Quantum Stochastic Energy Conservation

 Classical statistical equilibrium appears to be based on a no-flow situation with a balance of pressure (including -dV/dx in the case of a potential present). It may be noted, however, that like quantum mechanics, a single particle has different probabilities of being at one x versus another. Furthermore, the Maxwell-Boltzmann, Fermi-Dirac and Bose-Einstein distributions may be obtained using reaction balance which is linked to conservation of energy i.e. g(f(ei))g(f(ej)) = g(f(el))g(f(ek)) where ei+ej=el+ei=e. Taking ln of the former expression and linking to the latter yields f(ei), the probability distribution. g=1 for the MB case, f(e)/(1-f(ei)) for the FD and f(e)/(1+f(ei)) for the BE. 

    In classical mechanics, one has the conservation equation:  p(x)*p(x)/2m + V(x) = E. In quantum bound systems one has:   KE(ave)(x) + V(x) = E which appears to be very similar, although it is based on ensemble averages and there is a momentum distribution P(p/x) at each x. A question which arises is: What is the “conservation of energy” equation for a particular p of the P(p/x) distribution? In this note, we argue the “potential” term associated with a component p*p/2m is related to the part of the potential which contributes to the “creation” of p. In particular, we consider V(x)= Sum over k V(k) exp(ikx) where each k term renders a kick of momentum k. One may create a momentum p using k and p-k for each value of k. We further argue that:   p*p/2m P(p/x) + contributing portion of V(x) = E P(p/x)  ((1)) for all p. Thus, in quantum equilibrium, which involves a flow, there is a “constant E” type of equation i.e. ((1)), different in form but similar in idea, to that of classical statistical mechanical equilibrium. In previous notes, we referred to this as cycling, but in this note we wish to consider it in terms of dynamics. We further note that in both classical statistical mechanics and quantum mechanics, the energy-probability p equation makes use of the overall average energy which is bT for the classical case and E for the quantum. (b is constant)