= Boundary force in an infinite square well
{title2=$P_j=\hbar^2|\partial_x\psi(x_j)|^2/(2m)$}
For a normalized sufficiently regular solution of the <Schrödinger equation> on $[0,L]$ with Dirichlet walls, the instantaneous outward forces on the walls have magnitudes
$$
P_0=\frac{\hbar^2}{2m}|\psi'(0,t)|^2,\qquad P_L=\frac{\hbar^2}{2m}|\psi'(L,t)|^2.
$$
A wall displacement changes the kinetic-energy quadratic form by the boundary term $-P_j\,dL_j$, giving the local force. Equivalently, direct integration by parts in the time derivative of the mean particle momentum gives
$$
\frac{d\langle p\rangle}{dt}=\frac{\hbar^2}{2m}\left(|\psi'(0,t)|^2-|\psi'(L,t)|^2\right)=P_0-P_L.
$$
The force on the particle is the opposite of the net force $-P_0+P_L$ on the walls. For a stationary <energy eigenstate> each magnitude equals $2E_n/L$. For the <equal superposition of the first two box states>, writing $C=E_1/L$ and $\Omega=3E_1/\hbar$ gives $P_0=C(5+4\cos\Omega t)$ and $P_L=C(5-4\cos\Omega t)$. Both time averages equal $2\langle E\rangle/L$, the classical mean force on one wall at that energy, even though the quantum forces vary continuously in time.
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