Boundary force in an infinite square well ($P_j={\hslash }^2|{\partial }_x\psi (x_j){|}^2/(2m)$)

For a normalized sufficiently regular solution of the Schrödinger equation on $[0,L]$ with Dirichlet walls, the instantaneous outward forces on the walls have magnitudesA wall displacement changes the kinetic-energy quadratic form by the boundary term $-P_{j}d{L}_j$, giving the local force. Equivalently, direct integration by parts in the time derivative of the mean particle momentum givesThe 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 $\Omega =3E_1/\hslash$ gives $P_0=C(5+4\cos \Omega t)$ and $P_L=C(5-4\cos \Omega t)$. Both time averages equal $2⟨E⟩/L$, the classical mean force on one wall at that energy, even though the quantum forces vary continuously in time.