1212.6786 (K. R. W. Jones)

The classical Schrödinger equation    [PDF]

K. R. W. Jones

Using a simple geometrical construction based upon the linear action of the Heisenberg--Weyl group we deduce a new nonlinear Schr\"{o}dinger equation that provides an exact dynamic and energetic model of any classical system whatsoever, be it integrable, nonintegrable or chaotic. Within our model classical phase space points are represented by equivalence classes of wavefunctions that have identical position and momentum expectation values. Transport of these equivalence classes is effected in a manner that avoids dispersion and thereby leads to a system of wavefunction dynamics such that the expectation values track classical trajectories precisely for arbitrarily long times. Interestingly, the value of $\hbar$ proves immaterial for the purpose of constructing this alternative version of classical mechanics. The new feature which $\hbar$ does mediate concerns a surprising embedding of Berry's phase within ordinary classical mechanics. Some interesting problems are exposed concerning inclusion of the projection postulate within this model nonlinear system and we discover a remarkable route for the recovery of the ordinary linear theory.

View original: http://arxiv.org/abs/1212.6786