What is meant by a stationary state? What form does the wavefunction take in such a state? A particle has wavefunction $\psi(x, t)$, such that

$$\psi(x, 0)=\sqrt{\frac{1}{2}}\left(\chi_{1}(x)+\chi_{2}(x)\right)$$

where $\chi_{1}$ and $\chi_{2}$ are normalised eigenstates of the Hamiltonian with energies $E_{1}$ and $E_{2}$. Write down $\psi(x, t)$ at time $t$. Show that the expectation value of $A$ at time $t$ is

$$\langle A\rangle_{\psi}=\frac{1}{2} \int_{-\infty}^{\infty}\left(\chi_{1}^{*} \hat{A} \chi_{1}+\chi_{2}^{*} \hat{A} \chi_{2}\right) d x+\operatorname{Re}\left(e^{i\left(E_{1}-E_{2}\right) t / \hbar} \int_{-\infty}^{\infty} \chi_{1}^{*} \hat{A} \chi_{2} d x\right)$$