Thermodynamic Approach to Quantum Mechanics and the Question of Pressure

Quantum mechanics and classical statistical mechanics/thermodynamics are both statistical theories, but have key differences. We argue that one such key difference is nonlocality in the quantum world i.e a wavelength proportional to 1/p and a probability of exp(ipx) associated with p versus locality in the classical case. Quantum nonlocality tends to lead to spatial densities with humps even for a particle with a fixed energy in an infinite potential box. 

     Recently it has been suggested in (1) that one may think of a quantum particle in a box with infinite walls in terms of thermodynamical properties e.g. pressure which is taken to be constant throughout the box. From this assumption, one finds that quantum entropy is ln[ Average(nxnx + nyny+nznz)]. For a single quantum state, this seems to imply that entropy is ln[nxnx+nyny+nznz] ((1a)).

    In a previous note (2), we suggested that quantum entropy for a wavefunction (say an eigenstate) is  Integral dx W*(x)W(x) ln[W*W] +  Integral dp a*(p)a(p) ln[a*(p)a(p)] ((1b)). We found this is independent of L for a particle in a box with infinite walls so that displacing the wall adiabatically should mean that dS/dL = 0 which it does as L cancels from ((1)). 

     We argue in this note that quantum pressure should be linked to the density humps for a sin(pave x)sin(pave x) wavefunction even though pave is constant at all x points. Thus it seems that one would not consider a constant pressure for sin(pave x)=wavefunction.  Furthermore ((1b)) does not yield the same entropy as ((1a)). It is noted in (1) that ((1a)) is an exact expression for entropy and that the Maxwell-Boltzmann distribution entropy approximates it. In this note, we argue that a time reversal balanced scattering reaction leads to the Fermi-Dirac and Bose-Einstein distributions and these both lead to the Maxwell-Boltzmann distribution under certain conditions. (The Maxwell-Boltzmann distribution may also be obtained from time reversal balanced scattering with looser restrictions.)