The Principle of equal Probabilities of Quantum States

TL;DR

This paper derives the probability distribution of quanta among particles using quantum and classical principles, showing their equivalence and justifying classical assumptions through quantum mechanics.

Contribution

It introduces a new principle of equal probability of quantum states for indistinguishable quanta, connecting quantum statistics with classical Boltzmann law.

Findings

Derives $p()$ using quantum state distribution

Shows equivalence of quantum and classical statistical mechanics for the system

Demonstrates the classical Boltzmann law as a limit of quantum distribution

Abstract

The statistical problem of the distribution of $s$ quanta of equal energy $\epsilon_0$ and total energy $E$ among $N$ distinguishable particles is resolved using the conventional theory based on Boltzmann's principle of equal probabilities of configurations of particles distributed among energy levels and the concept of average state. In particular, the probability that a particle is in the \k{appa}-th energy level i.e. contains \k{appa} quanta, is given by $p(\kappa)=\displaystyle \frac{\displaystyle \binom{N+s-\kappa-2}{N-2}}{\displaystyle \binom{N+s-1}{N-1}} \;\;\; ; \;\;\; \kappa = 0, 1, 2, \cdots, s$ In this context, the special case ($N=4$, $s=4$) presented indicates that the alternative concept of most probable state is not valid for finite values of $s$ and $N$. In the present article we derive alternatively $p(\kappa)$ by distributing $s$ quanta over $N$ particles and by introducing a new principle of equal probability of quantum states, where the quanta are indistinguishable in agreement with the Bose statistics. Therefore, the analysis of the two approaches presented in this paper highlights the equivalence of quantum theory with classical statistical mechanics for the present system. At the limit $\epsilon_{o} \rightarrow 0 $; $s \rightarrow \infty $; $s \epsilon_{o} = E \sim$ fixed, where the energy of the particles becomes continuous, $p(\kappa)$ transforms to the Boltzmann law $P(\epsilon) = \displaystyle \frac{1}{\langle \epsilon \rangle}e^{-\frac{\epsilon}{\langle \epsilon \rangle}} \;\;\; ; \;\;\; 0\leq \epsilon < +\infty$ where $\langle \epsilon \rangle = E/N$. Hence, the classical principle of equal a priori probabilities for the energy of the particles leading to the above law, is justified here by quantum mechanics.