Skip to main content

Mathematics for Physics

79 Pauli Exclusion Principle

The quantum-mechanical rule that no two identical fermions, such as electrons, can occupy the same quantum state at the same time.

definition [d] (Pauli Exclusion Principle) From Sakurai and Napolitano: an immediate consequence of the electron being a fermion is that the electron must satisfy the Pauli exclusion principle, which states that no two electrons can occupy the same state.

where

  • electrons are fermions.
  • a quantum state is a complete set of quantum numbers for one electron in the system.

79.1 Elementary Example

79.1.1 Simple

In the helium ground configuration \(1s^{2}\), the two electrons share \(n=1\), \(\ell=0\) but must differ in spin.

\[ (n,\ell,m_{\ell},m_{s}) = (1,0,0,+\tfrac{1}{2}),\quad (1,0,0,-\tfrac{1}{2}) \]

where

  • \(m_{s}\) distinguishes the two occupied states.

79.1.2 General

No third electron can join the same \(1s\) orbital with either spin already used; it must enter a new orbital such as \(2s\).

\[ 1s^{2}\,2s^{1} \]

where

  • the third electron occupies a different spatial state.