= Solution
Set $\hat B=\hat a_x^\dagger\hat a_y-\hat a_y^\dagger\hat a_x$. Taking the <Hermitian adjoint> reverses the order of products, so $\hat B^\dagger=-\hat B$. <Complex conjugation> also reverses the sign of $-i$, and therefore
$$
\hat Q^\dagger=i\hbar\hat B^\dagger=-i\hbar\hat B=\hat Q.
$$
Thus $\hat Q$ is a <self-adjoint operator> on the usual <oscillator> states.
Using the coefficients in part (a), products containing two <position operators> or two <momentum operators> cancel in the difference. Cross-coordinate operators commute, leaving
$$
\hat B=2i\alpha\beta(\hat x\hat p_y-\hat y\hat p_x)
=\frac i\hbar(\hat x\hat p_y-\hat y\hat p_x).
$$
Hence
$$
\boxed{\hat Q=\hat x\hat p_y-\hat y\hat p_x=\hat L_z.}
$$
It is the component of <orbital angular momentum> perpendicular to the plane, with units $\mathrm{kg\,m^2\,s^{-1}}$ (equivalently $\mathrm{J\,s}$). The <rotational symmetry> of the <two-dimensional isotropic harmonic oscillator> makes this a conserved <observable>; directly, each term in $\hat B$ preserves $N_x+N_y$, so $[\hat H,\hat Q]=0$.
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