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Exclusor

Pauli Exclusion Principle / Degeneracy Pressure Physics

\[\psi(x_1,x_2) = \frac{1}{\sqrt{2}}\begin{vmatrix} \phi_a(x_1) & \phi_b(x_1) \\ \phi_a(x_2) & \phi_b(x_2) \end{vmatrix}\]
$29.99 USD
$29.99 USD
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Size guide S–5XL · inches

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. The physics panel below gives the equation, what each symbol means and the sources.

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Details

Size & fitS–5XL

Unisex heavy cotton (Gildan 5000), classic fit.

SizeWidthLengthSleeve
S182815.1
M202916.5
L223018
XL243119.5
2XL263221
3XL283322.4
4XL303423.7
5XL323525

크기 비교: M 사이즈는 폭이 20 in (50.80 cm)로, 약 3.143 × 1034 플랑크 길이입니다(lP = 1.616255 × 10−35 m). 줄자가 더 빠릅니다.

Measurements in inches, ±1.5 in tolerance.

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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.

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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.

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

Behind him is a neon atom: ten electrons, seated 1s² 2s² 2p⁶. Each box on his chart is one orbital: two electrons at most, one spin-up, one spin-down. Every seat in the second shell is taken. Hence the sign.

Wolfgang Pauli wrote the rule in 1925, before anyone knew what the second seat in each box was for. To account for atomic spectra he needed a fourth quantum number with only two values, and he 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 three lines on this page. Swap two identical electrons and their joint wavefunction changes sign. Put both in the same state, a equal to b, and the determinant has two identical columns: the wavefunction is zero. That state does not exist.

The badge says ½ because the rule binds him too. Markus Fierz in 1939 and Pauli in 1940 showed that relativistic quantum theory makes every particle of half-integer spin obey it.

Neon has no stable negative ion. An eleventh electron would have to sit in 3s, and neon will not hold it.

Scale it up. In 1926 Ralph Fowler showed that electrons packed this tightly push back: degeneracy pressure, the last formula. It holds up white dwarfs, to about 1.4 solar masses, Chandrasekhar's limit. Exclusion is not a force. 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

\[\psi(x_1,x_2) = \frac{1}{\sqrt{2}}\begin{vmatrix} \phi_a(x_1) & \phi_b(x_1) \\ \phi_a(x_2) & \phi_b(x_2) \end{vmatrix}\]
\[\psi(x_2,x_1) = -\,\psi(x_1,x_2)\]
\[\phi_a = \phi_b \;\Rightarrow\; \psi = 0\]
\[P = \frac{(3\pi^{2})^{2/3}}{5}\,\frac{\hbar^{2}}{m_e}\,n_e^{5/3}\]
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 \(\mathrm{m^{-3}}\)

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

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Field notes

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