Speculation on Quantum Thermodynamics and the Adiabatic Theorem

Addendum Feb. 3, 2022  We restrict S=entropy=Sx+Sp to pure energy eigenstates. I.e. Wn(x) used in Sx is an energy eigenstate.

  In (1) we suggested the following form for the entropy of a quantum bound state:

S = Integral dx W(x)W(x) ln[WW] + Integral dp a(p)a(p) ln[a(p)a(p)] ((1)) where W(x)= Sum over p a(p) exp(ipx). In principle this may be extended to nonpure bound states. In (2) we showed that this definition (which follows from a Shannon’s entropy with a particular choice of probability) ensures that an adiabatic change in the size of box with an infinite potential leads to no change in entropy as ln(1/L) from the first term of ((1)) cancels with ln(L) from the second for every energy level n. This seems to have some thermodynamic consequences. We revisit the particle in a box with infinite potential and also examine the quantum oscillator solution as the spring constant or mass is varied and find that it behaves in a similar way for all energy levels under an adiabatic transformation. We finally consider under what conditions such a transformation would lead to entropy changes.