Starting from the time-dependent Schrödinger equation, show that a stationary state $\psi(x)$ of a particle of mass $m$ in a harmonic oscillator potential in one dimension with frequency $\omega$ satisfies

$$-\frac{\hbar^{2}}{2 m} \frac{d^{2} \psi}{d x^{2}}+\frac{1}{2} m \omega^{2} x^{2} \psi=E \psi .$$

Find a rescaling of variables that leads to the simplified equation

$$-\frac{d^{2} \psi}{d y^{2}}+y^{2} \psi=\varepsilon \psi$$

Setting $\psi=f(y) e^{-\frac{1}{2} y^{2}}$, find the equation satisfied by $f(y)$.

Assume now that $f$ is a polynomial

$$f(y)=y^{N}+a_{N-1} y^{N-1}+a_{N-2} y^{N-2}+\ldots+a_{0}$$

Determine the value of $\varepsilon$ and deduce the corresponding energy level $E$ of the harmonic oscillator. Show that if $N$ is even then the stationary state $\psi(x)$ has even parity.