The Principle of equal Probabilities of Quantum States
- URL: http://arxiv.org/abs/2111.09246v1
- Date: Wed, 17 Nov 2021 17:23:46 GMT
- Title: The Principle of equal Probabilities of Quantum States
- Authors: Michalis Psimopoulos, Emilie Dafflon
- Abstract summary: Boltzmann law $P(epsilon) = frac1langle epsilon rangle-fracepsilonlangle epsilon rangle ; ; ;;;; 0leq epsilon +infty$ where $langle epsilon rangle = E/N$.
kappa quanta, is given by $p(kappa)=fracdisplaystyle binomN+s
- Score: 0.0
- License: http://creativecommons.org/licenses/by/4.0/
- 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.
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