Speculation on Quantum Bound Entropy For Crests and Thermodynamics

 Shannon’s entropy is defined for probabilities as - Sum over i P(i) ln(P(i)). For a probability density, such as spatial or momentum density which appear in quantum bound states i.e. P(x) = W*(x)W(x) or P(p) = a(p)* a(p) where W(x)=wavefunction = Sum over p a(p)exp(ipx), one defines:  Sx = - Integral dx P(x)) ln{P(x)} and Sp = -Integral dp P(p) ln(P(p)) (1) and integrates over al space or momentum.

  For quantum bound states, however, one may integrate Shannon’s entropy using P(x) and/or P(p) only over the region of a single crest. Such a crest is of the form of an unnormalized probability density which has zero probability at the endpoints. Thus the total P(x) and/or P(p) might consist of a set of crests with possibly different heights.

   Thus instead of integrating over all x one may consider integrating only over the base of a particular crest to find a local entropy density so to speak. A reason for doing this is that the overall entropy may not satisfy dE = TdS, but an entropy for a crest might. We investigate this for an infinite well and an oscillator.  For example, for a infinite potential well Sx over the full range 0 to L is independent of n, but by adding energy one creates pockets (crest entropies) which are very low as one has very high resolution.Thus we argue that a quantum bound system has an overall entropy, but also entropies for each crest (whether in x or p space). The latter may be linked to thermodynamics in a different manner than the former.