Schroedinger current for discontinuous states from the first passage time decomposition

TL;DR

This paper develops a method to calculate probability currents for discontinuous quantum states using first passage time decomposition, revealing how the current's behavior depends on the type of discontinuity.

Contribution

It generalizes previous approaches to compute probability currents for discontinuous states, accounting for different types of discontinuities and their effects on current behavior.

Findings

Current behaves as t^{1/2} for certain discontinuities

Current behaves as t^{3/2} for others

Current approaches a constant for some discontinuities

Abstract

We revisit the problem of calculating the probability current for discontinuous states, such that may arise in atom trapping or as a result of projective measurements. In the first passage time representation, the problem reduces to evaluation of a localised wave originating from the discontinuity, whose interference with the initial state determines the transfer of probability. Depending on the type of discontinuity, the current behaves as $t^{1/2}$, $t^{3/2}$ or $const(t)$. Our approach generalises earlier work on this subject.