General Proof That Normal Modes Become Quantum Eigenstates"
This proof generalizes the argument given in Exercise 9.1. Consider a set of $N$ particles $a=1, \ldots N$ with masses $m_{\alpha}$ interacting via a potential
$$
U=\frac{1}{2} \sum_{a, b} x_{a} V_{a, b} x_{b}
$$
where $x_{a}$ is the deviation of the position of particle $a$ from its equilibrium position and $V$ can be taken (without loss of generality) to be a symmetric matrix. (Here we consider a situation in $1 d$, however, we will see that to go to $3 \mathrm{~d}$ we just need to keep track of three times as many coordinates.)
(i) Defining $y_{a}=\sqrt{m_{a}} x_{a}$, show that the classical equations of motion may be written as
$$
\ddot{y}_{a}=-\sum_{b} S_{a, b} y_{b}
$$
where
$$
S_{a, b}=\frac{1}{\sqrt{m_{a}}} V_{a, b} \frac{1}{\sqrt{m_{b}}}
$$
Thus show that the solutions are
$$
y_{a}^{(m)}=e^{-i \omega_{m} t} s_{a}^{(m)}
$$
where $\omega_{m}$ is the $m^{t h}$ eigenvalue of the matrix $S$ with corresponding eigenvector $s_{a}^{(m)}$. These are the $N$ normal modes of the system.
(ii) Recall the orthogonality relations for eigenvectors of hermitian matrices
$$
\begin{aligned}
&\sum_{a}\left[s_{a}^{(m)}\right]^{*}\left[s_{a}^{(n)}\right]=\delta_{m, n} \\
&\sum_{m}\left[s_{a}^{(m)}\right]^{*}\left[s_{b}^{(m)}\right]=\delta_{a, b} .
\end{aligned}
$$
Since $S$ is symmetric as well as hermitian, the eigenvectors can be taken to be real. Construct the transformed coordinates
$$
\begin{aligned}
Y^{(m)} &=\sum_{a} s_{a}^{(m)} x_{a} \sqrt{m_{a}} \\
P^{(m)} &=\sum_{a} s_{a}^{(m)} p_{a} / \sqrt{m_{a}}
\end{aligned}
$$
show that these coordinates have canonical commutations
$$
\left[P^{(m)}, Y^{(n)}\right]=-i \hbar \delta_{n, m}
$$
and show that in terms of these new coordinates the Hamiltonian is rewritten as
$$
H=\sum_{m}\left[\frac{1}{2}\left[P^{(m)}\right]^{2}+\frac{1}{2} \omega_{m}^{2}\left[Y^{(m)}\right]^{2}\right]
$$
Conclude that the quantum eigenfrequencies of the system are also $\omega_{m}$. (Can you derive this result from the prior two equations?)