George Uhlenbeck (L) and Samuel Goudsmit (R) came up with the idea of quantum spin in the mid-1920s. This photo was taken with Hendrik Kramers (center) in 1928 (Public Domain)

Spin: The Quantum Property That Should Have Been Impossible

How a quantum number that made no physical sense turned out to be real… and irreplaceable.

Paul Halpern
Dec 5, 2017 · 7 min read

In the early 1920s, physicists were first working out the mysteries of the quantum Universe. Particles sometimes behaved as waves, with indeterminate positions, momenta, energies, and other properties. There was an inherent uncertainty to a great many properties that we could measure, and physicists raced to work out the rules.

Amidst this frenzy, a young Dutch researcher named George Uhlenbeck implored Paul Ehrenfest, his research supervisor at the University of Leiden, not to submit the paper he wrote with Samuel “Sam” Goudsmit about a new quantum number called spin. It was not correct, Uhlenbeck told him in a frenzy. Let’s just drop it and start over, he implored.

Uhlenbeck and Goudsmit, both then in their mid-20s, had just showed their joint result to the great Dutch physicist Hendrik Lorentz who had found what seemed like a major error. Electrons, he pointed out, couldn’t possibly rotate fast enough to generate the magnetic moment (interaction strength between a particle and an external magnetic field) that the duo had predicted. The particles would need to whirl faster than the sacred speed limit of light. How could they? The spin paper is unphysical, Uhlenbeck told Ehrenfest, and should not be published.

Electrons, like all spin-1/2 fermions, have two possible spin orientations when placed in a magnetic field (CK-12 Foundation / Wikimedia Commons).

Ehrenfest’s reply was curt. “It is too late,” he told Uhlenbeck. “I have already submitted the paper. It will be published in two weeks.” Then he added, “Both of you are young and can afford to do something stupid.”

Ehrenfest’s words certainly weren’t comforting. Surely, Uhlenbeck didn’t want to start off his career with a foolish error. Luckily, however, the spin quantum number, interpreted abstractly and having nothing whatsoever to do with rotation despite its name, has become an essential feature of modern physics. Electrons somehow acted in a magnetic field as if they were whirling, even thought they really couldn’t be. Uhlenbeck and Goudsmit’s roulette wheel bet on a weird new concept had paid off handsomely.

Thomas precession demonstrated with a gyroscope in space, as in the Gravity Probe B experiment (NASA).

One of the harshest critics of spin was the acerbic physicist Wolfgang Pauli. Pauli, like Ehrenfest was born in Vienna, and moved elsewhere for his career. Like Lorentz, Pauli believed at first that spin was unphysical. (In January 1925, German American researcher Ralph Kronig had made a similar suggestion to Pauli, which he had immediately rejected and was never published.) He changed his mind only after Llewellyn Thomas demonstrated a phenomenon called “Thomas precession” that examined spin using special relativity.

Wolfgang Pauli (L) and Paul Ehrenfest (R), only a few years before Ehrenfest would tragically commit suicide (CERN photo archives)

Pauli and Ehrenfest shared a blunt demeanor and willingness to criticize others in matters of science. They had first met in 1922 at the “Bohrfestspiele” (celebration of Niels Bohr’s work around the time of his Nobel Prize ) in Göttingen, Germany. Pauli, then in his early 20s, was already famous as a “wunderkind” for an excellent article about general relativity that appeared in a scientific encyclopedia edited by German physicist Arnold Sommerfeld. Ehrenfest and his wife had contributed a piece on statistical mechanics for the same volume. Pauli and Ehrenfest’s initial conversation centered on those respective works.

As physicist Oskar Klein reported: “On that occasion Ehrenfest stood a little away from Pauli, looked at him mockingly and said: ‘Herr Pauli, I like your article better than I like you! To which Pauli very calmly replied: ‘That is funny, with me it is just the opposite!’”

In an atom, each s orbital (red), each of the p orbitals (yellow), the d orbitals (blue) and the f orbitals (green) can contain only two electrons apiece: one spin up and one spin down in each one (Libretexts Library / NSF / UC Davis).

It was ironic that Pauli was initially opposed to spin, given that one of his key proposals — the exclusion principle — was one of the main motivators for its development. Introduced in early 1925, it stated that no two electrons (later extended to an entire class of particles called fermions) could occupy exactly the same quantum state. (Other types of particles, such as photons, that don’t obey that law are called bosons.)

Quantum states in atoms (such as hydrogen) can be characterized by quantum numbers denoting the properties of an electron occupying such a state. The principal quantum number, introduced by Bohr, described the energy of an electron due to its electric interaction with the nucleus. The second and third quantum numbers, introduced by Sommerfeld, pertained to aspects of an electron’s angular momentum (a measure of the shape and configuration of its orbit). Traditionally, each of those quantum numbers were integers — counting numbers denoting a finite set of possibilities, such as the seat and row numbers in an arena. In tandem, those three quantum numbers determine how the probability clouds representing the electrons position themselves in the “stadium” surrounding the nucleus. That intricate pattern, well known by chemists, helps explain the periodic table.

Hydrogen density plots for an electron in a variety of quantum states. While the three quantum numbers of charge and angular momentum in two different dimensions could explain a great deal, ‘spin’ must be added to explain the periodic table and the number of electrons in orbitals for each atom (PoorLeno / Wikimedia Commons).

However, as Goudsmit realized in May 1925, there was a problem with using pure integers to characterize the quantum states. If you did, the exclusion principle couldn’t be maintained. Two electrons in the ground state (lowest energy level) of an atom would have identical set of those three quantum numbers. Goudsmit found that by introducing a fourth quantum number, representing a kind of intrinsic or extra angular momentum, that could take on only one of two possible values — either +½ or -½ — he could preserve the Pauli exclusion principle. The ground state could still have two electrons, but their fourth quantum numbers would be opposite: if one was +½, the other would be -½.

In the absence of a magnetic field, the energy levels of various states within an atomic orbital are identical (L). If a magnetic field is applied, however (R), the states split according to the Zeeman effect. Here we see the Zeeman splitting of a P-S doublet transition (Evgeny at English Wikipedia).

Introducing a half-integer quantum number without physical justification was a rather audacious move. In a stadium concert, if an agency issued two tickets for the same seat A11 by labeling them A10½ & A11½ that would seem like chicanery. In hindsight, Goudsmit freely admitted that his physical understanding was not developed enough to justify such a move. He was working part time with Pieter Zeeman on atomic spectral lines, but had yet to see the connection. Zeeman had found extra spectral lines when an atom was placed in a magnetic field for which there was no explanation. Luckily Ehrenfest paired Goudsmit with Uhlenbeck, who knew a greater deal of foundational physics.

Graph showing the Zeeman splitting in Rb-87, the energy levels of the 5s orbitals, including fine structure and hyperfine structure (Danski14 / Wikimedia Commons).

As Goudsmit recalled, “Ehrenfest said: ‘You should work together with him for a while, then he may learn something about the new atomic structure and all that spectral business.’ What he clearly thought, of course, was: ‘Perhaps I might learn a little bit of real physics from Uhlenbeck.’”

Uhlenbeck learned from Goudsmit about the anomalous spectral lines as well as his theory of a half-integer quantum number and brilliantly connected the two ideas. The fourth quantum number, Uhlenbeck pointed out, made sense if the electron generated its own magnetic field like a spinning ball of charge. If it was a mini-magnet that could spin either clockwise or counterclockwise, it would have two different energy states in the presence of an external magnet — either aligned or anti-aligned — which would explain the split in spectral lines. Goudsmit was convinced. They wrote up their results and gave them to Ehrenfest, who promptly submitted them to a journal.

The visualization of an electron’s spin on the exterior wall of a building in Leiden (Vysotsky: Wikimedia)

The young physicists were lucky that Ehrenfest could be impulsive. If he had discussed the spin idea with others, probably few in the physics community (except, potentially, for Werner Heisenberg, who was also thinking about half-integer quantum numbers) would have supported it. But once it was published, and the spin idea was re-interpreted as an abstract quantum number, it seemed the perfect way of understanding Pauli’s exclusion principle. Integer “stadium seating” for electrons was out, half-integer was in.

Later on, in the late 1930s, Goudsmit (L) and Heisenberg (center) would find themselves on opposite sides in World War II. Goudsmit at one point planned to kidnap Heisenberg, but did not; Heisenberg worked on the atomic bomb for the Nazis, and did not rescue Goudsmit’s parents from the concentration camps. They were killed in 1943 (AIP EMILIO SEGRE VISUAL ARCHIVES, CRANE-RANDALL COLLECTION).

When they left Leiden, Uhlenbeck and Goudsmit conducted a different kind of experiment, dubbed the “Michigan experiment,” when they both took on roles as Assistant Professors at the University of Michigan at the same time. They even collaborated on training graduate students, including Dutch physicist Max Dresden (who would become the research supervisor of this author and carry on the pedagogical tradition handed down by Ehrenfest, Uhlenbeck, and Goudsmit.) Open-minded inquiry was the hallmark of that school of thought — which splendidly led to the important concept of spin.

George Uhlenbeck accepting the Medal of Science from Jimmy Carter (National Science Foundation).

Paul Halpern is the author of fifteen popular science books, including The Quantum Labyrinth: How Richard Feynman and John Wheeler Revolutionized Time and Reality.

Starts With A Bang is now on Forbes, and republished on Medium thanks to our Patreon supporters. Ethan has authored two books, Beyond The Galaxy, and Treknology: The Science of Star Trek from Tricorders to Warp Drive.

Starts With A Bang!

The Universe is out there, waiting for you to discover it.

Paul Halpern

Written by

Physicist and science writer. Author of Synchronicity: The Epic Quest to Understand the Quantum Nature of Cause and Effect

Starts With A Bang!

The Universe is out there, waiting for you to discover it.

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