A particle moves in one dimension in an infinite square-well potential $V(x)=0$ for $|x|<a$ and $\infty$ for $|x|>a$. Find the energy eigenstates. Show that the energy eigenvalues are given by $E_{n}=E_{1} n^{2}$ for integer $n$, where $E_{1}$ is a constant which you should find.

The system is perturbed by the potential $\delta V=\epsilon x / a$. Show that the energy of the $n^{\text {th }}$level $E_{n}$ remains unchanged to first order in $\epsilon$. Show that the ground-state wavefunction is

$$\psi_{1}(x)=\frac{1}{\sqrt{a}}\left[\cos \frac{\pi x}{2 a}+\frac{D \epsilon}{\pi^{2} E_{1}} \sum_{n=2,4, \ldots}(-1)^{A n} \frac{n^{B}}{\left(n^{2}-1\right)^{C}} \sin \frac{n \pi x}{2 a}+\mathcal{O}\left(\epsilon^{2}\right)\right]$$

where $A, B, C$ and $D$ are numerical constants which you should find. Briefly comment on the conservation of parity in the unperturbed and perturbed systems.