Relativistic Generalizations of the Quantum Harmonic Oscillator

Abstract

We investigate the dynamics of the spin-less relativistic particle subject to an external eld of a harmonic oscillator potential. The KleinGordon equation with one-and three-dimensional vector and scalar parabolic potentials is solved using the expansion of the wavefunction in properly selected basis-sets. The resonance states are determined using the complex coordinate rotation method. The analytic expressions for the rst-and secondorder relativistic energy corrections are derived perturbatively. The relativistic model of the harmonic oscillator in the momentum representation, originally proposed by Znojil, is also discussed.

First and second-order relativistic corrections to the two and higher-dimensional isotropic harmonic oscillator obeying the spinless Salpeter equation

“…in agreement with previous results [8,9]. For d=1, recall that l=0 and n=N/2 with N a non-negative integer so that E N N 6 6 3 in agreement with therelativistic correction to the usual (d=1) harmonic oscillator [9,23]. Our result for d=2 is novel and is given by…”

“…Note that the above second-order correction is positive and valid for any dimension d. It is novel and reduces to previous results [9] for d=1 and d=3 respectively. In one dimension recall that we set l=0 and n N 2 = where N is a non-negative integer and this yields .…”

“…In this paper we obtain explicit analytical formulas for the first and second order relativistic corrections to the isotropic QHO governed by the Salpeter Hamiltonian (1), that are valid in any spatial dimension d. The problem is solved in spherical coordinates and the results are expressed in terms of the dimension d and two quantum numbers n and ℓ which are non-negative integers. Our first-order correction formula reduces to that obtained in [8,9] for d=1 and d=3, and our second-order correction formula reduces to that obtained in [9] for d=1 and d=3. Our d=2 results are novel and we also solve the two-dimensional case using a completely independent method that makes use of ladder operators in polar coordinates.…”

“…The situation is different from the case of the Coulomb potential, which can be recovered from the tree level Feynman diagram between two charged particles in quantum electrodynamics (see [4] for a derivation of the Coulomb potential in quantum field theory). For the QHO, one can define a Dirac oscillator for spin 1/2 particles and a Klein-Gordon oscillator for spinless particles both of which have been studied by many authors [5][6][7][8][9]]. An interesting model called finite-difference relativistic quantum mechanics was solved in [3].…”

“…Expanding the above Hamiltonian operator in powers of are first and second-order perturbations to the Hamiltonian H 0 respectively. Using perturbation theory, analytical results for the Hamiltonian (1) were previously obtained in one and three dimensions: first-order relativistic corrections in three dimensions were obtained in [8] whereas first and second-order relativistic corrections in one and three dimensions were obtained in [9].…”