Transition time estimation for $\delta$-function coupling in two state problem: An analytically solvable model

TL;DR

This paper derives an exact analytical formula for transition time in a two-state scattering problem with delta function coupling, revealing how transition dynamics depend on potential energy, incident energy, and coupling strength.

Contribution

It introduces a simple analytical method to calculate transition time in a coupled two-state system with delta function interaction, providing explicit dependence on system parameters.

Findings

Transition time depends explicitly on potential energy, incident energy, and coupling strength.

Coupling potential can act as transparent or opaque depending on initial energy.

Analytical expression enables precise understanding of transition dynamics in two-state scattering.

Abstract

We propose a simple method to calculate transition time in a two-state scattering problem, where two constant potentials are coupled by a delta function potential $V_{12}=V_{21}=k_0 \delta(x)$. The exact analytical expression for the time of transition $\tau$ is derived. We notice $\tau$ explicitly depends on the second state's potential energy along with the incident energy and coupling strength. We also observe from the derived expression of $\tau$ that depending on the initial energy, the coupling potential could behave like a transparent or opaque medium to the incident wave in a single state equivalent description.