Consider a quantum-mechanical particle of mass $m$ moving in a potential well,

$$V(x)= \begin{cases}0, & -a<x<a \\ \infty, & \text { elsewhere }\end{cases}$$

(a) Verify that the set of normalised energy eigenfunctions are

$$\psi_{n}(x)=\sqrt{\frac{1}{a}} \sin \left(\frac{n \pi(x+a)}{2 a}\right), \quad n=1,2, \ldots$$

and evaluate the corresponding energy eigenvalues $E_{n}$.

(b) At time $t=0$ the wavefunction for the particle is only nonzero in the positive half of the well,

$$\psi(x)= \begin{cases}\sqrt{\frac{2}{a}} \sin \left(\frac{\pi x}{a}\right), & 0<x<a \\ 0, & \text { elsewhere }\end{cases}$$

Evaluate the expectation value of the energy, first using

$$\langle E\rangle=\int_{-a}^{a} \psi H \psi d x$$

and secondly using

$$\langle E\rangle=\sum_{n}\left|a_{n}\right|^{2} E_{n},$$

where the $a_{n}$ are the expansion coefficients in

$$\psi(x)=\sum_{n} a_{n} \psi_{n}(x)$$

Hence, show that

$$1=\frac{1}{2}+\frac{8}{\pi^{2}} \sum_{p=0}^{\infty} \frac{(2 p+1)^{2}}{\left[(2 p+1)^{2}-4\right]^{2}}$$