Abstract
We study the relativistic version of the d-dimensional isotropic quantum harmonic oscillator based on the spinless Salpeter equation. This has no exact analytical solutions. We use perturbation theory to obtain compact formulas for the first and second-order relativistic corrections; they are expressed in terms of two quantum numbers and the spatial dimension d. The formula for the first-order correction is obtained using two different methods and we illustrate how this correction splits the original energy into a number of distinct levels each with their own degeneracy. Previous authors obtained results in one and three dimensions and our general formulas reduce to them when d = 1 and d = 3 respectively. Our two-dimensional results are novel and we provide an example that illustrates why two dimensions is of physical interest. We also obtain results for the two-dimensional case using a completely independent method that employs ladder operators in polar coordinates. In total, three methods are used in this work and the results all agree.
Highlights
The harmonic oscillator plays a crucial role in quantum mechanics and quantum field theory
In this paper we obtain explicit analytical formulas for the first and second order relativistic corrections to the isotropic quantum harmonic oscillator (QHO) governed by the Salpeter Hamiltonian (1), that are valid in any spatial dimension d
H1 and H2 are the relativistic corrections (4) to the Hamiltonian and EN0 = (N +1) ω and EN0 = (N +1) ω are the energies of the non-relativistic two-dimensional isotropic harmonic oscillator (N and N can take on values of 0, 1, 2, 3,...)
Summary
The harmonic oscillator plays a crucial role in quantum mechanics and quantum field theory. 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. Though obtaining the energies and eigenfunctions for the (unperturbed) d-dimensional isotropic QHO is much easier in Cartesian coordinates, those eigenfunctions do not form the correct basis for use in nondegenerate perturbation theory. The effect of this correction is illustrated in an energy level diagram. The conclusion summarizes our final results and discusses a physical system whose relativistic corrections would be of interest to study in the future
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