2. Consider a particle of mass $m$ in an infinite square well with $V = 0$ in $0 < x < L$. For a definite energy $E_n$, the wave function is given by:
$\Psi_n(x, t) = \psi_n(x)e^{-i\omega_n t}$, where $n = 1, 2, 3, ...$
and where
$\psi_n(x) = \begin{cases} \sqrt{\frac{2}{L}}\sin(\frac{n\pi x}{L}) & 0 < x < L \\ 0 & \text{elsewhere} \end{cases}$ and $E_n = \hbar\omega_n = n^2\frac{\pi^2\hbar^2}{2mL^2}$.
(a) Show that
$\int_{-\infty}^{\infty} \psi_j^*(x)\psi_n(x) dx = \delta_{jn}$, where $\delta_{jn} = \begin{cases} 0 & \text{if } j \neq n \\ 1 & \text{if } j = n \end{cases}$.
Note that this result includes the normalization of $\psi_n(x)$!
Hint: consider the cases $j = n$ and $j \neq n$ separately, and use the trig identity
$2\sin\alpha \sin\beta = \cos(\alpha - \beta) - \cos(\alpha + \beta)$. (2 pts)
