Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

American Journal of Physics

Minimum uncertainty wave packet in relativistic quantum mechanics

Yukihisa Nogami, F. Masafumi Toyama

Published Jan 12, 2010• 2 citations
Publisher

Abstract

In nonrelativistic quantum mechanics in one dimension a wave packet can be constructed for which the minimum uncertainty relation for position and momentum, ΔxΔp=ℏ/2, holds exactly. The wave function of the wave packet is Gaussian and satisfies the Schrödinger equation for a harmonic oscillator potential. We illustrate a similar situation in relativistic quantum mechanics for the one-dimensional Dirac equation.

DOI
10.1119/1.3238469

Related papers

Uncertainty relations for angular momentum eigenstates in two and three spatial dimensions

Christian Bracher

I reexamine Heisenberg’s uncertainty relation for two- and three-dimensional wave packets with fixed angular momentum quantum numbers m or ℓ. A simple proof shows that the product of the average extent Δr and Δp of a two-dimensional wave packet in position and momentum space is bounded from below by ΔrΔp≥ℏ(|m|+1). The minimum uncertainty is attained by modified Gaussian wave packets that are special eigenstates of the two-dimensional isotropic harmonic oscillator, which include the ground states of electrons in a uniform magnetic field. Similarly, the inequality ΔrΔp≥ℏ(ℓ+3/2) holds for three-dimensional wave packets with fixed total angular momentum ℓ and the equality holds for a Gaussian radial profile. I also discuss some applications of these uncertainty relations.

American Journal of Physics•Feb 16, 2011•13 citations

The Wave Function and the Minimum Uncertainty Function of the Time Dependent Harmonic Oscillator(New Developments in Statistical Physics Similarities in Diversities,YITP Workshop)

Kyu Hwang Yeon, Chung‐In Um, Thomas F. George

The time dependent harmonic oscillator is solved explicitly for quantum mechanics by the operator method with an auxiliary condition as the classical solution. Two classical invariant quantities which determine whether or not the system is bound are derived by the classical equation of motion. We obtain the invariant operator from one classical invariant quantity. Its eigenfunction is related to the solution of Schrodinger equation of the system and its eigenvalue is related to another classical quantity. The wave function is evaluated exactly by the eigenfunction of the invariant operator but it is not the eigenfunction of the Hamiltonian of the system. The uncertainty which calculates with the wave function is not a minimum one. We will confirm that the function which holds minimum uncertainty is a eigenfunction of the Hamiltonian.

•Jul 20, 1993•0 citations

Investigation of an Equation with Logarithmic Coupling with Momentum in Dirac Equation

Mash-Huda Rahman Shipra, Syed Badiuzzaman Faruque

The free particle solutions of the relativistic Dirac equation are characterized by plane waves with infinite uncertainty in position. However, many practical implementations of the solutions require a wave packet configuration, which can be utilized to represent a localized Dirac particle. Unlike the traditional wave packet generation method by superposing multiple plane waves, this study of ours presents an alternative approach towards obtaining a wave packet solution of a free particle relativistic Dirac equation. In this paper, we present Dirac’s free particle equation with a modification in the generalized momentum. The modification is achieved by coupling the momentum with a spatially varying logarithmic function, and this alteration does not affect the relativistic dispersion relation of the particle. Moreover, a solution of this modified Dirac equation is provided as well, which has been calculated using a trial wave function. The wave function solution is carried out in one dimension, where it behaves as a wave packet for a given ratio of the envelope parameter to the reduced Planck's constant greater than unity, where the envelope parameter regulates the width of the wave packet. The solution, being subject to this constraint, represents a bound particle with spin and a continuous energy spectrum.

International Journal of High Energy Physics•Jan 1, 2020•0 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

Quantum Mechanics and Liouville's Equation

Michael Nauenberg

In non-relativistic quantum mechanics, the absolute square of Schrödinger's wave function for a particle in a potential determines the probability of finding it either at a position or momentum at a given time. In classical mechanics the corresponding problem is determined by the solution of Liouville's equation for the probability density of finding the joint position and momentum of the particle at a given time. Integrating this classical solution over either one of these two variables can then be compared with the probability in quantum mechanics. For the special case that the force is a constant, it is shown analytically that for an initial Gaussian probability distribution, the solution of Liouville's integrated over momentum is equal to Schrödinger's probability function in coordinate space, provided the coordinate and momentum initial widths of this classical solution satisfy the minimal Heisenberg uncertainty relation. Likewise, integrating Lioville's solution over position is equal to Schrödinger's probability function in momentum space. Quanta 2017; 6: 53–56.

Quanta•Sep 9, 2017•0 citations

Quantum uncertainty as an intrinsic clock

Etera R. Livine

Abstract In quantum mechanics, a classical particle is raised to a wave-function, thereby acquiring many more degrees of freedom. For instance, in the semi-classical regime, while the position and momentum expectation values follow the classical trajectory, the uncertainty of a wave-packet can evolve and beat independently. We use this insight to revisit the dynamics of a 1d particle in a time-dependent harmonic well. One can solve it by considering time reparameterizations and the Virasoro group action to map the system to the harmonic oscillator with constant frequency. We prove that identifying such a simplifying time variable is naturally solved by quantizing the system and looking at the evolution of the width of a Gaussian wave-packet. We further show that the Ermakov-Lewis invariant for the classical evolution in a time-dependent harmonic potential is actually the quantum uncertainty of a Gaussian wave-packet. This naturally extends the classical Ermakov-Lewis invariant to a constant of motion for quantum systems following Schrödinger equation. We conclude with a discussion of potential applications to quantum gravity and quantum cosmology.

arXiv:2212.09442•Journal of Physics A Mathematical and Theoretical•Oct 25, 2023•9 citations
In nonrelativistic quantum mechanics in one dimension a wave packet can be constructed for which the minimum uncertainty relation for position and momentum, ΔxΔp=ℏ/2, holds exactly. The wave function of the wave packet is Gaussian and satisfies the Schrödinger equation for a harmonic oscillator potential. We illustrate a similar situation in relativistic quantum mechanics for the one-dimensional Dirac equation.
I reexamine Heisenberg’s uncertainty relation for two- and three-dimensional wave packets with fixed angular momentum quantum numbers m or ℓ. A simple proof shows that the product of the average extent Δr and Δp of a two-dimensional wave packet in position and momentum space is bounded from below by ΔrΔp≥ℏ(|m|+1). The minimum uncertainty is attained by modified Gaussian wave packets that are special eigenstates of the two-dimensional isotropic harmonic oscillator, which include the ground states of electrons in a uniform magnetic field. Similarly, the inequality ΔrΔp≥ℏ(ℓ+3/2) holds for three-dimensional wave packets with fixed total angular momentum ℓ and the equality holds for a Gaussian radial profile. I also discuss some applications of these uncertainty relations.
The time dependent harmonic oscillator is solved explicitly for quantum mechanics by the operator method with an auxiliary condition as the classical solution. Two classical invariant quantities which determine whether or not the system is bound are derived by the classical equation of motion. We obtain the invariant operator from one classical invariant quantity. Its eigenfunction is related to the solution of Schrodinger equation of the system and its eigenvalue is related to another classical quantity. The wave function is evaluated exactly by the eigenfunction of the invariant operator but it is not the eigenfunction of the Hamiltonian of the system. The uncertainty which calculates with the wave function is not a minimum one. We will confirm that the function which holds minimum uncertainty is a eigenfunction of the Hamiltonian.
The free particle solutions of the relativistic Dirac equation are characterized by plane waves with infinite uncertainty in position. However, many practical implementations of the solutions require a wave packet configuration, which can be utilized to represent a localized Dirac particle. Unlike the traditional wave packet generation method by superposing multiple plane waves, this study of ours presents an alternative approach towards obtaining a wave packet solution of a free particle relativistic Dirac equation. In this paper, we present Dirac’s free particle equation with a modification in the generalized momentum. The modification is achieved by coupling the momentum with a spatially varying logarithmic function, and this alteration does not affect the relativistic dispersion relation of the particle. Moreover, a solution of this modified Dirac equation is provided as well, which has been calculated using a trial wave function. The wave function solution is carried out in one dimension, where it behaves as a wave packet for a given ratio of the envelope parameter to the reduced Planck's constant greater than unity, where the envelope parameter regulates the width of the wave packet. The solution, being subject to this constraint, represents a bound particle with spin and a continuous energy spectrum.
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.
In non-relativistic quantum mechanics, the absolute square of Schrödinger's wave function for a particle in a potential determines the probability of finding it either at a position or momentum at a given time. In classical mechanics the corresponding problem is determined by the solution of Liouville's equation for the probability density of finding the joint position and momentum of the particle at a given time. Integrating this classical solution over either one of these two variables can then be compared with the probability in quantum mechanics. For the special case that the force is a constant, it is shown analytically that for an initial Gaussian probability distribution, the solution of Liouville's integrated over momentum is equal to Schrödinger's probability function in coordinate space, provided the coordinate and momentum initial widths of this classical solution satisfy the minimal Heisenberg uncertainty relation. Likewise, integrating Lioville's solution over position is equal to Schrödinger's probability function in momentum space. Quanta 2017; 6: 53–56.
Abstract In quantum mechanics, a classical particle is raised to a wave-function, thereby acquiring many more degrees of freedom. For instance, in the semi-classical regime, while the position and momentum expectation values follow the classical trajectory, the uncertainty of a wave-packet can evolve and beat independently. We use this insight to revisit the dynamics of a 1d particle in a time-dependent harmonic well. One can solve it by considering time reparameterizations and the Virasoro group action to map the system to the harmonic oscillator with constant frequency. We prove that identifying such a simplifying time variable is naturally solved by quantizing the system and looking at the evolution of the width of a Gaussian wave-packet. We further show that the Ermakov-Lewis invariant for the classical evolution in a time-dependent harmonic potential is actually the quantum uncertainty of a Gaussian wave-packet. This naturally extends the classical Ermakov-Lewis invariant to a constant of motion for quantum systems following Schrödinger equation. We conclude with a discussion of potential applications to quantum gravity and quantum cosmology.