Classical limit of the canonical partition function

TL;DR

This paper investigates the classical limit of the quantum canonical partition function, establishing exact relations between symmetrized and non-symmetrized density matrices, and clarifying the role of the N! factor in statistical mechanics.

Contribution

It provides precise mathematical relations between symmetrized and non-symmetrized density matrices, offering a new physical interpretation of the N! factor in quantum statistics.

Findings

Derived exact relations between symmetrized and non-symmetrized density matrices.

Clarified the physical meaning of the N! factor in quantum statistical mechanics.

Established reverse relations for partition functions of fermions and bosons.

Abstract

We analyze the so-called classical limit of the quantum-mechanical canonical partition function. In order to do that, we define accurately the density matrix for symmetrized and antisymmetrized wave functions only (Bose-Einstein and Fermi-Dirac), and find an exact relation between them and the density matrix for non symmetrized functions (Maxwell-Boltzmann). Our results differ from the generally assumed in a numerical factor N!, for which we suggest a physical interpretation. We derive as well the reverse (and also exact) relation, to find the canonical partition function for non-symmetrized wave functions in terms of the corresponding function for fermions and bosons.