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Revista Mexicana de Física

Darboux-deformed barriers and resonances in quantum mechanics

Nicolás Fernández-García

Published Jan 8, 2007• 5 citations
OpenAlex

Abstract

"Scattering states in the continuum are used as Darboux transformation functions to deform square barrier potentials. The results includecomplex as well as real new potentials. It is shown that an appropriate superposition of Breit-Wigner distributions connects the transmissioncoefficient of one dimensional short range potentials."

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"Scattering states in the continuum are used as Darboux transformation functions to deform square barrier potentials. The results includecomplex as well as real new potentials. It is shown that an appropriate superposition of Breit-Wigner distributions connects the transmissioncoefficient of one dimensional short range potentials."
"We derive the expression V (x) u(x) = c δ(x − a) + v(x) u(x) (where V (x) is the potential, u(x) the wave function, c a constant and v(x) a finite potential function for x ≤ a), which is present in the one-dimensional Schr ̈ dinger equation on the whole real line when we have o an impenetrable barrier at x ≥ a, that is, an infinite step potential there. By studying the solution of this equation, we identify, connect and discuss three different Hamiltonian operators that describe the barrier. We extend these results by constructing an infinite square-well potential from two impenetrable barriers."
The properties of a prescription for the inner products of resonance (Gamow states), scattering (Dirac kets) and bound states for one-dimensional quantum barriers are worked out. The divergent asypmtotic behaviour of the Gamow states is regularized using a Gaussian convergence factor first introduced by Zel'dovich. With this prescription, most of these states (with discrete complex energies) are found to be orthogonal to each other and to the Dirac kets, except when they are neighbours, in which case the inner product is divergent. Therefore, as it happens for the continuum scattering states, the norm of the resonant ones remains non-calculable. Thus, they exhibit properties halfway between the (continuum real) Dirac-δ orthogonality and the (discrete real) Kronecker-δ orthogonality of the bound states.
Computer integration of Schrödinger’s time-independent wave equation with graphical display of solutions provides a valuable teaching aid, and is much used in the study of bound states. Much less attention has been paid to continuum solutions although these provide a richer set of phenomena. Programs are described that integrate the one-dimensional equation for potential steps and barriers, and model impedance matching, barrier penetration, and field emission. The radial equation for spherically symmetric potentials is also integrated and graphical demonstrations of scattering phase shifts, a simple model of the Ramsauer effect, and resonance scattering are provided. The programs have been used by undergraduates to investigate resonance scattering from spherical step and Woods–Saxon potentials, and a new power law relating resonance cross-section breadths and energies has been established.