An elementary proof of symmetrization postulate in quantum mechanics for a system of particles

TL;DR

This paper provides a mathematical proof that, under certain conditions, the wave function of identical particles in quantum mechanics must be either symmetric or antisymmetric, supporting the symmetrization postulate without considering spin.

Contribution

It offers an elementary proof of the symmetrization postulate for identical particles, assuming invariance of probability density and continuity conditions, ignoring spin effects.

Findings

Wave function must be symmetric or antisymmetric under particle exchange.

Proof relies on invariance of probability density and continuity conditions.

Applicable to connected configuration space with exchange-invariant potential.

Abstract

According to symmetrization postulate for a system of identical particles, wave function has to be completely symmetric or completely anti-symmetric. In this paper we want to mathematically justify this postulate ignoring the spin part of wave function in three dimension. For a system of N identical particles, if the solution to the governing Schrodinger equation meets these criteria: a) the probability density remains invariant when any two particle positions are exchanged over time, b) the wave function is continuous and has a continuous gradient, and the system exhibits the following characteristics: c) the configuration space, which is 3N dimensional, is connected, and d) the potential term in the Hamiltonian is invariant under the exchange of any two particle positions, then the wave function must be either totally symmetric or totally antisymmetric over time.