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

Quantum and classical probability distributions for position and momentum

Richard W. Robinett

Published Sep 1, 1995• 78 citations
Publisher

Abstract

The classical and quantum probability distributions for both position and momentum are compared for several model systems admitting bound states including the harmonic oscillator, the infinite well, and the linear confining potential (V(x)=F‖x‖). Examples corresponding to unbound systems, including the uniformly accelerating particle and the motion of a particle moving away from a point of unstable equilibrium, i.e., the ‘‘unstable oscillator’’ defined by V(x)=−kx2/2, are also considered. The quantum and classical distribution of kinetic and potential energy for the harmonic oscillator is briefly discussed.

DOI
10.1119/1.17807

Related papers

Visualizing classical and quantum probability densities for momentum using variations on familiar one-dimensional potentials

Richard W. Robinett

After briefly reviewing the definitions of classical probability densities for position, P CL ( x ), and for momentum, P CL ( p ), we present several examples of classical mechanical potential systems, mostly variations on such familiar cases as the infinite well and the uniformly accelerated particle for which the classical distributions can be easily derived and visualized. We focus especially on a simple potential which interpolates between the symmetric linear potential, V ( x ) = F | x |, and the infinite well, which can illustrate, in a mathematically straightforward way, how the divergent δ-function classical probability density for momentum for the infinite well can be seen to arise. Such examples can help students understand the quantum mechanical momentum-space wavefunction (and its corresponding probability density) in much the same way that other semiclassical techniques, such as the WKB approximation, can be used to visualize position-space wavefunctions.

arXiv:quant-ph/0307056•European Journal of Physics•Jan 30, 2002•11 citations

Quantum, classical and semiclassical momentum distributions: I. Theory and elementary examples

H. J. Korsch, B Schellhaaß

We discuss the basic properties of momentum distributions in quantum mechanics for elementary systems as well as their classical analogue. Semiclassical approximations can show a quantitative connection between the classical and quantum cases. We believe that such distributions provide a useful tool to improve the understanding of elementary quantum mechanics. Important differences between distributions in coordinate and momentum space are pointed out. Elementary examples (free and uniformly accelerating particle, harmonic oscillator and square-well potential) are discussed.

European Journal of Physics•Jan 1, 2000•11 citations

Average value of position for the anharmonic oscillator: Classical versus quantum results

Richard W. Robinett

The evaluation of the average value of the position coordinate, 〈x〉, of a particle moving in a harmonic oscillator potential (V(x)=kx2/2) with a small anharmonic piece (V(x)=−λkx3) is a standard calculation in classical Newtonian mechanics and statistical mechanics where the problem has relevance to thermal expansion. In each case, the calculation is most easily done using a perturbative expansion. In this note, we perform the same computation of 〈x〉 in quantum mechanics using time-independent perturbation theory and the ladder operator formalism to show how similar results are obtained. We also indicate how a semiclassical calculation using a classical probability distribution can also be used to obtain the same result.

American Journal of Physics•Mar 1, 1997•15 citations

Classical and quantum behavior of the harmonic and the quartic oscillators

David Brizuela

In a previous paper a formalism to analyze the dynamical evolution of classical and quantum probability distributions in terms of their moments was presented. Here the application of this formalism to the system of a particle moving on a potential is considered in order to derive physical implications about the classical limit of a quantum system. The complete set of harmonic potentials is considered, which includes the particle under a uniform force, as well as the harmonic and the inverse harmonic oscillators. In addition, as an example of anharmonic system, the pure quartic oscillator is analyzed. Classical and quantum moments corresponding to stationary states of these systems are analytically obtained without solving any differential equation. Finally, dynamical states are also considered in order to study the differences between their classical and quantum evolution.

arXiv:1411.1522•Physical review. D. Particles, fields, gravitation, and cosmology/Physical review. D. Particles and fields•Dec 18, 2014•26 citations

Comparing classical and quantum probability distributions for an asymmetric infinite well

M. A. Doncheski, Richard W. Robinett

We compare the classical and quantum mechanical position-space probability densities for a particle in an asymmetric infinite well. In an idealized system with a discontinuous step in the middle of the well, the classical and quantum probability distributions agree fairly well, even for relatively small quantum numbers, except for anomalous cases which arise due to the unphysical nature of the potential. We are able to derive upper and lower bounds on the differences between the quantum and classical results. We also qualitatively discuss the momentum-space probability densities for this system using intuitive ideas about the amount of time a classical particle spends in various parts of the well. This system provides an excellent example of a non-trivial, but tractable, quantum mechanical bound state problem where the correlations between the amplitude and curvature of quantum mechanical wavefunctions can be easily compared to classical intuition about particle motion, with quantitative success, but also warning of possible surprises in non-physical limiting cases.

arXiv:quant-ph/0307014•European Journal of Physics•Apr 5, 2000•27 citations

Quantum observables for probabilistic classical particles

Christof Wetterich

The classical observables of position and momentum are not well adapted to particles in a microphysical situation where typical probability distributions are characterized by a substantial dispersion. We propose the use of more robust quantum observables for probabilistic classical particles. The quantum observables are statistical observables which do not take fixed values for a given classical position and momentum. Solutions of the Liouville equation are discussed in the quantum formalism for classical statistics. Statistical observables are represented by non-commuting operators. No classical correlation function is defined for these observables and Bell's inequalities do not apply. We demonstrate for a general potential how a quantum system emerges from classical statistics. For the particular cases of a harmonic potential and a Coulomb potential we investigate subsystems which describe all features of a quantum particle. This covers the discrete energy spectrum of the hydrogen atom and quantum harmonic oscillator. We discuss the interference for the double-slit experiment. Conserved statistical observables may also be relevant for the probabilistic dynamics of dust or planets.

quant-pharXiv:2607.13937•Jul 16, 2026•0 citations
The classical and quantum probability distributions for both position and momentum are compared for several model systems admitting bound states including the harmonic oscillator, the infinite well, and the linear confining potential (V(x)=F‖x‖). Examples corresponding to unbound systems, including the uniformly accelerating particle and the motion of a particle moving away from a point of unstable equilibrium, i.e., the ‘‘unstable oscillator’’ defined by V(x)=−kx2/2, are also considered. The quantum and classical distribution of kinetic and potential energy for the harmonic oscillator is briefly discussed.
After briefly reviewing the definitions of classical probability densities for position, P CL ( x ), and for momentum, P CL ( p ), we present several examples of classical mechanical potential systems, mostly variations on such familiar cases as the infinite well and the uniformly accelerated particle for which the classical distributions can be easily derived and visualized. We focus especially on a simple potential which interpolates between the symmetric linear potential, V ( x ) = F | x |, and the infinite well, which can illustrate, in a mathematically straightforward way, how the divergent δ-function classical probability density for momentum for the infinite well can be seen to arise. Such examples can help students understand the quantum mechanical momentum-space wavefunction (and its corresponding probability density) in much the same way that other semiclassical techniques, such as the WKB approximation, can be used to visualize position-space wavefunctions.
We discuss the basic properties of momentum distributions in quantum mechanics for elementary systems as well as their classical analogue. Semiclassical approximations can show a quantitative connection between the classical and quantum cases. We believe that such distributions provide a useful tool to improve the understanding of elementary quantum mechanics. Important differences between distributions in coordinate and momentum space are pointed out. Elementary examples (free and uniformly accelerating particle, harmonic oscillator and square-well potential) are discussed.
The evaluation of the average value of the position coordinate, 〈x〉, of a particle moving in a harmonic oscillator potential (V(x)=kx2/2) with a small anharmonic piece (V(x)=−λkx3) is a standard calculation in classical Newtonian mechanics and statistical mechanics where the problem has relevance to thermal expansion. In each case, the calculation is most easily done using a perturbative expansion. In this note, we perform the same computation of 〈x〉 in quantum mechanics using time-independent perturbation theory and the ladder operator formalism to show how similar results are obtained. We also indicate how a semiclassical calculation using a classical probability distribution can also be used to obtain the same result.
In a previous paper a formalism to analyze the dynamical evolution of classical and quantum probability distributions in terms of their moments was presented. Here the application of this formalism to the system of a particle moving on a potential is considered in order to derive physical implications about the classical limit of a quantum system. The complete set of harmonic potentials is considered, which includes the particle under a uniform force, as well as the harmonic and the inverse harmonic oscillators. In addition, as an example of anharmonic system, the pure quartic oscillator is analyzed. Classical and quantum moments corresponding to stationary states of these systems are analytically obtained without solving any differential equation. Finally, dynamical states are also considered in order to study the differences between their classical and quantum evolution.
We compare the classical and quantum mechanical position-space probability densities for a particle in an asymmetric infinite well. In an idealized system with a discontinuous step in the middle of the well, the classical and quantum probability distributions agree fairly well, even for relatively small quantum numbers, except for anomalous cases which arise due to the unphysical nature of the potential. We are able to derive upper and lower bounds on the differences between the quantum and classical results. We also qualitatively discuss the momentum-space probability densities for this system using intuitive ideas about the amount of time a classical particle spends in various parts of the well. This system provides an excellent example of a non-trivial, but tractable, quantum mechanical bound state problem where the correlations between the amplitude and curvature of quantum mechanical wavefunctions can be easily compared to classical intuition about particle motion, with quantitative success, but also warning of possible surprises in non-physical limiting cases.
The classical observables of position and momentum are not well adapted to particles in a microphysical situation where typical probability distributions are characterized by a substantial dispersion. We propose the use of more robust quantum observables for probabilistic classical particles. The quantum observables are statistical observables which do not take fixed values for a given classical position and momentum. Solutions of the Liouville equation are discussed in the quantum formalism for classical statistics. Statistical observables are represented by non-commuting operators. No classical correlation function is defined for these observables and Bell's inequalities do not apply. We demonstrate for a general potential how a quantum system emerges from classical statistics. For the particular cases of a harmonic potential and a Coulomb potential we investigate subsystems which describe all features of a quantum particle. This covers the discrete energy spectrum of the hydrogen atom and quantum harmonic oscillator. We discuss the interference for the double-slit experiment. Conserved statistical observables may also be relevant for the probabilistic dynamics of dust or planets.