Fermat's Principle, Particles with Rest Mass and Quantum Mechanics Part II

    In Part I we noted that Fermat’s least time principle was equivalent to conservation of momentum in the direction of an invariant spatial variable and that this principle is contained in the dispersion relation E=pc for a light which is strictly a wave relationship. We then argued that this wave relationship should somehow be extended to a particle with rest mass because such a particle may bounce off a mirror like light. To make the connection we wrote  0 = -Et+(px)x +(py)y for light with r/t=c and  constant = -Et+(px)x + (py)y for a free particle with rest mass. In both cases Fermat’s principle is contained and is equivalent to conservation of momentum in the x direction. The form -Et+px, however, suggests a wavelength proportional to 1/p and a frequency proportional to 1/E for both light and a free particle with rest mass. We argue in this note that the wavelength represents spatial information which may show conservation (e.g. of momentum in a particular direction). Thus the focus of Part I was to link a free particle with rest mass to the wave formalism of light.

     Newtonian mechanics describes a free particle with rest mass as well as one moving in a potential without any need for a wavelength or frequency. In fact, for light E=pc means p→pn and c→c/n where n is the index of refraction means that E remains constant. For a particle with rest mass (nonrelativistic)  E = pp/2m + V(x). A change in p means one needs the potential to absorb or yield energy to keep E constant which is different from light. Given that Newtonian mechanics describes particles with rest mass one may ask: Why bother to try to link a free particle with light and Fermat’s principle as done in Part I? The answer is that a free particle (with rest mass) diffracts or shows interference just like light in experiments. Furthemore -Et+px which suggests a wavelength proportional to 1/p and frequency proportional to 1/E is equivalent to the free particle action: -mo t sqrt(1-vv/cc) (relativistic) and .5 mo t vv (nonrelativistic) with v=x/t. Thus if Lagrangian/Hamiltonian formalism is identical to Newtonian mechanics we argue that the idea of wavelength and frequency is already present. It is not manifest because one deals with averages in pp/2m + V(x) = E.

    To see how wavelength may be incorporated in Newtonian conservation of energy, consider a free particle in a potential i.e. E= pp/2m + V(x). Next consider knocking out the particle so it has constant momentum. We argue that such a particle should exhibit wave-like behavour i.e be described in a manner linking it with light/Fermat’s principle e.g. be described by exp(ipx). Fermat’s principle already suggests that d/dx → p so -id/dx exp(ipx) = p exp(ipx) is an eigenfunction. We then argue that if exp(ipx) describes the particle when it is knocked out of the bound state, this information should still exist while it is within the bound state. Thus we suggest the form:  W(x)= Sum over p a(p)exp(ipx) as describing the particle within a bound state. On average one has Newton’s conservation of energy result, but the idea that a particle may be knocked out and should behave as a wave is also present. Furthermore the various exp(ipx) interfere so the wave nature is also present in the bound state.