---
title: "Pauli: count the states before filling the shells"
description: "How did sorting spectral lines lead to a rule about which electron states are allowed?"
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last_modified: "2026-10-15"
---

[Quantum physics: reading the evidence](https://sandboxphysics.com/stories/quantum-evidence/)

Nobel A · Evidence and control

# Pauli: count the states before filling the shells

Sandbox Physics · 2026-10-15

How did sorting spectral lines lead to a rule about which electron states are allowed?

## The puzzle was a pattern in the allowed states

Why do atomic shells close, instead of every electron settling into the same lowest arrangement? Drawing more planetary orbits could not answer that. The periodic pattern of chemistry, multiplet spectra and magnetic line splitting all had to fit one way of counting electron states.

The ETH archive ties Pauli’s fourth quantum number to multiplet structure and the anomalous Zeeman effect. Each spectral line marks a jump between two terms, so counting states means first sorting the terms. The evidence was an ordered network of lines, not a direct image of which electron places were filled.

Read the evidence: ETH Library, The Pauli Principle — Pauli archive and institutional account

## The rule came before an intuitive picture

Looking back, Pauli recalls rejecting the idea that alkali doublets come from the angular momentum of the atomic core. Instead he gave the electron a two-valued property that classical physics could not describe. Stoner’s comparison between the number of states counted in spectra and the closed subgroups helped link the two counts.

The exclusion rule then allowed at most one electron in each fully specified one-electron state. The rule came before spin was accepted as its interpretation. Projecting modern spin labels onto every step of the discovery hides why the fourth degree of freedom looked so hard to understand at first.

Read the evidence: W. Pauli (1946), Exclusion Principle and Quantum Mechanics — Nobel lecture, December 13; readable copy ETH Library, The Pauli Principle — Pauli archive and institutional account

## The complete state includes spin

Two electrons can share a spatial orbital if their spins differ, because they then occupy different spin-orbitals. Likewise, two states need not differ in location to be distinct. What counts is the full spatial and spin description.

A single determinant treats the electrons as independent particles. It is useful, but it does not describe every interacting atom exactly. Correlated electron states can need a superposition of several determinants, and the superposition is still antisymmetric. The exclusion requirement holds in that better calculation; the simple picture of filling orbitals may not.

Read the evidence: W. Pauli (1946), Exclusion Principle and Quantum Mechanics — Nobel lecture, December 13; readable copy

## The formula cancels when two electrons share a spin-orbital

The normalized antisymmetric state below is built from two orthonormal spin-orbitals. Each label stands for a complete set of position and spin. Swapping the labels flips the sign of the state and leaves every probability unchanged. The labels name the arguments of the state; they do not mark which electron is which.

Put the same spin-orbital in both slots and the numerator cancels to zero everywhere, so no normalized state is left. Once antisymmetry is assumed, this algebra shows exclusion directly. It does not explain, within nonrelativistic mechanics alone, why electrons must belong to this symmetry class.

$$
\Psi(1,2)=\frac{\phi_a(1)\phi_b(2)-\phi_b(1)\phi_a(2)}{\sqrt{2}}
$$

Read the evidence: W. Pauli (1946), Exclusion Principle and Quantum Mechanics — Nobel lecture, December 13; readable copy

## Spin–statistics adds assumptions, not a repulsive force

Pauli’s 1940 paper takes up the link between spin and statistics in relativistic theory. Its abstract names the conditions it relies on, including positive energy and observables that commute at spacelike separation. This is a further theoretical argument, not a tidier version of the earlier spectral counting.

So exclusion should not be drawn as a new classical force that pushes labeled particles apart. Coulomb interactions, antisymmetry and the set of available states together shape how matter behaves. The occupation rule alone is not enough to calculate a real material’s pressure, bonding or energy spectrum.

Read the evidence: W. Pauli (1940), The Connection Between Spin and Statistics, Phys. Rev. 58, 716 — author abstract

## Use the Labs to see what each model assumes

The exchange Lab shows how distinguishability changes the comparison; the occupation Lab compares quantum state counting in an ideal gas. Neither rebuilds the spectral evidence that led Pauli to his rule. When a model forbids an occupation in its code, it is showing an assumption and what follows from it.

Historically the path ran from sorting spectral terms to a rule on occupation, then to antisymmetric many-electron states and a relativistic spin–statistics argument. Taking these steps one at a time shows what each one explains, and makes clear that no single experiment established them all.

Read the evidence: ETH Library, The Pauli Principle — Pauli archive and institutional account W. Pauli (1940), The Connection Between Spin and Statistics, Phys. Rev. 58, 716 — author abstract

## Continue with the models

These teaching Labs produce simulated data, which are separate from the historical observations discussed above.

- [Compare identical atoms with distinguishable ones](https://sandboxphysics.com/labs/identical-atom-exchange/)
- [See how particles fill the energy levels of a quantum gas](https://sandboxphysics.com/labs/quantum-gas-occupation/)

## Sources and reading notes

The history draws on ETH’s Pauli archive and a readable copy of Pauli’s Nobel lecture, given in December 1946 for the 1945 prize. For the 1940 spin–statistics paper we read only the author abstract. The two-orbital formula is a modern illustration, not the argument Pauli made in 1925.

1. [ETH Library, The Pauli Principle — Pauli archive and institutional account](https://library.ethz.ch/en/collections-and-archives/platforms/virtual-exhibitions/wolfgang-pauli-and-modern-physics/the-pauli-principle.html)
2. [W. Pauli (1946), Exclusion Principle and Quantum Mechanics — Nobel lecture, December 13; readable copy](https://gilles.montambaux.com/files/histoire-physique/Pauli-1946-nobel-lecture.pdf)
3. [W. Pauli (1940), The Connection Between Spin and Statistics, Phys. Rev. 58, 716 — author abstract](https://journals.aps.org/pr/abstract/10.1103/PhysRev.58.716)

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