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

Abstract

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 a D 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.

Resonances and antibound states for the Pöschl–Teller potential: Ladder operators and SUSY partners

“…In order to obtain a Hamiltonian of a hierarchy, one uses in general an eigenfunction without zeros of the initial Hamiltonian, often corresponding to the ground state. However, one rarely uses an antibound state or a Gamow state to construct supersymmetric partners [23,24,25]. With these ideas in mind, we give some examples in which we build supersymmetric partners of Pöschl-Teller potentials using wave functions of antibound and Gamow states.…”

Leaky Modes of Waveguides as a Classical Optics Analogy of Quantum Resonances

“…On the other hand, as the scattering by quantum wells attenuates the outgoing wave packets only because of the multiple reflections at the well boundaries, it is expected that the resonance condition implies a maximum in the time spend by the projectile in traversing the zone of influence of the scatterer [6]. Recent theoretical research includes the case of a rectangular well that is embedded in an environment formed of a zero potential energy (flat potential) at the right and a parabolic potential at the left of the well [22][23][24]. Thus, the connection between time delay and 'time of capture' associated to the scattering processes would be verified in the laboratory by using semiconductor materials.…”

Negative Time Delay for Wave Reflection from a One-dimensional Semi-harmonic Well

“…This last is a rectangular well in a semi-harmonic background integrated by zero potential energy (flat potential) at the right and a harmonic-like potential at the left of the well. Our model corresponds to a system (the rectangular well) embedded in an environment (the parabolic plus flat potentials), and the issue is the study of the modifications on the physical properties of the system due to the environment [17]. For instance, the number N + 1 of bound states ψ n (x), n = 0, 1, .…”

“…In this context, remark that the wells of unit area V 0 = a + b admit only one bound state and constitute a family of compact support functions which converge to the delta well in the sense of distribution theory [18]. Then, the single bound state (dimensionless) energy E 0 = −0.25 of the delta well becomes less negative E 0 = −0.0797104 in the presence of the semi-harmonic background [17] (compare with [19]). On the other hand, the isolated resonances of a rectangular well are easily identified by expressing the transmission amplitude T as a superposition of Breit-Wigner distributions [20].…”