On the Time-Dependent Analysis of Gamow Decay

TL;DR

This paper critically examines Gamow's complex eigenvalue approach to quantum decay, demonstrating how Gamow functions can approximate the evolution of trapped wave functions during exponential decay in a one-dimensional model.

Contribution

It clarifies the role of non-square-integrable Gamow functions in describing time evolution of quantum decay within a specific potential model.

Findings

Gamow functions approximate wave function evolution during decay

The approximation remains valid during exponential decay phase

The study provides a rigorous link between resonances and decay dynamics

Abstract

Gamow's approach to exponential decay of meta-stable particles via complex 'eigenvalues' (resonances) of a Hamiltonian is scrutinized. We explain the sense in which the non-square-integrable 'eigenfunctions' that belong to these resonances (Gamow functions) are relevant for the time-evolution of square-integrable wave functions. For concreteness we study a one dimensional square-well potential with a trapping region K and the evolution of wave functions, whose support is initially inside of K. It is shown that the sum over the first few time-evolved Gamow functions restricted to K yields an approximation for the evolution of these initial wave functions within the trapping region. The approximation is good for all times for which exponential decay prevails.