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The boundary force in an infinite square well is obtained from the kinetic momentum flux at each wall. With $C_1={\hslash }^2{\pi }^2/(2m{L}^3)=E_1/L$, the derivatives of the state in part (c) areThe outward force magnitudes are $P_j=({\hslash }^2/2m)|{\Psi }^{\prime }(x_j,t){|}^2$. Taking positive force to mean rightwards, the forces exerted by the particle on the walls areBoth point outwards at all times, but their magnitudes oscillate out of phase. This force formula follows directly by differentiating the mean particle momentum and integrating the interior Schrödinger equation by parts: $d⟨p⟩/dt=({\hslash }^2/2m)(|{\Psi }^{\prime }(0){|}^2-|{\Psi }^{\prime }(L){|}^2)$. As a check, the net force on the particle is $8C_1\cos \Omega t=m{d}^2⟨x⟩/d{t}^2$, the opposite of the net force on the two walls.

A classical particle of energy $E=⟨E⟩$ transfers momentum $2mv$ in each collision with a particular wall, and returns to that wall after $2L/v$. Its mean outward force on that wall isThus the time averages agree separately at each wall, although the classical force consists of collision impulses and the quantum expectation varies continuously. Constant mean energy alone does not make the instantaneous quantum wall forces constant; their unequal instantaneous values supply the oscillation of $⟨x⟩$.