In this problem, we will work to understand the time evolution of wavefunctions that are localized in position in the infinite square well.

(a) It will prove simplest later to re-write the infinite square well in a way that is symmetric for \(x \rightarrow-x\). So, for an infinite square well in which the potential is 0 for \(x \in[-\pi / 2, \pi / 2]\), find the energy eigenstates \(\psi_{n}(x)\) and the corresponding energy eigenvalues \(E_{n}\).

(b) Now, let's consider the initial wavefunction \(\psi(x)\) that is a uniform bump in the middle of the well:

\[\psi(x)=\left\{\begin{array}{cc}\sqrt{\frac{2}{\pi}}, & |x|<\pi / 4 \tag{5.73}\\0, & |x|>\pi / 4\end{array}\right.\]

From this initial wavefunction, determine the wavefunction at a general time \(t, \psi(x, t)\).

(c) Does this wavefunction "leak" into the regions where it was initially 0 ? Let's take the inner product of this time-dependent wavefunction with the wavefunction \(\chi(x)\) that is uniform over the well:

\[\begin{equation*}\chi(x)=\frac{1}{\sqrt{\pi}} \tag{5.74}\end{equation*}\]

What is \(\langle\chi \mid \psiangle\), as a function of time?

(d) What is the first time derivative of the inner product at \(t=0\) :

\[\begin{equation*}\left.\frac{d\langle\chi \mid \psiangle}{d t}\right|_{t=0} ? \tag{5.75}\end{equation*}\]

Can you think about what this means in the context of the Schrödinger equation?
(e) On the state \(\psi(x, t)\), what are the expectation values of position and momentum for all time, \(\langle\hat{x}angle\) and \(\langle\hat{p}angle\) ?