Double Dispersion Relations in Quantum Statistical Mechanics

A. K. Rajagopal, Morrel H. Cohen

Walter Felipe Wreszinski, G. Scharf

Massimo V. Fischetti, William G. Vandenberghe

Francesco R. Ruggeri

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.

Anders Andersen, Jacob Madsen, Christian Reichelt +5 more

In a thought-provoking paper, Couder and Fort [Phys. Rev. Lett. 97, 154101 (2006)] describe a version of the famous double-slit experiment performed with droplets bouncing on a vertically vibrated fluid surface. In the experiment, an interference pattern in the single-particle statistics is found even though it is possible to determine unambiguously which slit the walking droplet passes. Here we argue, however, that the single-particle statistics in such an experiment will be fundamentally different from the single-particle statistics of quantum mechanics. Quantum mechanical interference takes place between different classical paths with precise amplitude and phase relations. In the double-slit experiment with walking droplets, these relations are lost since one of the paths is singled out by the droplet. To support our conclusions, we have carried out our own double-slit experiment, and our results, in particular the long and variable slit passage times of the droplets, cast strong doubt on the feasibility of the interference claimed by Couder and Fort. To understand theoretically the limitations of wave-driven particle systems as analogs to quantum mechanics, we introduce a Schrödinger equation with a source term originating from a localized particle that generates a wave while being simultaneously guided by it. We show that the ensuing particle-wave dynamics can capture some characteristics of quantum mechanics such as orbital quantization. However, the particle-wave dynamics can not reproduce quantum mechanics in general, and we show that the single-particle statistics for our model in a double-slit experiment with an additional splitter plate differs qualitatively from that of quantum mechanics.

Bassano Vacchini