# Clas. Lim. of Schröd. eq. -> Hamilton Jacobi: Phase=Action - Direction of Prob. Flux?

1. Sep 30, 2012

### tobe

Hi,

reading Sakurai pages 102-103 (see http://www.scribd.com/doc/3035203/J-J-Sakurai-Modern-Quantum-Mechanics [Broken] ) I found one thing hard to understand:

If the Phase times Planck's constant equals Hamilton's Principal function in the classical limit (i.e. the action for the physically realized path), why is then the "velocity" (eq. (2.4.23)), respectively the probability flux j directed in the direction of INCREASING Action?
It would be the same in classical mechanics if you identify gradient of S with the momentum vector.

Isn't that in opposition to Hamilton's principle that the action has to be minimized (or at least stationary) on the actual physically realized "path"(I know path is the wrong word for quantum mechanics)?
I guess locally increasing doesn't mean that you cannot still achieve a global minimum somehow (my mathematics background is rather poor, so it would be nice if somebody could help me out here, because I find that hard to imagine).

Cheers,
tobe

Last edited by a moderator: May 6, 2017