= Equal superposition of the first two box states
For an <infinite square well> on $[0,L]$, choose $\phi_n=\sqrt{2/L}\sin(n\pi x/L)$ and $E_n=n^2E_1$, $E_1=\pi^2\hbar^2/(2mL^2)$. The state $\psi=(\phi_1e^{-iE_1t/\hbar}+\phi_2e^{-iE_2t/\hbar})/\sqrt2$ has constant <expectation value> $\langle E\rangle=5E_1/2$, but
$$
\langle x\rangle=\frac L2-\frac{16L}{9\pi^2}\cos\left(\frac{3E_1t}{\hbar}\right).
$$
Indeed, the diagonal position <matrix elements> are $L/2$, whereas the off-diagonal element is
$$
\langle1|x|2\rangle=\frac2L\int_0^L x\sin\frac{\pi x}L\sin\frac{2\pi x}L\,dx=-\frac{16L}{9\pi^2}.
$$
A different initial relative phase shifts the oscillation in time. The oscillation comes from <quantum interference>, while the constant mean energy follows from the fixed populations of the two <energy eigenstates>.
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