A simple formula for calculating quantal expectation values without the use of wave functions

Martin Kretzschmar

Rouslan L. Stratonovich

Daniel F. Styer

The Ehrenfest equations govern the time evolution of quantal expectation values, and they have a widely known generalization that governs the time evolution of quantal uncertainties. Unfortunately, these results are usually applied only to free particles. Here, these methods are reviewed and applied to the simple harmonic oscillator, revealing a simple and beautiful picture of wave-packet motion with an obvious classical limit.

T. S. Shankara

Per Olof Fröman

Simple phase-integral formulas, not involving wave functions, are derived for quantal matrix elements associated with bound states of a quantal particle in a smooth single-well potential. In these formulas one uses an arbitrary order of the phase-integral approximation generated from an unspecified base function.

Andrei Olegovich Barvinsky, G. A. Vilkovisky