Bound State Quantum Mechanics as a Statistical Theory in Momentum Space?

Classical statistical mechanics appears to be a statistical theory in momentum space with the Maxwell-Boltzmann distribution as the driver. The exp(-V(x)/T) spatial dependency follows from the Maxwell-Boltzmann distribution. The reason the MB distribution “exists” in momentum space is that it can be derived solely from two particle scattering with f(v1)f(v2)=f(v1prime)f(v2prime) together with conservation of momentum and kinetic energy i.e. elastic scattering. Maximization of entropy is not needed to derive the MB distribution, although it can be derived in such a manner. Thus, classical statistical mechanics is a theory of many particle scattering in momentum space subject elastic scattering or conservation of kinetic energy. Bound state quantum mechanics, on the other hand, is usually based on the time-independent Schrodinger equation which is a spatial differential equation. Thus, the focus is on space and spatial derivatives. The wavefunction W(x)= Sum over p fp sin(px) (or cos(px) depending on the symmetry) links momentum and space, but one is really concerned with W(x). Density W(x)W(x) is strictly a spatial observable. Furthermore, there is interference because sin(px) can have positive and negative values and so matters are complicated. For example, the average kinetic energy [Sum over p p*p/2m fp sin(px)]/W(x) at a point x, involves some sin(px) values being positive and others negative, which is difficult to interpret. In this note, we suggest considering bound state quantum mechanics as a statistical theory in momentum space. Instead of two particle collisions, collisions are occurring with the potential in a similar manner. In quantum mechanics fp, the momentum wavefunction is the square root of momentum probability. This allows fp to be positive or negative, unlike f(v) in classical statistical mechanics. These positive and negative values, present even when there is no potential, allow for periodicity, which seems to be central to quantum mechanics. We try to show in this note, that bound state quantum mechanics is also based on conservation of energy at each p level. It is interesting to note that for the ground state of the quantum oscillator, where there are no humps (the solution is Gaussian in both space and momentum space), the quantum statistical theory yields the same results as classical statistical mechanics