The case of escape probability as linear in short time

TL;DR

This paper rigorously proves that the short-time escape probability of a quantum particle from a compact initial state with boundary discontinuity is linear in time, with implications for quantum decay, Zeno effect, and measurement.

Contribution

It introduces a novel rigorous derivation of the linear short-time escape probability including boundary layer effects, a previously unestablished result in quantum dynamics.

Findings

Escape probability is linear in short time.

Boundary layer effects are crucial for accurate calculation.

Results have broad implications for quantum measurement and decay processes.

Abstract

We derive rigorously the short-time escape probability of a quantum particle from its compactly supported initial state, which has a discontinuous derivative at the boundary of the support. We show that this probability is liner in time, which seems to be a new result. The novelty of our calculation is the inclusion of the boundary layer of the propagated wave function formed outside the initial support. This result has applications to the decay law of the particle, to the Zeno behavior, quantum absorption, time of arrival, quantum measurements, and more, as will be discussed separately.