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Europhysics Letters (EPL)

On Relativistic Singular Harmonic-Oscillator Potentials

Francisco Domínguez-Adame, Enrique Maciá

Published Apr 15, 1989• 11 citations
Publisher

Abstract

The Dirac equation for a harmonic-oscillator potential plus a Lorentz scalar delta-shaped term has been solved. This singular potential changes the usual oscillator energy levels. Unlike the nonrelativistic treatment, the particle cannot fall into the centre for any negative value of the delta-function strength. The apparent occurrence of double degenerate energy levels for some limiting cases is found.

DOI
10.1209/0295-5075/8/8/001

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The Dirac equation for a harmonic-oscillator potential plus a Lorentz scalar delta-shaped term has been solved. This singular potential changes the usual oscillator energy levels. Unlike the nonrelativistic treatment, the particle cannot fall into the centre for any negative value of the delta-function strength. The apparent occurrence of double degenerate energy levels for some limiting cases is found.
For the Dirac equation in one space dimension, we examine the possibility of harmonic oscillator (HO) potentials. A HO potential is such that it has only bound states and the energy levels are equally spaced. Regarding their Lorentz transformation properties, there are three types of potentials for the Dirac equation: vector, scalar, and pseudoscalar. HO potentials are possible for the scalar and pseudoscalar types, but not for the vector type. We show how HO potentials can be constructed by means of the ``inverse scattering method.'' We also examine the behavior of a wave packet in the HO potentials so constructed. The behavior is, in most of the cases, very similar to that of Schr\"odinger's coherent wave packet of the nonrelativistic harmonic oscillator.
The importance of the energy spectrum of bound states and their restrictions in quantum mechanics due to the different methods have been used for calculating and determining the limit of them. Comparison of Schrödinger-like equation obtained by Dirac equation with the nonrelativistic solvable models is the most efficient method. By this technique, the exact relativistic solutions of Dirac equation for Hartmann and Ring-Shaped Oscillator Potentials are accessible, when the scalar potential is equal to the vector potential. Using solvable nonrelativistic quantum mechanics systems as a basic model and considering the physical conditions provide the changes in the restrictions of relativistic parameters based on the nonrelativistic definitions of parameters.
Bound state solutions of the Schrödinger equation have been investigated for n -dimensional ( n ⩾ 2) harmonic oscillator potential decorated with any finite number ( P ) of Dirac delta functions. The potential is radially symmetric and given as , where σ i s are arbitrary real numbers, r 1 < r 2 < ⋅⋅⋅ < r P and r i ∊ (0, + ∞). We have demonstrated that addition of Dirac delta functions lifts the accidental degeneracies of n -dimensional harmonic oscillator energy levels and leaves only the degeneracy due to the radial symmetry. Explicit forms of bound state eigenfunctions and the eigenvalue equation are given for n , l values, where n is the space dimension and l is the degree of n -dimensional spherical harmonics. We have shown that, for given n and l , there are a countably infinite number of bound state energy levels which are continuous functions of ω, σ i s and at most P of them can be negative.
The Klein Gordon equation for the harmonic oscillator may be written as (E-.5kx2)2=p2+mo2 ((1)) where mo is the rest mass and p the relativistic momentum. In the literature, it is argued that the resulting x4 term dominates for large x and prevents the Klein Gordon equation from being solved for bound states. As a result, it is replaced with an equation of the form: (E2-mo2) = (p-imwx)(p+imw) ((2)) where p=-ihbar d/dx in one dimension which may be solved exactly. Recently, the authors of (1) have used perturbation theory to calculate energy changes to ((1)) by using -.5V2 - e2 +2eV as a perturbative potential where V=.5kx2 and E=e+mo. They utilise the unperturbed wavefunction exp(-ax2) so large x4 values are damped. Their results are close to the exact calculations of ((2)) for the case of k=1 and m=1. In this note, we argue that for an oscillator there are classical turning points at which the quantum average kinetic energy -1/2m d/dx d/dx W should be zero and small nearby even if the average kinetic energy near x=0 corresponds to relativistic momenta. Thus, it is argued there are different spatial regions, each with its own quantum equation. The Schrodinger equation may be applied to regions near the classical turning point moving to infinite. In such a case, there is no issue with an x4 term. Near x=0, the full Klein Gordon equation ((1)) may be applied. There are questions of then matching the two sets of solutions. In this note, we attempt to investigate this scenario and compare results with those of (1) and (2). (1) and (2) suggest similar relativistic corrections to the nonrelativistic ground state oscillator energy of more than 20%. In this note, we suggest a smaller value.