Quantum and Classical Momentum Distributions II. Morse and Coulomb Potential

孙国华, 董世海

Hans Jürgen Korsch, B. Schnellhaß

H. J. Korsch, B Schellhaaß

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.

J Stanek, Jerzy Konarski

In this paper a theoretical study of the Coulomb problem in quantum phase-space is reported. The Langer coordinate transformation is used to map the Coulomb problem into a one-dimensional Morse oscillator. As a result the Wigner distribution functions for the Morse oscillator are obtained. The form of these functions is presented for a few principal quantum number n and the angular momentum quantum number l . The results obtained correspond to the solution of the Coulomb problem in the spherical coordinate system.

Kenneth G. Kay

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.

Benjamin Withers, H. R. Glyde