Consider a particle confined in a one-dimensional infinite potential well: $V(x)=\infty$ for $|x| \geqslant a$ and $V(x)=0$ for $|x|<a$. The normalised stationary states are

$$\psi_{n}(x)= \begin{cases}\alpha_{n} \sin \left(\frac{\pi n(x+a)}{2 a}\right) & \text { for }|x|<a \\ 0 & \text { for }|x| \geqslant a\end{cases}$$

where $n=1,2, \ldots$.

(i) Determine the $\alpha_{n}$ and the stationary states' energies $E_{n}$.

(ii) A state is prepared within this potential well: $\psi(x) \propto x$ for $0<x<a$, but $\psi(x)=0$ for $x \leqslant 0$ or $x \geqslant a$. Find an explicit expansion of $\psi(x)$ in terms of $\psi_{n}(x) .$

(iii) If the energy of the state is then immediately measured, show that the probability that it is greater than $\frac{\hbar^{2} \pi^{2}}{m a^{2}}$ is

$$\sum_{n=0}^{4} \frac{b_{n}}{\pi^{n}}$$

where the $b_{n}$ are integers which you should find.

(iv) By considering the normalisation condition for $\psi(x)$ in terms of the expansion in $\psi_{n}(x)$, show that

$$\frac{\pi^{2}}{3}=\sum_{p=1}^{\infty} \frac{A}{p^{2}}+\frac{B}{(2 p-1)^{2}}\left(1+\frac{C(-1)^{p}}{(2 p-1) \pi}\right)^{2}$$

where $A, B$ and $C$ are integers which you should find.