Abstract
A recently proposed approach to relativistic quantum mechanics (Grave de Peralta, Poveda, Poirier in Eur J Phys 42:055404, 2021) is applied to the problem of a particle in a quadratic potential. The methods, both exact and approximate, allow one to obtain eigenstate energy levels and wavefunctions, using conventional numerical eigensolvers applied to Schrödinger-like equations. Results are obtained over a nine-order-of-magnitude variation of system parameters, ranging from the non-relativistic to the ultrarelativistic limits. Various trends are analyzed and discussed—some of which might have been easily predicted, others which may be a bit more surprising.
Highlights
The harmonic oscillator (HO) is a fundamental problem in quantum mechanics, which serves as an elementary description of physical phenomena as varied as the vibrational modes of molecules and solids, or the modes of the electromagnetic field
We take the solutions of the non-relativistic quantum HO problem to be the eigenstate solutions of the following ordinary differential equation—i.e., the stationary Schrödinger equation: p 2 + 1 kx2 2m 2 ψn(x) = Enψn(x)
The above defines what we call the “self-consistent” (SC)-Grave de Peralta-Poveda-Poirier” (GPPP) approximation. It is socalled because the self-consistent GPPP” (SC-GPPP) gGP values may be solved for in self-consistent fashion [16]—similar to, but more straightforwardly than, the method outlined at the end of Sect. 2.1
Summary
Foundations of Physics (2022) 52:29 prominent feature of the quantum HO is its spaced energy levels, bringing a direct linkage to the concept of quantum field excitation. The solutions as described above are in close analogy with the behavior of a classical relativistic particle in a quadratic potential. The period of the oscillation increases with energy, due to time dilation along the world line—indicating that the behavior of a particle in a quadratic potential loses its “harmonic” character in the relativistic regime [10]. Equation (6) above (but generalized for arbitrary four-vector potentials) is often encountered in textbooks, as a covariant relativistic eigenvalue equation It is often dismissed as “impractical”, due to the squareroot operator. In order to avoid the problematic square-root operator, the solutions of this equation have been obtained using non-relativistic-like effective hamiltonians [13, 14] or the auxiliary fields formalism [15], with which the present method bears some resemblance. The previously developed approximate methods rely on a nonlinear Schrödinger or
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