AcademyAtomic Quantum Structure
Academy
Exclusion Principle
Level 1 - Physics topic page in Atomic Quantum Structure.
Principle
The Pauli exclusion principle says no two identical fermions in the same system can occupy the same quantum state. For electrons in atoms, no two electrons can have the same four quantum numbers.
This principle explains shell structure, chemical periodicity, and the two-electron capacity of each orbital.
Notation
\(n\)
principal quantum number
1
\(\ell\)
orbital angular momentum quantum number
1
\(m_\ell\)
magnetic quantum number
1
\(m_s\)
spin projection quantum number
1
\(N\)
number of states or electrons
1
Method
Derivation 1: Label a one-electron state
An atomic electron state is specified by four quantum numbers.
State label
\[(n,\ell,m_\ell,m_s)\]
Spin choices
\[m_s=\pm\frac12\]
Derivation 2: Count orbital occupancy
A spatial orbital has fixed \(n\), \(\ell\), and \(m_\ell\). The only remaining distinction is spin.
One spatial orbital
\[m_s=+\frac12\quad\mathrm{or}\quad-\frac12\]
Maximum orbital occupancy
\[N_{\mathrm{orbital}}=2\]
Derivation 3: Count shell capacity
Counting all allowed \(\ell\), \(m_\ell\), and spin states in shell \(n\) gives \(2n^2\) states.
Spatial orbitals in shell
\[N_{\mathrm{orbitals}}=n^2\]
Electron capacity
\[N_e=2n^2\]
Rules
Electron state label
\[(n,\ell,m_\ell,m_s)\]
Spin choices
\[m_s=\pm\frac12\]
Orbital capacity
\[N_{\mathrm{orbital}}=2\]
Shell capacity
\[N_e=2n^2\]
Examples
Question
Why can a
\[1s\]
orbital hold only two electrons?Answer
The
\[1s\]
orbital fixes \[n=1\]
\[\ell=0\]
and \[m_\ell=0\]
Only two spin values remain, so only two electrons can occupy it.Checks
- Pauli exclusion applies to identical fermions.
- Electrons in the same orbital must have opposite spin projections.
- A subshell with \(2\ell+1\) orbitals holds \(2(2\ell+1)\) electrons.
- Shell filling follows from quantum-state counting.