Impulsive forces and the Heisenberg equations of motion for a particle in a box
The quantum mechanical problem of a particle bouncing between two walls is formulated both in terms of the Heisenberg equations of motion and the Schrödinger equation. The reason for considering the Heisenberg equations is to understand the quantal nature of the impulsive forces. It is shown that these two formulations are compatible if, in addition to the classical impulsive forces, there are singular forces proportional to [Formula: see text] and [Formula: see text], i.e., forces of quantum origin. When these forces are added to the free-particle Hamiltonian then the total Hamiltonian is self-adjoint, and from it one can derive the boundary condition that must be imposed on the wave function. As an application of this formulation one can study the quantum mechanics of classical dynamical systems expressible as difference equations, e.g., the problem of a particle trapped between two walls moving relative to each other. The total Hamiltonian can also be used to study the question of the separability of the wave equation for such a motion.