Beyond the Schrödinger Equation: Quantum Motion with Traversal Time Control

We study a quantum particle, for which the duration τ it spends in some region of space is controlled by a meter, e.g., a Larmor clock. The particle is described by a wave function Ψ(x,t∣τ), with |Ψ(x,t∣τ)|² giving the distribution of the meter's readings at location x. The wave function satisfies the “clocked” Schrödinger equation, which we solve numerically for the cases of bound motion and wave packet scattering. The method is shown to be a natural extension of the conventional quantum mechanics.