(a) A particle of mass $m$ in one space dimension is confined to move in a potential $V(x)$ given by

$$V(x)= \begin{cases}0 & \text { for } 0<x<a \\ \infty & \text { for } x<0 \text { or } x>a\end{cases}$$

The normalised initial wavefunction of the particle at time $t=0$ is

$$\psi_{0}(x)=\frac{4}{\sqrt{5 a}} \sin ^{3}\left(\frac{\pi x}{a}\right)$$

(i) Find the expectation value of the energy at time $t=0$.

(ii) Find the wavefunction of the particle at time $t=1$.

[Hint: It may be useful to recall the identity $\sin 3 \theta=3 \sin \theta-4 \sin ^{3} \theta$.]

(b) The right hand wall of the potential is lowered to a finite constant value $U_{0}>0$ giving the new potential:

$$U(x)= \begin{cases}0 & \text { for } 0<x<a \\ \infty & \text { for } x<0 \\ U_{0} & \text { for } x>a\end{cases}$$

This potential is set up in the laboratory but the value of $U_{0}$ is unknown. The stationary states of the potential are investigated and it is found that there exists exactly one bound state. Show that the value of $U_{0}$ must satisfy

$$\frac{\pi^{2} \hbar^{2}}{8 m a^{2}}<U_{0}<\frac{9 \pi^{2} \hbar^{2}}{8 m a^{2}}$$