= Solution
For a state in the domain of the operator, its <expectation value> is
$$
\langle\mathcal O\rangle=\frac{\int\Psi^*(x,t)(\mathcal O\Psi)(x,t)\,dx}{\int|\Psi(x,t)|^2dx}.
$$
For a normalized state the denominator is one. Use an orthonormal <energy eigenstate> basis, choosing orthogonal vectors within degenerate eigenspaces. Inserting an expansion into the time-dependent <Schrödinger equation> gives $i\hbar\dot a_n(t)=E_na_n(t)$. Consequently
$$
\boxed{\Psi(x,t)=\sum_n a_n e^{-iE_nt/\hbar}u_n(x)}.
$$
The phases preserve normalization. Projection onto a specified basis eigenstate gives probability $\boxed{|a_p|^2}$, independent of time. If the measurement distinguishes only the energy and $E_p$ is degenerate, sum $|a_n|^2$ over all $n$ with $E_n=E_p$.
Using orthonormality to eliminate all cross terms,
$$
\boxed{\langle H\rangle=\sum_n|a_n|^2E_n}.
$$
This is time independent whenever the expectation is defined. The time-independent Hamiltonian changes only relative phases, not the energy probabilities.
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