ABSTRACT
The analysis of the Helmholtz equation is shown to lead to an
exact Hamiltonian system describing in terms of ray trajectories
a very wide family of wave-like phenomena (including diffraction
and interference) containing the eikonal approximation as a simple
limiting case. Since, moreover, the time-independent Schro¨dinger
equation is itself a Helmholtz-like equation, the same mathematics
holding for a classical optical beam turns out to apply to a quantum
particle beam moving in a stationary force field, and the so-called
quantumpotential of the Bohmian theory turns out to be a particular
case of a general wave-like property. A suitable set of Hamiltonian
equations is shown to provide both the exact and deterministic
trajectories of the particles and their dynamical laws, reducing to the
laws of classical mechanics in the eikonal limit. Wave-like features in
Hamiltonian form are presented in some typical numerical cases.
