References: Shankar, R. (1994), Principles of Quantum Mechanics, Plenum Press. Section 5.2, Exercise 5.2.4.
One way of comparing the classical and quantum pictures of a particle in an infinite square well is to calculate the force exerted on the walls by the particle. If a particle is in state , its energy is
If the particle remains in this state as the walls are slowly pushed in, so that slowly decreases, then its energy will increase, meaning that work is done on the system. The force is the change in energy per unit distance, so the force required is
If we treat the system classically, then a particle with energy between the walls is effectively a free particle in this region (since the potential there), so all its energy is kinetic. That is
The classical particle bounces elastically between the two walls, which means its velocity is exactly reversed at each collision. The momentum transfer in such a collision is
The time between successive collisions on the same wall is
Thus the average force exerted on one wall is
Comparing with 2, we see that the quantum and classical forces in this case are the same.