Abstract
The analysis of the Helmholtz equation is shown to lead to an exact Hamiltonian system describing in terms of ray trajectories, for a stationary refractive medium, a very wide family of wave-like phenomena (including diffraction and interference) going much beyond the limits of the geometrical optics (“eikonal”) approximation, which is contained as a simple limiting case. Due to the fact, moreover, that the time independent Schrödinger 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 leads to a system of Hamiltonian equations providing exact and deterministic particle trajectories and dynamical laws, and containing the laws of Classical Mechanics in the eikonal limit.
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Orefice, A., Giovanelli, R. & Ditto, D. Complete Hamiltonian Description of Wave-Like Features in Classical and Quantum Physics. Found Phys 39, 256–272 (2009). https://doi.org/10.1007/s10701-009-9280-2
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DOI: https://doi.org/10.1007/s10701-009-9280-2