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arXiv:quant-ph/0012004

Singular potentials and absorption problem in Quantum Mechanics

Vladimir D. Skarzhinsky, Juergen Audretsch

Published Dec 1, 2000• 4 citations
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Abstract

We discuss a possible approach to the absorption problem in Quantum Mechanics based on using of singular attractive potentials in the corresponding Schrödinger equations. Possible criteria for selection of exact solutions of these equations are considered and it is shown that different models of absorption can be realized by a special choice of exact solutions. As an example, the motion of charged particles in the Aharonov-Bohm (AB) and the scalar attractive $ρ^{-2}$ potentials is investigated in detail. Other attractive potentials are briefly considered.

DOI
10.48550/arxiv.quant-ph/0012004

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We discuss a possible approach to the absorption problem in Quantum Mechanics based on using of singular attractive potentials in the corresponding Schrödinger equations. Possible criteria for selection of exact solutions of these equations are considered and it is shown that different models of absorption can be realized by a special choice of exact solutions. As an example, the motion of charged particles in the Aharonov-Bohm (AB) and the scalar attractive ρ−2ρ^{-2}ρ−2 potentials is investigated in detail. Other attractive potentials are briefly considered.
A new approach to find exact solutions to one--dimensional quantum mechanical systems is devised. The scheme is based on the introduction of a potential function for the wavefunction, and the equation it satisfies. We recover known solutions as well as to get new ones for both free and interacting particles with wavefunctions having vanishing and non--vanishing Bohm potentials. For most of the potentials, no solutions to the Schrödinger equation produce a vanishing Bohm potential. A (large but) restricted family of potentials allows the existence of particular solutions for which the Bohm potential vanishes. This family of potentials is determined, and several examples are presented. It is shown that some quantum, such as accelerated Airy wavefunctions, are due to the presence of non--vanishing Bohm potentials. New examples of this kind are found and discussed.
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Aharonov–Bohm effect is a quantum mechanical phenomenon that attracted the attention of many physicists and mathematicians since the publication of the seminal paper of Aharonov and Bohm [1] in 1959. We consider different types of Aharonov–Bohm effects such as the magnetic AB effect, electric AB effect, combined electromagnetic AB effect, AB effect for the Schrödinger equations with Yang–Mills potentials, and the gravitational analog of AB effect. We shall describe different approaches to prove the AB effect based on the inverse scattering problems, the inverse boundary value problems in the presence of obstacles, spectral asymptotics, and the direct proofs of the AB effect.
We investigate the electronic absorption induced by a complex potential in one-dimensional quantum mechanics. By solving the Schr\"odinger equation for spatially localized and extended complex potentials, we derive the electronic absorption coefficient \ensuremath{\alpha}. Neither a single nor a double Dirac-\ensuremath{\delta} potential can represent a totally absorbing potential (\ensuremath{\alpha}=1) for any choice of the complex amplitude of the potential. Maximum absorption coefficients for these potential types are 0.5 and 2(\ensuremath{\surd}2-1)\ensuremath{\simeq}0.828, respectively. On the contrary, wall and well potentials do account for total absorption at a particular limit of the potential parameters. \textcopyright{} 1996 The American Physical Society.