Classical Maximum Entropy Probabilities Versus Quantum Dynamical Probabilities

There is a focus in classical statistical mechanics on maximizing an entropy function subject to constraints to obtain probability values. For a coin or die probabilities are given, yet one may still calculate a Shannon’s entropy. In the case of a Maxwell-Boltzmann (MB) gas, one may obtain probabilities using energy conservation and products of probabilities and then show this is equivalent to maximizing an entropy function subject to an energy constraint. We argue that the energy conservation approach yields an energy conservation equation and a product of probabilities representing a certain total energy equated to a product of different probabilities representing the same total energy. The two equations are linked to solve for probability in terms of energy and this is formally the same as a maximization problem subject to conservation of energy. A point we make is that this approach is based on a kind of static conservation, even in the presence of a potential V(x). In quantum mechanics, we argue one one has fluxes even though there is also conservation of energy. For light moving in the positive x direction which reflects and refracts (with each photon retaining its energy) there is both energy conservation and flux. The energy conservation equation in terms of flux/intensity terms is given in (1) as: AA/c1 = BB/c1 + CC/c2 where A,B,C refer to the incident, reflected and refracted rays. This single equation cannot solve for probabilities B,C in terms of A and there is not a separate product of probabilities equation as in the MB case. Rather quantum mechanics seems to be solved using square root fluxes as dynamical probabilities. In particular, the energy conservation equation is equivalent to two square root flux equations A+B=C and p1A -p1B = p2C where E=pc. Thus probabilities are determined through dynamics not static conservation or entropy maximization. In an earlier note we concluded that exp(ipx), a quantum free particle wavefunction, is also like a square root flux, and also acts as a dynamic probability. These ideas may be extended to a quantum bound state which we discuss in the note. Thus the approach to finding probabilities in the classical statistical mechanics and quantum realms seem to be completely different. One uses static conservation laws in the former and square root flux continuity/dynamical probability in the latter