A Grasp of Identity

TL;DR

This paper revisits the quantum treatment of identical particles, analyzing symmetrized, antisymmetrized, and Maxwell-Boltzmann solutions, revealing implications for particle individuality and statistical mechanics.

Contribution

It introduces a new solution to the Schrödinger equation with Maxwell-Boltzmann statistics, challenging traditional views on particle identity in quantum mechanics.

Findings

Symmetrized and antisymmetrized eigenfunctions blur particle identity.

A new solution preserves particle individuality with non-univocal properties.

Extensivity of the partition function arises from particle identity.

Abstract

We revisit the treatment of identical particles in quantum mechanics. Two kinds of solutions of Schr\"{o}dinger equation are found and analyzed. First, the known symmetrized and antisymmetrized eigenfunctions. We examine how the very concept of particle is blurred whithin this approach. Second, we propose another kind of solution with no symmetries that we identify with Maxwell-Boltzmann statistics. In it, particles do preserve their individuality, as they are provided with individual energy and momenta. However, these properties cannot be univocally ascribed; moreover, particles do not possess distinctive positions. Finally, we explore how these results affect the calculation of canonical partition function, and we show that extensivity arises as a consequence of identity.