A particle of unit mass moves in one dimension in a potential

$$V=\frac{1}{2} \omega^{2} x^{2}$$

Show that the stationary solutions can be written in the form

$$\psi_{n}(x)=f_{n}(x) \exp \left(-\alpha x^{2}\right)$$

You should give the value of $\alpha$ and derive any restrictions on $f_{n}(x)$. Hence determine the possible energy eigenvalues $E_{n}$.

The particle has a wave function $\psi(x, t)$ which is even in $x$ at $t=0$. Write down the general form for $\psi(x, 0)$, using the fact that $f_{n}(x)$ is an even function of $x$ only if $n$ is even. Hence write down $\psi(x, t)$ and show that its probability density is periodic in time with period $\pi / \omega$.