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Physical Review A

Continuum coupling in one-dimensional scattering using a transformed harmonic oscillator basis

I. Martel, Francisco Pérez-Bernal, M. Rodríguez-Gallardo, J. M. Arias, Joaquín Gómez-Camacho

Published Apr 24, 2002• 8 citations
Publisher

Abstract

The coupling to the continuum is studied in a one-dimensional problem that describes the interaction of a weakly bound composite object with a wall in a semiclassical approach. A transformed harmonic oscillator basis is introduced to provide an appropriate discrete and finite basis for treating the continuum part of the spectrum. The convergence of the scattering magnitudes is investigated as the number of states in the basis is increased. The role of bound-to-continuum and continuum-to-continuum coupling is investigated.

DOI
10.1103/physreva.65.052708

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The coupling to the continuum is studied in a one-dimensional problem that describes the interaction of a weakly bound composite object with a wall in a semiclassical approach. A transformed harmonic oscillator basis is introduced to provide an appropriate discrete and finite basis for treating the continuum part of the spectrum. The convergence of the scattering magnitudes is investigated as the number of states in the basis is increased. The role of bound-to-continuum and continuum-to-continuum coupling is investigated.
We propose a model for the quantum harmonic oscillator on a discrete lattice which can be written in a supersymmetric form, in contrast with the more direct discretization of the harmonic oscillator. Its ground state is easily found to be annihilated by the annihilation operator defined here, and its excitation spectrum is obtained numerically. We then define an operator whose continuum limit corresponds to an angular momentum in terms of the creation–annihilation operators of our model. Coherent states with the correct continuum limit are also constructed. The versatility of the model is then used to calculate, in a simple way, the generalized position-dependent scattering length for a particle colliding with a single static impurity in a periodic potential and the exact ground state of an interacting many-body problem in a one-dimensional ring.
Studies the coupling of N discrete levels to a common continuum. The discrete levels, which can include both genuine bound states and autoionisation states, are assumed to be coupled to the continuum and to each other in such a way as to permit a rotating wave approximation in the Schrodinger equation. A general solution for the model is derived with the aid of Laplace transforms and the Weisskopf-Wigner approximation. The author discusses general properties of the discrete level populations and the electron spectrum like population trapping, zeros, crossings, resonances and confluences. As examples the author treats cases where two or three discrete levels interact with a continuum.
An exactly solvable model of the one-dimensional quantum harmonic oscillator confined in a box with infinite walls is constructed. We have found explicit expressions of the non-equidistant energy spectrum as well as stationary states wavefunctions in both momentum and position configuration spaces. It is shown that they are expressed through continuous q-Hermite polynomials. We have also found an explicit expression for the kernel of the finite-continuous Fourier transform between these two representation spaces.
The quantum theory of a damped harmonic oscillator has been continuously studied since 1939. For quantization, an oscillation oscillator usually uses a reservoir that includes a number of finite and discrete oscillators. However, a set of discrete oscillators is unable to study the quantum theory of an Ohmic damping (damping proportional to velocity). In this research, a continuous set of harmonic oscillators has been used to investigate Ohmic damping. Using a continuum reservoir instead of a discrete reservoir will enrich the results of the dynamic system. The results of this research can be used to study nano-mechanical systems and opto-mechanical systems.