3. Pauli Exclusion Principle

Quantum state specified by all quantum numbers:

• Position/momentum: $\vec{r}$ or $\vec{p}$
• Energy level: $n$
• Angular momentum: $\ell, m_\ell$
• Spin: $m_s$

Two electrons can have same spatial wave function if opposite spins:

• 1925: Wolfgang Pauli proposed principle to explain atomic spectra
• Problem: Why do atoms build up shells rather than all electrons falling to ground state?
• Solution: Exclusion principle forces electrons into different states
• Nobel Prize 1945: Pauli honored for exclusion principle

Ground state: Both electrons in 1s orbital

Allowed because different spins:

Total spin: $S = 0$ (singlet, antiparallel spins)

Excited state: One electron promoted to 2s

Two possibilities:

• Singlet: $S = 0$, spins antiparallel (parahelium)
• Triplet: $S = 1$, spins parallel (orthohelium)

Maximum electrons per shell/subshell:

Factor of 2 from two spin states, $(2\ell + 1)$ from magnetic quantum numbers

• s-orbital ($\ell = 0$): max 2 electrons
• p-orbital ($\ell = 1$): max 6 electrons
• d-orbital ($\ell = 2$): max 10 electrons
• f-orbital ($\ell = 3$): max 14 electrons

Electrons fill orbitals from lowest to highest energy:

Note: 4s fills before 3d due to shielding effects

Combined with Pauli exclusion: Explains periodic table structure

For filling degenerate orbitals (same $n, \ell$):

1. Maximum total spin: Fill with parallel spins first (minimize electron-electron repulsion)
2. Maximum orbital angular momentum: Maximize $L$ for given $S$
3. J coupling:
   • Less than half full: $J = |L - S|$
   • More than half full: $J = L + S$

• Noble gases: Filled shells → chemically inert
• Alkali metals: One electron beyond filled shell → reactive
• Halogens: One electron short of filled shell → reactive
• Valence: Number of unpaired electrons determines bonding
• Transition metals: Partially filled d-shells → complex chemistry

Without Pauli exclusion:

• All electrons would collapse to ground state
• Atoms would be ~10,000 times smaller
• No chemistry, no life

With Pauli exclusion:

• Electrons forced into higher energy shells
• Atoms have finite size
• Matter has structure and rigidity

White dwarfs:

• Electron degeneracy pressure prevents gravitational collapse
• Mass limit (Chandrasekhar): $M \lesssim 1.4 M_\odot$

Neutron stars:

• Neutron degeneracy pressure supports even denser object
• Mass limit: $M \lesssim 2-3 M_\odot$
• Beyond this → black hole