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AIP conference proceedings

A fermion in a scalar inversely linear potential

Antonio S. de Castro

Published Jan 1, 2004• 0 citations
Publisher

Abstract

The problem of a fermion subject to a a scalar inversely linear potential in a two‐dimensional world is mapped into a Sturm‐Liouville problem for nonzero eigenenergies. This mapping gives rise to an effective Kratzer potential and exact bounded solutions are found in closed form. The normalizable zero‐eigenmode solution is also found. A few unusual results are revealed.

DOI
10.1063/1.1843695

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The problem of a fermion subject to a a scalar inversely linear potential in a two‐dimensional world is mapped into a Sturm‐Liouville problem for nonzero eigenenergies. This mapping gives rise to an effective Kratzer potential and exact bounded solutions are found in closed form. The normalizable zero‐eigenmode solution is also found. A few unusual results are revealed.
Exact Solution of a Spin-12\frac{1}{2}21​ Particle for a Linear Potential
The problem of a spin-12\frac{1}{2}21​ particle moving in a linear potential field in two-dimensions is searched to obtain for nonzero energy eigenvalues and the corresponding normalized eigenfunctions. The zero-mode (E=0E=0E=0) eigenfunctions are also studied and it is seen that they are normalizable. The variation of the non-zero eigenfunctions and also eigenvalues are according to the position and potential parameter γγγ, respectively, given in the text.
Exactly solvable models play an extremely important role in many fields of quantum physics. In this study, the Schrödinger equation is applied for a solution of a two--dimensional (2D) problem for two particles interacting via Kratzer, and modified Kratzer potentials. We found the exact bound state solutions of the two--dimensional Schrödinger equation with Kratzer--type potentials and present analytical expressions for the eigenvalues and eigenfunctions. The eigenfunctions are given in terms of the associated Laguerre polynomials.
We study the Dirac equation in 3+1 dimensions with a general combination of scalar, vector and tensor interactions with arbitrary strengths, all of them described by central Coulomb potentials acting on a particular plane of motion. For the tensor coupling a constant term is also included, since this gives rise to an effective Coulomb potential, which is necessary for the formation of bound states in a pure tensor coupling configuration. The exact bound-state solutions for this generalized Coulomb problem are computed by exploiting the freedom in choosing the coefficients of the \textit{Ansätze} for the radial functions, which leads to wave functions in terms of generalized Laguerre polynomials. From the quantization condition, the exact energy spectrum is also determined and its dependence on the parameters of the potentials is discussed. We show that similar features of the equations for the problem in the plane and the spherically symmetric problem allow a simple and direct mapping between them, thereby providing the solution to the spherical Coulomb problem. Our results are validated by showing that the solutions correctly encompass several previous solutions available in the literature for particular cases of this problem, for which we further develop the analysis of the parameters. We also derive two new particular cases not yet reported in the literature: the case of breaking of spin and pseudospin symmetries by the addition of a Coulomb plus constant tensor potential and the problem of a scalar plus tensor Coulomb potentials.
We investigate bound states of a non-relativistic scalar particle in a three-dimensional helically twisted (torsional) geometry, considering both the free case and the presence of external radial interactions. The dynamics is described by the Schrödinger equation on a curved spatial background and, when included, by minimal coupling to a magnetic vector potential incorporating an Aharonov--Bohm flux. After separation of variables, the problem reduces to a one-dimensional radial eigenvalue equation governed by an effective potential that combines torsion-induced Coulomb-like and centrifugal-like structures with magnetic/flux-dependent terms and optional model interactions. Because closed-form analytic solutions are not reliable over the parameter ranges required for systematic scans, we compute spectra and eigenfunctions numerically by formulating the radial equation as a self-adjoint Sturm--Liouville problem and solving it with a finite-difference discretization on a truncated radial domain, with explicit convergence control. We analyze four representative scenarios: (i) no external potential, (ii) Cornell-type confinement, (iii) Kratzer-type interaction, and (iv) the small-oscillation regime around the minimum of a Morse potential. We present systematic trends of the low-lying levels as functions of the torsion parameter, magnetic field, and azimuthal sector, and we show that geometric couplings alone can produce effective confinement even in the absence of an external interaction.