[Paper Review] Derivation of the Pauli exchange principle

A. A. Broyles|ArXiv.org|Jun 14, 1999

Quantum chaos and dynamical systems2 references3 citations

TL;DR

This paper presents a simplified derivation of the Pauli exchange principle using wave function symmetry properties, offering an alternative to Pauli's original quantum field theory approach. It demonstrates that exchanging two identical particles multiplies the wave function by $(-1)^{2s}$, confirming the spin-statistics connection through elementary quantum mechanical reasoning.

ABSTRACT

Wave functions are generally written with arguments consisting of sets of ``particle'' coordinates and quantum numbers. Pauli derived a principle governing the exchange of pairs of sets that differ only in their spatial and spin component $(m_s)$ coordinates. This principle states that an exchange of two of these sets produces the same wave function except for its being multiplied by a factor of $(-1)^{2s}$. Pauli's proof is based upon quantum field operators and is difficult to understand. A much simpler proof, making use of properties of wave functions, is presented here.

Motivation & Objective

• To provide a more accessible derivation of the Pauli exchange principle than Pauli's original quantum field theory approach.
• To establish the exchange symmetry of wave functions for identical particles using only properties of wave functions.
• To clarify the connection between particle spin and the sign change under exchange, i.e., $(-1)^{2s}$.
• To make the foundational principle of quantum statistics more transparent for students and researchers.
• To demonstrate that the exchange symmetry can be derived without advanced field-theoretic formalism.

Proposed method

• Analyzes wave functions defined over sets of particle coordinates and quantum numbers, including spatial and spin ($m_s$) components.
• Considers the effect of exchanging two identical particles by swapping their respective coordinate and quantum number sets.
• Applies the requirement of indistinguishability and symmetry under exchange to constrain the form of the wave function.
• Uses the fact that the wave function must remain unchanged (up to a phase) under particle exchange to derive the $(-1)^{2s}$ factor.
• Relies on the properties of permutation symmetry and the behavior of wave functions under transposition of particle labels.
• Demonstrates that the sign factor $(-1)^{2s}$ arises naturally from the symmetry of the many-body wave function under particle exchange.

Experimental results

Research questions

• RQ1 Can the Pauli exchange principle be derived without invoking quantum field theory?
• RQ2 What is the role of spin quantum number $s$ in determining the sign of the wave function under particle exchange?
• RQ3 How does the symmetry of the wave function relate to the statistics of identical particles?
• RQ4 What are the implications of wave function symmetry for fermions and bosons?
• RQ5 Can the exchange factor $(-1)^{2s}$ be derived from wave function properties alone?

Key findings

• The Pauli exchange principle can be derived using only the symmetry properties of many-body wave functions.
• Exchanging two identical particles multiplies the wave function by $(-1)^{2s}$, where $s$ is the spin quantum number.
• The derivation confirms that fermions ($s = 1/2, 3/2, \dots$) transform with a sign change under exchange, while bosons ($s = 0, 1, 2, \dots$) remain invariant.
• The result is consistent with the spin-statistics theorem, but derived through elementary quantum mechanical reasoning.
• The method avoids the complexity of second quantization and provides a more intuitive understanding of exchange symmetry.
• The wave function's behavior under exchange is fully determined by the spin quantum number, without requiring field-theoretic formalism.