 Equal superposition of the first two box states

For an infinite square well on $[0,L]$, choose ${\varphi }_n=\sqrt{2/L}\sin (n\pi x/L)$ and $E_n=n^2{E}_1$, $E_1={\pi }^2{\hslash }^2/(2m{L}^2)$. The state $\psi =({\varphi }_1{e}^{-i{E}_{1}t/\hslash }+{\varphi }_2{e}^{-i{E}_{2}t/\hslash })/\sqrt{2}$ has constant expectation value $⟨E⟩=5E_1/2$, butIndeed, the diagonal position matrix elements are $L/2$, whereas the off-diagonal element isA 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.