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arXiv:hep-ph/0501268 · Journal of Mathematical Physics

Relativistic harmonic oscillator

Zhifeng Li, Jin-Jin Liu, Wolfgang Lucha B, Wen-Gan Ma, Franz F. Schöberl

Published Oct 1, 2005• 50 citations
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Abstract

We study the semirelativistic Hamiltonian operator composed of the relativistic kinetic energy and a static harmonic-oscillator potential in three spatial dimensions and construct, for bound states with vanishing orbital angular momentum, its eigenfunctions in “compact form,” i.e., as power series, with expansion coefficients determined by an explicitly given recurrence relation. The corresponding eigenvalues are fixed by the requirement of normalizability of the solutions.

DOI
10.1063/1.2054648

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We study the semirelativistic Hamiltonian operator composed of the relativistic kinetic energy and a static harmonic-oscillator potential in three spatial dimensions and construct, for bound states with vanishing orbital angular momentum, its eigenfunctions in “compact form,” i.e., as power series, with expansion coefficients determined by an explicitly given recurrence relation. The corresponding eigenvalues are fixed by the requirement of normalizability of the solutions.
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.
Abstract We consider the time-independent Schrödinger equation for the harmonic oscillator potential and determine its bound states and energy levels in one spatial dimension using two approaches: the algebraic and the analytic method. Using the former, we introduce ladder operators: the annihilation and creation operators, as well as the occupation-number operator, and determine the ground state of the harmonic oscillator to be a Gaussian wave packet, while the latter method yields the general bound state solutions in terms of the Hermite polynomials. We further discuss the zero-point energy and uncertainty relation for the quantum harmonic oscillator and make a comparison to the classical harmonic oscillator. Finally, we examine the three-dimensional harmonic oscillator, which leads us to the description of systems with multiple degrees of freedom via the tensor product
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.
The quantization of orbital angular momentum can be introduced simply through the model of the two- and three-dimensional simple harmonic oscillator. Most of the analysis is carried out in the familiar Cartesian coordinate system in which the simple harmonic oscillator Schrödinger equation is separable. Angular momentum eigenfunctions are constructed by superposition, and the resulting functions easily transformed into spherical coordinates. The angle-dependent parts of the solutions are the spherical harmonics, which are the orbital angular momentum eigenfunctions for all central potentials including the Coulomb potential of the hydrogen atom. The method also introduces the separation of variables technique and provides practice in applying the concepts of completeness, orthogonality, superposition of states, and alternative representations.