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When you write the fourth quantum number, s = ±1/2, you are using an idea that two graduate students in Leiden nearly withdrew before it was printed. George Uhlenbeck and Samuel Goudsmit proposed in October 1925 that the electron spins. This post follows how they got there, why a great physicist told them it could not be literally true, and a short calculation that shows his point.
Our posts on the Pauli exclusion principle and the Stern-Gerlach experiment cover the rule and the beam experiment. This one is about the two people who gave the fourth quantum number a physical picture.
Who were Uhlenbeck and Goudsmit?
MacTutor at the University of St Andrews records that George Uhlenbeck was born on 6 December 1900 in Batavia, Java (now Jakarta), and died on 31 October 1988. After a job in Italy he came close to becoming a historian. Paul Ehrenfest, keen to keep him in physics, made him his assistant, and very soon afterwards, working with a graduate student named Samuel Goudsmit, Uhlenbeck discovered electron spin.
The two were a contrast. In a 1971 lecture to the Dutch Physical Society, published by Leiden's Instituut-Lorentz in English translation, Goudsmit said he knew the spectra in great detail but not the deeper physics, while Uhlenbeck knew classical physics well but had never heard of the spectroscopists Goudsmit kept quoting.
How did the idea come up?
Pauli had published his principle early in 1925. To make it work he needed a fourth quantum number for each electron, but nobody knew what it described. Goudsmit recalled explaining Pauli's rule to Uhlenbeck, who replied, in the translated text: "But don't you see what this implies? It means that there is a fourth degree of freedom for the electron. It means that the electron has a spin, that it rotates." Goudsmit's answer was to ask what a degree of freedom was.
Goudsmit then saw that a magnetic electron would explain the complicated Zeeman patterns and Landé's formulas. As the Leiden history page puts it, they proposed that each electron spins with an angular momentum of one half of h-bar and carries a magnetic moment of one Bohr magneton. Their note in Naturwissenschaften is dated Saturday 17 October 1925.
Why did Lorentz object?
Uhlenbeck took the idea to Hendrik Lorentz. The Leiden page records that Lorentz pointed out that a spinning electron would be incompatible with classical electrodynamics. Uhlenbeck asked Ehrenfest not to submit the paper. Ehrenfest replied that it had already gone, and added: "You are both young enough to be able to afford a stupidity!"
You can see the problem with a rough estimate. Treat the electron as a ring of charge with all its mass at the classical electron radius, r = 2.82 × 10⁻¹⁵ m, and give it angular momentum ħ/2.
Set up. For mass m at radius r moving at speed v, L = mvr. So v = L/(mr).
Numbers. ħ/2 = 5.27 × 10⁻³⁵ J s, m = 9.11 × 10⁻³¹ kg.
Divide. v = 5.27 × 10⁻³⁵ / (9.11 × 10⁻³¹ × 2.82 × 10⁻¹⁵) ≈ 2.1 × 10¹⁰ m/s.
Compare. That is about 70 times the speed of light.
A literal spinning ball cannot work. Spin is real angular momentum, but it is a quantum property with no classical picture.
What was the missing factor of 2?
Goudsmit recalled that the day after publication a letter arrived from Heisenberg praising their "courageous note" and asking what they had done with "the factor 2". Worked out with spin, the size of the fine-structure splitting came out wrong by a factor of 2. In Copenhagen, L. H. Thomas showed that the missing factor was a relativistic effect, published in Nature in April 1926. Goudsmit said Pauli, who had resisted spin, later sent a postcard saying he had seen Thomas's work and believed it. The pair's fuller account appeared in Nature in February 1926.
The Leiden page adds an uncomfortable detail. Ralph Kronig had the idea months earlier, put it to Pauli, and was told it was "indeed very clever but of course has nothing to do with reality". Kronig did not publish.
What should you take into exam work?
The spin quantum number has two values, +1/2 and −1/2. It is the reason each orbital holds two electrons. Our post on quantum numbers and atomic structure works through the counting.
Common mistake: treating spin as a ball turning on its axis. Use the picture to remember the name, not to calculate.
Common mistake: using e/2m for spin. That ratio is for orbits. Spin gives twice it.
What can a student take from their way of working?
Pool different knowledge. The idea needed both of them: one knew the spectra, the other the physics.
Let the data lead. Spin was accepted because it explained Zeeman patterns, not because it was easy to picture.
Have someone who lets you try. Goudsmit remembered Ehrenfest saying they had no reputation yet, so nothing to lose.
If magnetic moments and angular momentum still blur together, the Fermi Concept Library covers every concept in physics, chemistry, biology, mathematics and computing, one idea at a time.
Next step: redo the surface-speed estimate with a radius ten times smaller and work out how the speed changes, before you calculate it.
