Bouncing wave packets, Ehrenfest theorem, and uncertainty relation based upon a new concept for the momentum of a particle in a box

For a particle in a box, the operator −i∂x is not self-adjoint and thus does not qualify as the physical momentum. As a result, in general the Ehrenfest theorem is violated. Based upon a recently developed new concept for a self-adjoint momentum operator, we reconsider the theorem and find that it is now indeed satisfied for all physically admissible boundary conditions. We illustrate these results for bouncing wave packets which first spread, then shrink, and return to their original form after a certain revival time. We derive a very simple form of the general Heisenberg–Robertson–Schrödinger uncertainty relation and show that our construction also provides a physical interpretation for it. •A new momentum operator is shown to satisfy the Ehrenfest theorem.•A quantum-mechanical boundary force is identified via the Ehrenfest Theorem.•Wavepackets are wrapped onto a finite interval and studied using the new momentum.•A version of the uncertainty relation is derived for non-Hermitian operators.•The uncertainty relation for the usual momentum has a physical interpretation.