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Journal of Yiyang Teachers College

A Relativistic Revision of the Energy in a Quantum Harmonic Oscillator

Qiao Chu

Published Jan 1, 1999• 0 citations
OpenAlex

Abstract

Thc perturbational theory is applied to give two degrees relativistic corrections forthe energy of a quantum oscillator and the physis ideas of correction results have been analysed.

Related papers

Relativistic Harmonic Oscillator and the Klein Gordon Equation

Francesco R. Ruggeri

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.

•Dec 31, 2019•0 citations

NUMERICAL CALCULATIONS OF THE SMALL RELATIVISTIC CORRECTIONS OF ANISOTROPIC QUANTUM HARMONIC OSCILLATOR ENERGY EIGENVALUES: THE TWO- AND THREE-DIMENSIONAL CASE

Gheorghe Ardelean, Ion I. Cotǎescu

In this paper the small relativistic correction for the energy eigenvalues of the two- and three-dimensional anisotropic quantum harmonic oscillator are calculated, using as eigenstates [Formula: see text], for different values of the relativistic parameters βi ≡ ħwi / m0c2 with i = 1, 2 and 3.

International Journal of Modern Physics C•Aug 1, 1996•0 citations

NUMERICAL CALCULATIONS OF THE SMALL RELATIVISTIC CORRECTIONS OF QUANTUM HARMONIC OSCILLATOR ENERGY EIGENVALUES

Gheorghe Ardelean

The relativistic correction of the energy eigenvalues of quantum harmonic oscillator (QHO) are calculated using [Formula: see text] as eigenstates, for different values of the relativistic parameter α ≡ ħω/m0c2.

International Journal of Modern Physics C•Dec 1, 1996•0 citations

One Correction of Quantum Resonator Energy Level

Xuehua Jiang

Relativistic effect taken into consideration and a certain quantum resonator energy level being corrected by perturbation theory, the energy level structural formula which is the same as the result obtained by using perturbation items is obtained, but this method is more easier, which offers a new correction for analyzing the energy level of some quantum resonator.

•Jan 1, 2004•0 citations

A Non-relativistic Approach to Relativistic Quantum Mechanics: The Case of the Harmonic Oscillator

Luis A. Poveda, Luis Grave de Peralta, Jacob Pittman +1 more

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.

Foundations of Physics•Feb 1, 2022•11 citations

Closed form perturbative relativistic modifications to wave-packet dynamics in the quantum harmonic oscillator

Jian Carlo Ramos, Sujoy K. Modak

We derive closed form expressions of weak relativistic corrections to the wave-packet dynamics of the quantum harmonic oscillator within a perturbative framework. General expressions are derived for the leading-order relativistic contributions to wave-packet parameters, such as the time-dependent widths, variances, and uncertainty relations. Specific calculations are performed for unsqueezed, minimum-uncertainty Gaussian wave packets, valid up to leading order in ($1/c^{2}$). When applied to electron wave packets, the results indicate that relativistic effects become non-negligible for keV-scale harmonic confinement energies: the deviations in variances reach $0.15\% - 1.5\%$ for an electron wave packet confined within the $1-10 \mathrm{keV}$ energy range. We also show that the standard saturation of the uncertainty relation remains unaffected by the leading-order relativistic effect.

quant-pharXiv:2603.07954•Journal of Physics A Mathematical and Theoretical•Mar 10, 2026•0 citations
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 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.
We derive closed form expressions of weak relativistic corrections to the wave-packet dynamics of the quantum harmonic oscillator within a perturbative framework. General expressions are derived for the leading-order relativistic contributions to wave-packet parameters, such as the time-dependent widths, variances, and uncertainty relations. Specific calculations are performed for unsqueezed, minimum-uncertainty Gaussian wave packets, valid up to leading order in (1/c21/c^{2}1/c2). When applied to electron wave packets, the results indicate that relativistic effects become non-negligible for keV-scale harmonic confinement energies: the deviations in variances reach 0.15%−1.5%0.15\% - 1.5\%0.15%−1.5% for an electron wave packet confined within the 1−10keV1-10 \mathrm{keV}1−10keV energy range. We also show that the standard saturation of the uncertainty relation remains unaffected by the leading-order relativistic effect.