Solution
= Solution
For unit mass,
$$
H_A=\frac12P_A^2+\frac12\omega^2X_A^2
=\hbar\omega\left(a_A^\dagger a_A+\frac12\right),
$$
where
$$
X_A=\sqrt{\frac{\hbar}{2\omega}}(a_A+a_A^\dagger),
\qquad
P_A=-i\sqrt{\frac{\hbar\omega}{2}}(a_A-a_A^\dagger).
$$
The Hilbert space is $L^2(\mathbb R)$, equivalently the Fock-space completion of the orthonormal number states $|n\rangle$, $n\geq0$.