Quantum Interference and Statistical Mechanics Part 2

 In a previous note, we argued that a quantum wavefunction W(x) is a relative conditional probability at x such that:   W(x+dx)=W(x)exp[i dx d lnW/dx (1/i)] ((1)) where <p ave at x>= -i d/dx lnW and <pp ave at x>= - d/dx d/dx W/W. The latter may be utilized in an energy conservation equation to yield either the Schrodinger or Klein-Gordon equation. W(x) in general is periodic with positive and negative values. Alternatively, one may write: W(x)=Sum over p a(p)exp(ipx) which implies a particle may be in an instantaneous free momentum p state.

These states “interfere” not just at the exp(ipx) level, but also when forming averages of p or pp/2m because exp(ipx)=cos(px) + i sin(px) and both functions take on positive and negative values. One may argue that “coarse graining” occurs removing extra information contained in the exp(ipx)’s. We wish to investigate this further in this note. In particular, we wish to show that ((1)) together with KEave(x) + V(x) = E seems to force interference upon exp(ipx) components.