Bound State Quantum Mechanics as a Statistical Theory

 In this note, we try to develop bound state quantum mechanics directly from statistical arguments. The key idea, we argue, is that a statistical treatment of a system considers “free particles” together with a stochastic equilibrium driving mechanism. If an acceleration driving potential exists (as in classical statistical mechanics), it alters the results, but does not change the main scenario of free particles interacting through a stochastic mechanism at a point because one does not follow particles in time, noting how they accelerate as they travel. (For example, a Maxwell-Boltzmann gas distribution is based on two body elastic scattering whether there is a potential or not. The potential affects the density, but not the relative momentum probabilities at any x.) For the quantum case, the potential V(x) is the stochastic equilibrium driver, but on average kinetic energy plus potential must equal total energy. Again, one does not follow the quantum particle in time trying to see if or how it accelerates.