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

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 τ_D^a coincides with the phase time τ_W of Eisenbud and Wigner, calculated as the energy derivative of the reflected wave phase shift. Analytical expressions are derived for the phase time τ_W of the semi-harmonic delta well and barrier potentials.

Keywords

• exactly solvable potentials
• scattering process
• resonances
• Eisenbud-Wigner phase time
• dwell time