= Solution
{c}
Inverting the definitions of the <creation and annihilation operators> gives
$$
X_j=\sqrt{\frac{\hbar}{2m\omega}}(a_j+a_j^*),\qquad
P_j=-i\sqrt{\frac{m\omega\hbar}{2}}(a_j-a_j^*).
$$
Substituting in the <orbital angular momentum> $L$ gives
$$
L=-\frac{i\hbar}{2}\left[(a_1+a_1^*)(a_2-a_2^*)-(a_2+a_2^*)(a_1-a_1^*)\right].
$$
Operators belonging to distinct modes commute. Therefore the terms with two <annihilation operators> and those with two <creation operators> cancel, and $a_1^*a_2=a_2a_1^*$, $a_2^*a_1=a_1a_2^*$. The surviving terms give the <orbital angular momentum in oscillator ladder operators>:
$$
\boxed{L=i\hbar(a_1a_2^*-a_2a_1^*).}
$$
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