Rectangular Potentials in a Semi-Harmonic Background: Spectrum, Resonances and Dwell Time

TL;DR

This paper analyzes the energy spectrum, resonances, and dwell times of particles in semi-harmonic rectangular potentials, deriving analytical expressions and connecting dwell time with phase time in scattering processes.

Contribution

It provides analytical solutions for energy levels and dwell times in semi-harmonic rectangular potentials, linking dwell time with phase time and exploring capture phenomena.

Findings

Dwell time equals phase time in semi-harmonic potentials.

Analytical expressions for phase time in delta potentials are derived.

Scattering states include particle capture scenarios.

Abstract

We study the energy properties of a particle in one dimensional semi-harmonic rectangular wells and barriers. The integration of energies is obtained by solving a simple transcendental equation. Scattering states are shown to include cases in which the impinging particle is 'captured' by the semi-harmonic rectangular potentials. The 'time of capture' is connected with the dwell time of the scattered wave. Using the particle absorption method, it is shown that the dwell time $\tau^a_D$ coincides with the phase time $\tau_W$ of Eisenbud and Wigner, calculated as the energy derivative of the reflected wave phase shift. Analytical expressions are derived for the phase time $\tau_W$ of the semi-harmonic delta well and barrier potentials.