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European Journal of Physics

Quantum, classical and semiclassical momentum distributions: II. Morse and Coulomb potentials

H. J. Korsch, B Schellhaaß

Published Jan 1, 2000• 9 citations
Publisher

Abstract

The basic properties of momentum distributions in quantum mechanics for elementary systems as well as their (semi-)classical analogue derived in a preceding paper are illustrated by a detailed analysis of the Morse oscillator and the three-dimensional Coulomb potential, where the distributions can be calculated in closed form.
DOI
10.1088/0143-0807/21/1/311

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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.
Wave functions for the Coulomb, Morse, Rosen-Morse, and Eckart systems are expressed exactly and explicitly in terms of the corresponding classical trajectories. This is accomplished by a technique that maps previously obtained classical representations for wave functions of the three-dimensional isotropic harmonic oscillator to the wave functions for the present systems. By design, the classical behavior enters these expressions in a manner that allows them to be used as uniform semiclassical approximations for other systems. The construction of a hydrogen orbital from the corresponding classical orbits is illustrated for a particular case. The classical representation of the Eckart wave function is analyzed to understand how ordinary, real, classical trajectories that are confined to one side of a barrier are able to describe tunneling through the barrier.
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.
Using an analytic approximation to the quantum momentum previously derived to calculate single particle reduced density matrices in the semiclassical approximation, we have been able to construct equations of motion with which to perform trajectory calculations. Information calculated along these trajectories is used to calculate single particle reduced density matrices for one-dimensional systems. Charge densities for N fermions in a harmonic oscillator potential calculated using this quantum momentum approximation are in excellent agreement with the exact quantum mechanical calculations.
Semiclassical angular momentum matrices are calculated using a new contour-integral formula for matrix-elements. WKB spherical harmonic functions are found to be exaclty orthonormal with the contour-integral inner-product. Matrix-elements obtained from these wave-functions are accurate to about 2 % and the matrices obey the commutation relations expected of quantum angular momentum operators. The semiclassical wave-functions are related to a superposition of allowed classical orbits and this illustrates the connection to the Feynman pathsummation formulation of quantum mechanics for electrons in a spherically symmetric potential.