Momentum ConservationIncluded In Fermat's Least Time Principle and Wave Mechanics

According to (1) Fermat’s minimum time principle is identical to Hyugen’s wave principle. In a previous note (2) we considered two ray light systems (in particular Snell’s law and reflection from a mirror moving at constant velocity v) and argued that Fermat’s principle is equivalent to a hypothetical velocity approach where H sin(AA) = v(A relative)  and H sin(BB = v(B relative). Here H is the hypothetical velocity which acts as a hypotenuse and AA and BB are the incident and reflected rays in the case of a moving mirror (measured relative to the normal) and the incident and refracted rays for Snell’s law. The hypothetical velocity approach, however, is exactly in the form needed for conservation of momentum parallel to the medium surface which holds for both the moving mirror and Snell’s law if:  1/wavelength(A) is proportional to 1/v(A relative) and the same result for B. This is equivalent to considering a fixed frequency f which applies to A and B (and is not 1/energy proportional to wavelength) so that vrel(A) = (same frequency) wavelength(A) and the same for B. As a result, conservation of momentum parallel to the medium surface is implicitly included in Fermat’s principle.

   In the case of quantum mechanics of a free particle represented by exp(ipx) where p is momentum, conservation of momentum is also built in because  exp(ipx) is orthogonal to exp(ip1x). For <exp(ip1x) A W> where W is a wavelength only results consistent with conservation of momentum occur as is well-known in the literature. We argue that quantum mechanics of a free particle is statistical and linked to the idea of an equilibrium for which one does not consider time. Thus for exp(-iEt+ipx), t and x may be considered as independent. This implies that exp(ipx) must carry the idea of conservation of momentum by itself because one is not following a particle via x(t). “X’ may take on any value so this is an equilibrium type of scenario and the orthogonality of different exp(ipx) vectors ensures conservation of momentum. Thus a wave approach again implicitly contains the notion of conservation of momentum.