Quantum Mechanics, Statistical Indistinguishability and Brownian Motion
| Main Author: | Francesco R. Ruggeri |
|---|---|
| Format: | info publication-preprint Journal |
| Terbitan: |
, 2021
|
| Subjects: | |
| Online Access: |
https://zenodo.org/record/4421282 |
Daftar Isi:
- Brownian motion is formulated in the form d/dt (partial) spatial density = D d/dx d/dx density. This motion is generalized in the Ornstein-Uhlenbeck case (1) to: d/dt(partial) density(v) = bd/dv {v density(v) + kT d/dv density(v)/m} which yields the Maxwell-Boltzmann equilibrium distribution Cexp(- .5mvv/T). It is also generalized in the Smoluchowski case (1) to: d/dt (partial) density = d/dx (density dV/dx + kTd/dx density)/mb which yields the equilibrium result (d/dt partial density =0) of Cexp(-V(x)/T) where V(x) is the potential. In a previous note (2), we argued that for quantum mechanical equilibrium one may try to apply Brownian motion to the relative conditional probability i.e. the wavefunction W(x,t). In such a case, the equilibrium requirement is no longer d/dt partial density=0. In this note, we try to consider the nonrelativistic Brownian motion equation for W(x,t) in terms of statistical indistinguishability. We argue that statistical indistinguishability occurs at two levels. First, there is an inherent indistinguishability for a quantum particle with constant p to be at any x, yet there is a probability flow proportional to p. This suggests exp(ipx) as a solution, but exp(ipx) is also a solution of a Brownian motion equation with d/dt partial W(x,t) = -iE. For the case of a potential V(x), a second stochastic process occurs. By taking the time independent Schrodinger equation and replacing W(x,t) and V(x) by Fourier series we find: E ap = ap pp/2m + Sum over k Vk a(p-k) ((1)) This suggests that V(x) is an average describing stochastic hits which give rise to a momentum distribution at each x. Each constant momentum is associated with exp(ipx) and a weight a(p) and undergoes double stochastic motion (due to pp/2m Brownian motion and V(x)). Statistical indistinguishability suggests that for any p, the two Brownian motions are distinguishable only by a(p) which we argue is the statement of ((1)). We also show that statistical indistinguishability occurs in classical statistical mechanics. First, the full MB distribution Cexp(-.5/m pp/T - V/T)=Cexp(-E/T) utilizes statistical indistinguishability at different x points. Secondly, the term {v density + kT d/dv density/m} in the Ornstein Uhlenbeck case follows from statistical indistinguishability, as will be shown.