The wave function $\Psi(x, t)$ is a solution of the time-dependent Schrödinger equation for a particle of mass $m$ in a potential $V(x)$,

$$H \Psi(x, t)=i \hbar \frac{\partial}{\partial t} \Psi(x, t),$$

where $H$ is the Hamiltonian. Define the expectation value, $\langle\mathcal{O}\rangle$, of any operator $\mathcal{O}$.

At time $t=0, \Psi(x, t)$ can be written as a sum of the form

$$\Psi(x, 0)=\sum_{n} a_{n} u_{n}(x),$$

where $u_{n}$ is a complete set of normalized eigenfunctions of the Hamiltonian with energy eigenvalues $E_{n}$ and $a_{n}$ are complex coefficients that satisfy $\sum_{n} a_{n}^{*} a_{n}=1$. Find $\Psi(x, t)$ for $t>0$. What is the probability of finding the system in a state with energy $E_{p}$ at time $t$ ?

Show that the expectation value of the energy is independent of time.