Maximum Entropy and Resistance Balance in Quantum Mechanics

Classical statistical mechanics often utilizes the idea of maximizing entropy subject to a physical constraint For example, for a Maxwell-Boltzmann gas, this constraint is Sum over i ei p(ei) where ei=pp/2m (no V(x)). We have argued previously that there may be many possible constraints, i.e. an average of any power of ei could be a physical constraint. We stated that a conserved quantity associated with interactions present (creating equilibrium) should be used. Here we examine the constraint in terms of mechanical resistance i.e. pressure. 

   We further consider the free particle quantum wavefunction exp(-iEt+ipx) in terms of resistances in time and space. We argue that the form exp(-ipx)exp(ipx) = 1 removes the stochasticity associated with motion due to the resistance (impulse) p Thus 1 or 1/L (L=length) seems to be a smoothed over picture associated with a constant flux and overlooks the inherent stochasticity associated with motion.  This stochasticity is crucial when describing impulse hit reactions (n-slits, index of refraction interfaces, interactions with V(x)), but if a particle does not interact in space, then the W*(x)W(x) picture seems to be used to describe the v(average)=x/t scenario. The issue is that for a quantum bound state, there is also an rms velocity based on a wavefunction and one may create an average W(x)W(x)  (W(x) is real for a bound state), but like exp(-ipx)exp(ipx), this bypasses some of the stochasticity in the problem, we argue. It only demonstrates stochasticity associated with differences in momentum.