Two Kinds of Entropy in Quantum Mechanics?

 In a previous note (1) we suggested the existence of a quantum entropy associated with a free quantum particle with momentum p and wavefunction exp(ipx) due to the presence of a wavelength. The usual approach of using P(x)=W*(x)W(x) in Shannon’s entropy does not capture this entropy as it yields a constant for all p values. Thus we suggested using |cos(px)| C where C is a normalization constant over half a wavelength. This value may be used in Shannon’s entropy formula integrated over a half wavelength to give an entropy per crest. We then extended the idea to a quantum bound state.

     In this note we wish to examine the relationship of this entropy to momentum i.e. energy flow. In (1) we noted that this entropy does not satisfy the first law of thermodynamics. Here we argue that it also does not satisfy the second law using ideas from refraction.

   Finally we note that quantum thermodynamical calculations using temperature seem to ignore this entropy. In fact, these calculations ignore the usual P(x)=W*W or P(p) = a*(p)a(p) Shannon’s entropies for a pure state suggesting that only some “uncertainty” is included in typical entropy calculations while other uncertainty may apparently be ignored.