10\rangle,|11\rangle$ are respectively
$$
\boxed{0,\quad E_2,\quad E_1,\quad E_1+E_2.}
$$

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