Statistical Periodicity in Classical and Quantum Physics Part II

 In a previous note (1), we argued that sound wave equations (with certain boundary conditions) derived from the Boltzmann transport equation conservation relations match the form of a quantum mechanical solution of a particle in a box with an infinite potential at the walls. Furthermore, we argued that in both the sound and quantum pictures there exists a dispersion velocity u= d/d/x ln(density) (classical) and u= -i d/dx ln(W) where W is the wavefunction in quantum mechanics. Both represent forward backward motion at different x points. Thus, there seems to be a link between classical sound and quantum mechanics at least in terms of some density type solutions and a backwards forwards motion.

   It is also known, however, that for the ground state of a ground oscillator, the quantum solution is of the same form as the static Maxwell-Boltzmann (MB) solution  C exp(- mvv/2mT - .5kxx/T). In this case, quantum mechanics still has a dispersion velocity with forward backward motion, but statistical mechanics has a zero u(x,t). The Boltzmann transport conservation relations yield the MB solution on the basis of a balance of pressure and force. This leads to the question: Why do the two approaches yield the same solution in this one case, even though the underlying physics seems to be quite different?  We argue that even though u(x,t)=0 for the MB static solution, there is still dispersive flow which is forward at the left endpoint and backwards at the right. Thus, in both the case of sound and the MB solution with a potential there may be dispersion which has links to quantum mechanics.