Time Evolution of a Decaying Quantum State: Evaluation and Onset of Non-Exponential Decay

TL;DR

This paper generalizes a method to analyze the time evolution of decaying quantum states, predicting when non-exponential decay begins, with applications to systems like Gaussian wave functions tunneling through potential barriers.

Contribution

It extends an existing method to more general potentials and initial states, enabling analysis of long-time decay behavior in quantum systems.

Findings

Identifies the timescale for the onset of non-exponential decay.

Demonstrates the method with Gaussian wave functions and Eckart's potential.

Provides a framework for predicting decay behavior in quantum systems.

Abstract

The method developed by Van Dijk, Nogami and Toyama for obtaining the time-evolved wave function of a decaying quantum system is generalized to potentials and initial wave functions of non-compact support. The long time asymptotic behavior is extracted and employed to predict the timescale for the onset of non-exponential decay. The method is illustrated with a Gaussian initial wave function leaking through Eckart's potential barrier on the halfline.