Quantum and Classical 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.

Benjamin Withers, H. R. Glyde

Paul E. Sokol, Richard N. Silver, J. W. Clark

Richard A. Campos

We explore, using elementary examples of harmonic-oscillator superposition states, the quantum-mechanical measures of covariance and correlation coefficient for two noncommuting observables. A Schrödinger cat state which provides perfect correlation or anticorrelation of the position and momentum operators is presented. We employ quantum and classical phase-space distributions as graphical illustrations of correlation coefficients for the two theories.

Mark S. Ashbaugh

The author presents an elementary example which shows that position and momentum distributions do not uniquely determine the wavefunction. In particular, the example shows that a wavefunction consisting of two gaussian wavepackets with momenta k 1 and k 2 has the same position and momentum distributions as the wavefunction obtained by interchanging k 1 and k 2 . This interpretation relies on viewing the wavefunction as that of a free particle but, of course, the abstract result is simply that for f in L 2 (R) knowledge of mod f(x) mod and mod f(k) mod is not always sufficient to determine f(x) (or f(k); here f denotes the Fourier transform of f). In addition, the bearing of the example presented here upon certain interesting questions raised by Vogt (1978) is discussed.

M. Fannes, André F. Verbeure, Ricardo Weder