Ehrenfest Theorem for the Hamilton-Jacobi Equation

Abstract

A possibl e way fro m qua ntum me chanics to classical mechanics can b e achie ved w ith an exp onential substitutio n used in the Schr odin ger equation , and then considerin g the classical limit . T his gives a picture of classical Ûuid and an ensemble of classical tra jectories. In di˜erence from this approach to the classical limit, w hile utilisi ng the same substitution , w e assume a minimum uncertainty w ave packet. It is show n that this approach to the classical limit of quantum mechanics yields a single traj ectory traced by the centroid of the minimum uncertainty w ave packet. T he momentum and the centroid of such packet satisf y the classical H amilton { J acobi equation. PACS numb ers: 03.65.Bz

What is the limit ℏ → of quantum theory?

“…Using these facts in Ehrenfest's equations, Eqs. (15) and (16), we obtain immediately the integral equation for deterministic potentials…”

“…We have just shown that the first (more critical) condition is fulfilled; Ehrenfest-like relations corresponding to the second condition exist in almost all statistical theories. For the PHJ these take exactly the same form as in QT, namely [15,16]…”