Complex Probabilities with a Phase in Bound State Quantum Mechanics

Bound state quantum mechanics is formulated as a statistical theory in which averages seem to be given by the recipe:  <B> = integral dx W(x) B W(x) where W(x) is the wavefunction and B an operator. If B is strictly a function of x, the <B>= Integral dx d(x) B(x), where d(x)=W(x)W(x) is the spatial density and the recipe is exactly of the form of statistical mechanics. In the case that momentum is present, it is represented by ihbar d/dx operating on W(x) and so matters differ from classical statistical mechanics. In this note, we try to express bound state quantum mechanics in terms of complex probabilities (with phase) and show how absolute P(x), P(p) probabilities and also conditional P(x/p) and P(p/x) appear in the theory. We then suggest that it is P(x and p) that is important and show this has implications on entropy. In an earlier note, we described an entropy in such terms, but instead of using exp(ipx) as part of the probability, sin(px) was used to keep probabilities real. In this note, we suggest probabilities should be kept complex and shows this leads to a real expression for entropy. For the case of the ground state oscillator, this expression collapses to a sum of momentum entropy 2fp ln(fp) and spatial entropy 2W(x) ln(W(x)), where fp is the Fourier transform of W(x).