= Solution
The dimensionless coordinate for the <quantum harmonic oscillator> gives $d/dx=\sqrt{m\omega/\hbar}\,d/d\xi$. The <time-independent Schrödinger equation> therefore becomes
$$
-\psi_{\xi\xi}+\xi^2\psi=\frac{2E}{\hbar\omega}\psi.
$$
For $\psi=e^{-\xi^2/2}f$, direct <differentiation> yields
$$
\psi_{\xi\xi}=e^{-\xi^2/2}\left[f''-2\xi f'+(\xi^2-1)f\right].
$$
Cancelling the nonzero exponential reduces the equation to the <Hermite differential equation>
$$
f''-2\xi f'+(\epsilon-1)f=0,\qquad \boxed{\epsilon=\frac{2E}{\hbar\omega}.}
$$
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