Exclusor: Pauli Exclusion Principle / Degeneracy Pressure
Quantum Foundations arc, page 7 of 193. Back to The Pixelated Physics Universe
Who they are
Exclusor is a Pixelated Physics character, one of our quantum designs. No two identical fermions can occupy the same quantum state. Exclusor, Pauli’s bouncer, keeps neon’s shells full. This is a pixel-art unisex T-shirt.
Exclusor is a character of the Quantum Foundations arc: drawn in pixels, ruled by one piece of real physics, and never allowed to break it.
The physics
Electrons are fermions, so their joint wavefunction must change sign when two are swapped. If two electrons tried to occupy the same state, the wavefunction would vanish, so they can't. That exclusion produces the shell structure of atoms and the degeneracy pressure that holds up white dwarfs.
$$ \begin{gathered} \psi(x_1,x_2) = \frac{1}{\sqrt{2}}\begin{vmatrix} \phi_a(x_1) & \phi_b(x_1) \\ \phi_a(x_2) & \phi_b(x_2) \end{vmatrix} \\ \psi(x_2,x_1) = -\,\psi(x_1,x_2) \\ \phi_a = \phi_b \;\Rightarrow\; \psi = 0 \\ P = \frac{(3\pi^{2})^{2/3}}{5}\,\frac{\hbar^{2}}{m_e}\,n_e^{5/3} \end{gathered} $$
| Symbol | Meaning | SI unit |
|---|---|---|
| $\psi(x_1,x_2)$ | joint wavefunction of two identical electrons (unit for two particles in 3-D) | m^{-3} |
| $x_1, x_2$ | coordinates of electrons 1 and 2: position plus spin projection (unit of the position part) | m |
| $\phi_a, \phi_b$ | single-electron states a and b, each fixing orbital and spin (unit in 3-D) | m^{-3/2} |
| $P$ | electron degeneracy pressure (non-relativistic, zero temperature) | Pa |
| $\hbar$ | reduced Planck constant | J·s |
| $m_e$ | electron mass | kg |
| $n_e$ | number density of electrons | m^{-3} |
Sources
- Pauli (1925) Über den Zusammenhang des Abschlusses der Elektronengruppen im Atom mit der Komplexstruktur der Spektren, Z. Phys. 31, 765
- Uhlenbeck & Goudsmit (1925) Ersetzung der Hypothese vom unmechanischen Zwang durch eine Forderung bezüglich des inneren Verhaltens jedes einzelnen Elektrons, Naturwissenschaften 13, 953
- NobelPrize.org — The Nobel Prize in Physics 1945 (Wolfgang Pauli, "for the discovery of the Exclusion Principle")
- Fierz (1939) Über die relativistische Theorie kräftefreier Teilchen mit beliebigem Spin, Helv. Phys. Acta 12, 3 (E-Periodica)
- Pauli (1940) The Connection Between Spin and Statistics, Phys. Rev. 58, 716
- Feynman Lectures on Physics Vol. III Ch. 4: Identical Particles
- Encyclopaedia Britannica — Pauli exclusion principle
- NIST Handbook of Basic Atomic Spectroscopic Data — Neon (Ne I ground state 1s² 2s² 2p⁶ ¹S₀)
- Ning & Lu (2022) Electron Affinities of Atoms and Structures of Atomic Negative Ions, J. Phys. Chem. Ref. Data 51, 021502
- Andersen, Haugen & Hotop (1999) Binding Energies in Atomic Negative Ions: III, J. Phys. Chem. Ref. Data 28, 1511
- Fowler (1926) On Dense Matter, MNRAS 87, 114
- Chandrasekhar (1931) The Maximum Mass of Ideal White Dwarfs, ApJ 74, 81
- Encyclopaedia Britannica — Chandrasekhar limit
- Dyson & Lenard (1967) Stability of Matter. I, J. Math. Phys. 8, 423
- Lieb & Thirring (1975) Bound for the Kinetic Energy of Fermions Which Proves the Stability of Matter, PRL 35, 687
Crossovers
Exclusor meets The Condensate Queen (Bose–Einstein Condensate). Exclusor lets no two identical fermions share a state. The Condensate Queen gathers bosons into one state by the thousand. Spin decides which ruler a particle obeys: half-integer spin answers to Exclusor, integer spin to the Queen.
Exclusor meets The Limit Keeper (Chandrasekhar Limit). Exclusor's rule gives a white dwarf its degeneracy pressure. The Limit Keeper shows where that pressure stops being enough: once the electrons become relativistic, gravity wins above about 1.4 solar masses.
Exclusor meets Marina Dirac (Dirac Sea). Marina Dirac's sea of filled negative-energy states only stays filled because of Exclusor: no electron can drop into a state that is already taken. Knock one out and the hole behaves like a positron.
Wear the lore
- Exclusor – Pauli Exclusion Principle / Degeneracy Pressure Physics T-Shirt
- Related infographic poster: The Spin-Orbit Staircase – Plate 050 Physics Poster
Concept keywords: Pauli Exclusion Principle / Degeneracy Pressure, Exclusor, Quantum Foundations, physics lore, Pixelated Physics