= Midpoint current of a two-mode box state
{title2=$j(L/2,t)=2\pi\hbar\sin(\Omega t)/(mL^2)$}
For the <equal superposition of the first two box states> with zero initial relative <quantum phase>, put $\Omega=(E_2-E_1)/\hbar$. The <probability> in the left half is $P_{\mathrm L}=1/2+4\cos(\Omega t)/(3\pi)$. The <continuity equation> and zero wall <probability current> imply
$$
j(L/2,t)=-P_{\mathrm L}'(t)=\frac{2\pi\hbar}{mL^2}\sin\Omega t.
$$
Direct substitution of $\psi(L/2,t)=L^{-1/2}e^{-iE_1t/\hbar}$ and $\partial_x\psi(L/2,t)=-2\pi L^{-3/2}e^{-iE_2t/\hbar}$ into the <probability current> verifies the sign. Positive current means probability leaves the left half. The <quantum interference> controls transport even though each component has stationary <probability density>.
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