A one-dimensional quantum mechanical particle has normalised bound state energy eigenfunctions $\chi_{n}(x)$ and corresponding non-degenerate energy eigenvalues $E_{n}$. At $t=0$ the normalised wavefunction $\psi(x, t)$ is given by

$$\psi(x, 0)=\sqrt{\frac{5}{6}} e^{i k_{1}} \chi_{1}(x)+\sqrt{\frac{1}{6}} e^{i k_{2}} \chi_{2}(x)$$

where $k_{1}$ and $k_{2}$ are real constants. Write down the expression for $\psi(x, t)$ at a later time $t$ and give the probability that a measurement of the particle's energy will yield a value of $E_{2}$.

Show that the expectation value of $x$ at time $t$ is given by

$$\langle x\rangle=\frac{5}{6}\langle x\rangle_{11}+\frac{1}{6}\langle x\rangle_{22}+\frac{\sqrt{5}}{3} \operatorname{Re}\left[\langle x\rangle_{12} e^{i\left(k_{2}-k_{1}\right)-i\left(E_{2}-E_{1}\right) t / \hbar}\right]$$

where $\langle x\rangle_{i j}=\int_{-\infty}^{\infty} \chi_{i}^{*}(x) x \chi_{j}(x) d x$.