Adriano Orefice, Raffaele Giovanelli, Domenico Ditto
The behavior of classical monochromatic waves in stationary media is shown to be ruled by a frequency-dependent "Wave Potential" function, encoded in the structure of the Helmholtz equation. We present here, in Hamiltonian form, an exact, ray-based kinematic treatment (reducing to the usual geometrical optics approximation when the Wave Potential is neglected) where a mutual ray-coupling, due to the Wave Potential and acting normally to the ray trajectories, turns out to be the one and only cause of wave-like phenomena such as diffraction and interference. Recalling, then, that the time-independent Schroedinger equation (associating the motion of mono-energetic particles with stationary monochromatic "matter waves") is itself a Helmholtz-like equation, the ray-based kinematic treatment developed in the classical case is extended - without resorting to statistical concepts - to the exact, trajectory-based Hamiltonian quantum dynamics of point-like mono-energetic particles. The particle trajectories and motion laws turn out to be coupled, in this case, by a suitable, energy-dependent Wave Potential (assuming the form, but not the statistical nature, of Bohm's energy-independent "Quantum Potential"), and their numerical computation is shown to be possible without requiring the simultaneous solution of the time-dependent Schroedinger equation. The time-independent Schroedinger equation is argued to be the basic tool providing both the quantum dynamical ground on which the statistical descriptions associated with the time-dependent Schroedinger equation are based, and the most natural transition from classical to quantum dynamics.
View original:
http://arxiv.org/abs/1207.0130
No comments:
Post a Comment