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

European Journal of Physics

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

H. J. Korsch, B Schellhaaß

Published Jan 1, 2000• 11 citations
Publisher

Abstract

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.

DOI
10.1088/0143-0807/21/1/310

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 and classical probability distributions for position and momentum

Richard W. Robinett

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.

American Journal of Physics•Sep 1, 1995•78 citations

Classical particles in quantum mechanics

George Patsakos

An analysis of the change in the concept of a particle required by the passage from classical to quantum theory suggests a ’’classical particle representation’’ for the latter which should allow purely quantum-mechanical results to be cast into a form suited to interpretation in semiclassical terms. Application of this representation to the examples of the free particle and the particle in a constant force field essentially reproduces the usual arguments as to how classical trajectories can be generated in the quantum theory, although from a somewhat different point of view than that usually presented. Applying the representation to the energy eigenfunctions of the one-dimensional harmonic oscillator provides an interesting semiclassical interpretation of these functions and their relationship to the quantization rules of the old quantum theory. Some speculations regarding other possible applications of this approach are also presented.

American Journal of Physics•Feb 1, 1976•8 citations

Quantum mechanics and semiclassical approximations for a harmonic oscillator in half space

Peng Ni, Quanhui Liu, Zhu Shu-jun

The elementary quantum-mechanical results for the harmonic oscillator in half space are carried out, including expectation values for position, momentum and their square, the uncertainty relation in the eigenstates. Since the result of expectation value for position is quite complicated, the Heisenberg correspondence principle is used to give one of its approximate expressions, and the expression proves to be very accurate by numerical calculations.

College Physics•Jan 1, 2004•0 citations

Simple quantum systems in the momentum representation

H. N. Núñez‐Yépez, E. Guillaumı́n-España, R. P. Martı́nez-y-Romero +1 more

The momentum representation is seldom used in quantum mechanics courses. Some students are thence surprised by the change in viewpoint when, in doing advanced work, they have to use the momentum rather than the coordinate representation. In this work, we give an introduction to quantum mechanics in momentum space, where the Schrödinger equation becomes an integral equation. To this end we discuss standard problems, namely, the free particle, the quantum motion under a constant potential, a particle interacting with a potential step, and the motion of a particle under a harmonic potential. What is not so standard is that they are all conceived from momentum space and hence they, with the exception of the free particle, are not equivalent to the coordinate space ones with the same names. All the problems are solved within the momentum representation making no reference to the systems they correspond to in the coordinate representation.

arXiv:physics/0001030•Jan 17, 2000•3 citations

Introduction to Elementary Quantum Mechanics and Stability of the First Kind

Élliott H. Lieb, Robert Seiringer

In this second chapter we will review the basic mathematical and physical facts about quantum mechanics and establish physical units and notation. Those readers already familiar with the subject can safely jump to the next chapter. An attempt has been made to make the presentation in this chapter as elementary as possible, and yet present the basic facts that will be needed later. There are many beautiful and important topics which will not be touched upon such as self-adjointness of Schrödinger operators, the general mathematical structure of quantum mechanics and the like. These topics are well described in other works, e.g.. Much of the following can be done in a Euclidean space of arbitrary dimension, but in this chapter the dimension of the Euclidean space is taken to be three-which is the physical case-unless otherwise stated. We do this to avoid confusion and, occasionally, complications that arise in the computation of mathematical constants. The interested reader can easily generalize what is done here to the R d , d > 3 case. Likewise, in the next chapters we mostly consider N particles, with spatial coordinates in R 3 , so that the total spatial dimension is 3 N . A Brief Review of the Connection Between Classical and Quantum Mechanics Considering the range of validity of quantum mechanics, it is not surprising that its formulation is more complicated and abstract than classical mechanics. Nevertheless, classical mechanics is a basic ingredient for quantum mechanics. One still talks about position, momentum and energy which are notions from Newtonian mechanics.

Cambridge University Press eBooks•Nov 5, 2009•0 citations
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.
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.
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.
An analysis of the change in the concept of a particle required by the passage from classical to quantum theory suggests a ’’classical particle representation’’ for the latter which should allow purely quantum-mechanical results to be cast into a form suited to interpretation in semiclassical terms. Application of this representation to the examples of the free particle and the particle in a constant force field essentially reproduces the usual arguments as to how classical trajectories can be generated in the quantum theory, although from a somewhat different point of view than that usually presented. Applying the representation to the energy eigenfunctions of the one-dimensional harmonic oscillator provides an interesting semiclassical interpretation of these functions and their relationship to the quantization rules of the old quantum theory. Some speculations regarding other possible applications of this approach are also presented.
The momentum representation is seldom used in quantum mechanics courses. Some students are thence surprised by the change in viewpoint when, in doing advanced work, they have to use the momentum rather than the coordinate representation. In this work, we give an introduction to quantum mechanics in momentum space, where the Schrödinger equation becomes an integral equation. To this end we discuss standard problems, namely, the free particle, the quantum motion under a constant potential, a particle interacting with a potential step, and the motion of a particle under a harmonic potential. What is not so standard is that they are all conceived from momentum space and hence they, with the exception of the free particle, are not equivalent to the coordinate space ones with the same names. All the problems are solved within the momentum representation making no reference to the systems they correspond to in the coordinate representation.
In this second chapter we will review the basic mathematical and physical facts about quantum mechanics and establish physical units and notation. Those readers already familiar with the subject can safely jump to the next chapter. An attempt has been made to make the presentation in this chapter as elementary as possible, and yet present the basic facts that will be needed later. There are many beautiful and important topics which will not be touched upon such as self-adjointness of Schrödinger operators, the general mathematical structure of quantum mechanics and the like. These topics are well described in other works, e.g.. Much of the following can be done in a Euclidean space of arbitrary dimension, but in this chapter the dimension of the Euclidean space is taken to be three-which is the physical case-unless otherwise stated. We do this to avoid confusion and, occasionally, complications that arise in the computation of mathematical constants. The interested reader can easily generalize what is done here to the R d , d > 3 case. Likewise, in the next chapters we mostly consider N particles, with spatial coordinates in R 3 , so that the total spatial dimension is 3 N . A Brief Review of the Connection Between Classical and Quantum Mechanics Considering the range of validity of quantum mechanics, it is not surprising that its formulation is more complicated and abstract than classical mechanics. Nevertheless, classical mechanics is a basic ingredient for quantum mechanics. One still talks about position, momentum and energy which are notions from Newtonian mechanics.