Consider a quantum system with Hamiltonian $H$ and wavefunction $\Psi$ obeying the time-dependent Schrödinger equation. Show that if $\Psi$ is a stationary state then $\langle Q\rangle_{\Psi}$ is independent of time, if the observable $Q$ is independent of time.

A particle of mass $m$ is confined to the interval $0 \leqslant x \leqslant a$ by infinite potential barriers, but moves freely otherwise. Let $\Psi(x, t)$ be the normalised wavefunction for the particle at time $t$, with

$$\Psi(x, 0)=c_{1} \psi_{1}(x)+c_{2} \psi_{2}(x)$$

where

$$\psi_{1}(x)=\left(\frac{2}{a}\right)^{1 / 2} \sin \frac{\pi x}{a}, \quad \psi_{2}(x)=\left(\frac{2}{a}\right)^{1 / 2} \sin \frac{2 \pi x}{a}$$

and $c_{1}, c_{2}$ are complex constants. If the energy of the particle is measured at time $t$, what are the possible results, and what is the probability for each result to be obtained? Give brief justifications of your answers.

Calculate $\langle\hat{x}\rangle_{\Psi}$ at time $t$ and show that the result oscillates with a frequency $\omega$, to be determined. Show in addition that

$$\left|\langle\hat{x}\rangle_{\Psi}-\frac{a}{2}\right| \leqslant \frac{16 a}{9 \pi^{2}} .$$