Pauli Exclusion Principle

The principle

No two identical fermions can occupy the same quantum state simultaneously.

"Quantum state" specifies all quantum numbers: position/momentum, spin, energy, etc. For an electron in an atom — n (principal), ℓ (orbital), m_ℓ (magnetic), m_s (spin = ±½). Each state holds at most one electron.

Application to atoms

Electrons fill orbitals from bottom up:

Shell n	Subshells	Max electrons
1	1s (2)	2
2	2s (2), 2p (6)	8
3	3s (2), 3p (6), 3d (10)	18
4	4s (2), 4p (6), 4d (10), 4f (14)	32
n general	Combine all	2n²

Filling order (Aufbau): 1s 2s 2p 3s 3p 4s 3d 4p 5s 4d 5p 6s 4f 5d 6p 7s 5f 6d 7p — based on energy levels.

Periodic table structure

Row	Filling	Elements
1	1s (2)	H, He
2	2s, 2p (8)	Li-Ne
3	3s, 3p (8)	Na-Ar
4	4s, 3d, 4p (18)	K-Kr
5	5s, 4d, 5p (18)	Rb-Xe
6	6s, 4f, 5d, 6p (32)	Cs-Rn

JavaScript — Pauli exclusion

// Maximum electrons in shell n
function maxElectrons(n) {
  return 2 * n * n;
}

[1, 2, 3, 4, 5].forEach(n => {
  console.log(`Shell n=${n}: max ${maxElectrons(n)} electrons`);
});

// Electron configuration of common elements
function electronConfig(Z) {
  // Aufbau filling order (simplified)
  const orbitals = [
    { name: '1s', max: 2 },
    { name: '2s', max: 2 },
    { name: '2p', max: 6 },
    { name: '3s', max: 2 },
    { name: '3p', max: 6 },
    { name: '4s', max: 2 },
    { name: '3d', max: 10 },
    { name: '4p', max: 6 },
    { name: '5s', max: 2 },
    // ...etc
  ];
  
  let remaining = Z;
  const config = [];
  for (const orbital of orbitals) {
    if (remaining === 0) break;
    const filled = Math.min(remaining, orbital.max);
    config.push(`${orbital.name}^${filled}`);
    remaining -= filled;
  }
  return config.join(' ');
}

// Hydrogen (Z=1): 1s¹
// Carbon (Z=6): 1s² 2s² 2p²
// Iron (Z=26): 1s² 2s² 2p⁶ 3s² 3p⁶ 4s² 3d⁶
console.log(`H: ${electronConfig(1)}`);
console.log(`C: ${electronConfig(6)}`);
console.log(`Ne: ${electronConfig(10)}`);
console.log(`Na: ${electronConfig(11)}`);

// White dwarf: degeneracy pressure
function chandrasekharLimit() {
  // Mass limit for white dwarf supported by electron degeneracy pressure
  return 1.4;  // solar masses
}

console.log(`Chandrasekhar limit: ${chandrasekharLimit()} M_sun`);
// White dwarfs heavier than this collapse to neutron stars

// Density of state-filling
function fermiTemperature(electronDensity, m_electron = 9.11e-31) {
  // T_F = E_F / k_B
  // E_F ≈ ℏ²(3π²n)^(2/3) / (2m)
  const h_bar = 1.055e-34;
  const k_B = 1.38e-23;
  const E_F = h_bar * h_bar * Math.pow(3 * Math.PI * Math.PI * electronDensity, 2/3) / (2 * m_electron);
  return E_F / k_B;
}

// Free electrons in copper: ~8 × 10²⁸ /m³
console.log(`Copper Fermi T: ${fermiTemperature(8e28).toExponential(2)} K`);
// ~80,000 K — much higher than room T → electrons strongly degenerate

Where Pauli exclusion matters

• Atomic structure. All atoms with multiple electrons.
• Chemistry. Bonds, periodic table, every chemical reaction.
• Solid state physics. Metals, semiconductors, insulators — all rely on Pauli filling.
• Stars. White dwarf and neutron star structure; supernova mechanics.
• Quantum gases. Fermi gas of neutrons, electrons, atoms.
• Plasma. Hot dense plasmas show electron degeneracy.
• Quantum computing. Distinguish fermionic (electron) qubits from bosonic ones.

Common mistakes

• Applying Pauli to bosons. Bosons (photons, gluons, He-4) DON'T obey exclusion. Multiple bosons can share state.
• Treating spin as classical. Spin is intrinsic angular momentum, quantized in units of ℏ/2 for fermions. Two spin states (up/down) double available states.
• Confusing "different state" with "different position." Two electrons in same orbital can have different spins (different m_s) — counts as different states.
• Forgetting it's exact. Pauli is fundamental; not a probabilistic statement. Truly NEVER two identical fermions in same state.
• Thinking Pauli applies between non-identical particles. Different particle types are distinguishable; can share quantum numbers. Only identical fermions excluded.
• Confusing degeneracy with random arrangements. Degeneracy pressure (white dwarf) is from filling states — not "regular" pressure.