Wavefunction and the Identity of a Momentum State in a Quantum Bound State

Addendum July 19, 2021  The Fermi-Dirac and Bose-Einstein statistical cases, like the Maxwell-Boltzmann case, may be formulated in terms of a reaction balance between a regular two body elastic collision and its time reversed reaction:  e1+e2 = e3 + e4 and f(e1)[1-/+f(e3)]f(e2)[1-/+f(e4)] = f(e3)[1-/+ f(e1)] f(e4)[1-/+ f(e2)]. Both the Fermi-Dirac and Bose-Einstein cases reduce to the Maxwell-Boltzmann case under certain conditions.

  An unusual feature of bound state quantum mechanics is that there is a spatial density (or probability to be at x = P(x)), just as in classical statistical mechanics, but also a wavefunction W(x) such that P(x)=W(x)W(x). In other words, W(x) for the bound state is like a square root of probability. Furthermore, there is a P(p)=a(p)a(p) where W(x)=Sum over p a(p)exp(ipx). In this note, we try to argue that P(p), which is based on the measurement of momenta of particles knocked out of an ensemble of identical single particle bound states, does not have physical meaning within the bound state, unlike P(p,x) which has physical meaning for a Maxwell-Boltzmann gas both inside the gas and for particles released from a small hole in the gas wall. This forces one to create another function to “link” the two physically observed probabilities P(x) and P(p).