Transition Decomposition of Quantum Mechanical Evolution

TL;DR

This paper introduces a method to decompose quantum states into past and future asymptotes using Lyapunov operators, aiding the analysis of resonance phenomena like scattering off a potential barrier.

Contribution

It presents a novel decomposition technique for quantum states based on Lyapunov operators, providing new insights into resonance behavior and decay laws.

Findings

Decomposition captures resonance behavior accurately.

Past asymptote exhibits exponential decay.

Method applicable to scattering problems.

Abstract

We show that the existence of the family of self-adjoint Lyapunov operators introduced in [J. Math. Phys. 51, 022104 (2010)] allows for the decomposition of the state of a quantum mechanical system into two parts: A past time asymptote, which is asymptotic to the state of the system at t goes to minus infinity and vanishes at t goes to plus infinity, and a future time asymptote, which is asymptotic to the state of the system at t goes to plus infinity and vanishes at t goes to minus infinity. We demonstrate the usefulness of this decomposition for the description of resonance phenomena by considering the resonance scattering of a particle off a square barrier potential. We show that the past time asymptote captures the behavior of the resonance. In particular, it exhibits the expected exponential decay law and spatial probability distribution.