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Exclusor

Pauli Exclusion Physics

\[\psi(x_1,x_2) = \tfrac{1}{\sqrt{2}}\,\bigl[\,\psi_{ab} - \psi_{ba}\,\bigr]\]
$21.99 USD
$21.99 USD
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Why don’t atoms collapse into one blob? Exclusor guards the door. Identical fermions such as electrons have an antisymmetric wavefunction, ψ(x1,x2) = −ψ(x2,x1), so no two can share the same quantum state. Electrons fill shells in order, giving chemistry its periodic table. Pixel-art black ceramic mug in 11 or 15 oz; equation, symbols and sources in the physics panel below.

Shipping calculated at checkout by destinationMade to order, ships in ~10 days

Details

Size & material11oz · 15oz

Black glossy ceramic mug with a C-handle. Lead- and BPA-free.

  • 11oz: 0.33 l
  • 15oz: 0.44 l
CareDishwasher & microwave safe

Dishwasher safe (top rack) or hand wash. Microwave safe.

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Shipping calculated at checkout by destination.

Made to order: about 10 days of production and handling before it ships.

Delivery times & where we ship

Returns & replacements21 days

Every item is printed to order, so we can’t accept returns or exchanges for change of mind, or if you ordered the wrong size or colour.

Print defect, misprint, damage or the wrong item? We’ll send a free replacement, or a refund if you prefer. Report it within 21 days of delivery with a photo; no need to send it back.

Full refund & replacement policy

The physics

For two identical fermions, the wavefunction must flip sign when the particles are swapped. If both were in the same state, the two terms would cancel to zero, so no two fermions can share a quantum state.

Exclusor works one door, and he does not check names. He checks states.

His line is NOT IN MY STATE. A quantum state for an electron means everything about it: which orbital it occupies and which way its spin points. Two electrons can share one orbital, one spin-up and one spin-down. What they cannot share is the whole state.

Wolfgang Pauli stated the rule in 1925, to account for atomic spectra. He needed a fourth quantum number with only two values and ruled that no two electrons in an atom could share all four. Later that year George Uhlenbeck and Samuel Goudsmit identified it as electron spin. Pauli received the 1945 Nobel Prize.

The modern form is the first four lines on this mug. Swap two identical electrons and their joint wavefunction changes sign. Put both in the same state, a equal to b, and the two terms in the first line are identical and cancel: the wavefunction is zero. That state does not exist. Markus Fierz in 1939 and Pauli in 1940 showed that relativistic quantum theory makes every particle of half-integer spin obey it.

That is why electrons fill atoms shell by shell instead of piling into the lowest one, and why chemistry has a periodic table. Scale it up and the same rule pushes back: in 1926 Ralph Fowler showed that tightly packed electrons exert degeneracy pressure, the last formula. It holds up white dwarfs, up to Chandrasekhar's limit of about 1.4 solar masses. Exclusion is not a force, but together with electric attraction it keeps bulk matter from collapsing, as Dyson and Lenard proved in 1967, and Lieb and Thirring, far more simply, in 1975.

Exclusor never shoves. He just knows who is already inside.

Equations

\[\psi(x_1,x_2) = \tfrac{1}{\sqrt{2}}\,\bigl[\,\psi_{ab} - \psi_{ba}\,\bigr]\]
\[\psi_{ab} = \phi_a(x_1)\,\phi_b(x_2)\]
\[\psi(x_2,x_1) = -\,\psi(x_1,x_2)\]
\[\phi_a = \phi_b \;\Rightarrow\; \psi = 0\]
\[P = \frac{(3\pi^{2})^{2/3}}{5}\,\frac{\hbar^{2}}{m_e}\,n_e^{5/3}\]
SymbolMeaningUnit
\(\psi(x_1,x_2)\) joint wavefunction of two identical electrons (unit for two particles in 3-D) \(\mathrm{m^{-3}}\)
\(x_1, x_2\) coordinates of electrons 1 and 2: position plus spin projection (unit of the position part) \(\mathrm{m}\)
\(\phi_a, \phi_b\) single-electron states a and b, each fixing orbital and spin (unit in 3-D) \(\mathrm{m^{-3/2}}\)
\(P\) electron degeneracy pressure (non-relativistic, zero temperature) \(\mathrm{Pa}\)
\(\hbar\) reduced Planck constant \(\mathrm{J·s}\)
\(m_e\) electron mass \(\mathrm{kg}\)
\(n_e\) number density of electrons \(\mathrm{m^{-3}}\)
\(\psi_{ab}, \psi_{ba}\) product states with electron 1 in a and 2 in b, and the swap \(\mathrm{m^{-3}}\)

Sources

  1. Pauli (1925) Über den Zusammenhang des Abschlusses der Elektronengruppen im Atom mit der Komplexstruktur der Spektren, Z. Phys. 31, 765
  2. Uhlenbeck & Goudsmit (1925) Ersetzung der Hypothese vom unmechanischen Zwang durch eine Forderung bezüglich des inneren Verhaltens jedes einzelnen Elektrons, Naturwissenschaften 13, 953
  3. NobelPrize.org — The Nobel Prize in Physics 1945 (Wolfgang Pauli, "for the discovery of the Exclusion Principle")
  4. Fierz (1939) Über die relativistische Theorie kräftefreier Teilchen mit beliebigem Spin, Helv. Phys. Acta 12, 3 (E-Periodica)
  5. Pauli (1940) The Connection Between Spin and Statistics, Phys. Rev. 58, 716
  6. Feynman Lectures on Physics Vol. III Ch. 4: Identical Particles
  7. Encyclopaedia Britannica — Pauli exclusion principle
  8. NIST Handbook of Basic Atomic Spectroscopic Data — Neon (Ne I ground state 1s² 2s² 2p⁶ ¹S₀)
  9. Ning & Lu (2022) Electron Affinities of Atoms and Structures of Atomic Negative Ions, J. Phys. Chem. Ref. Data 51, 021502
  10. Andersen, Haugen & Hotop (1999) Binding Energies in Atomic Negative Ions: III, J. Phys. Chem. Ref. Data 28, 1511
  11. Fowler (1926) On Dense Matter, MNRAS 87, 114
  12. Chandrasekhar (1931) The Maximum Mass of Ideal White Dwarfs, ApJ 74, 81
  13. Encyclopaedia Britannica — Chandrasekhar limit
  14. Dyson & Lenard (1967) Stability of Matter. I, J. Math. Phys. 8, 423
  15. Lieb & Thirring (1975) Bound for the Kinetic Energy of Fermions Which Proves the Stability of Matter, PRL 35, 687

Source Pauli, W. (1925). Über den Zusammenhang des Abschlusses der Elektronengruppen im Atom mit der Komplexstruktur der Spektren. Zeitschrift für Physik 31, 765–783.doi.org/10.1007/BF02980631

Field notes

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