Table of Contents
Introduction The Original Setup The Surprising Result Modern Interpretation Sequential Measurements Modern Variants Misconceptions FAQ Sources Introduction
The Stern-Gerlach experiment, performed in 1922 by Otto Stern and Walther Gerlach at the University of Frankfurt, demonstrated that the angular momentum of atoms is quantized — meaning it can only take discrete values, not a continuous range. It also gave the first direct evidence of what would later be called electron spin. The experiment is one of the cleanest demonstrations of quantum behavior and is now standard pedagogy in every quantum mechanics course.
This article walks through the original experiment, why its result was startling, its modern interpretation in terms of spin, sequential measurements that illustrate quantum measurement theory, and modern variants of the technique. Every nontrivial claim is sourced.
The Original Setup
Stern and Gerlach sent a beam of silver atoms through a region with a strongly inhomogeneous magnetic field [1 ]. The atoms were produced by heating silver in a furnace; only atoms moving in a specific direction passed through a narrow slit and entered the field region.
Why Silver
Silver atoms have a single valence electron in an s orbital — orbital angular momentum L = 0. Total angular momentum comes only from the single unpaired electron's spin. The experiment was therefore sensitive to a single quantum-mechanical degree of freedom.
The Magnetic Field
The field had a strong gradient — meaning the field strength varied with position. An atom with magnetic moment μ in such a field experiences a force F = ∇(μ·B). The direction of the force depends on the orientation of the magnetic moment relative to the gradient.
The Detection
After passing through the field, the silver atoms hit a photographic plate. Stern and Gerlach looked for a pattern of deposition.
The Surprising Result
Classical Expectation
Classically, silver atoms in random thermal motion would have magnetic moments pointing in all possible directions. The force from the field gradient would push them by amounts proportional to the cosine of the angle between their moment and the gradient. The deposition pattern should be a continuous spread between the two extremes.
What They Observed
Exactly two spots — not a continuous distribution. The silver atoms split into two beams, deflected up or down with no intermediate values. The angular momentum (and thus magnetic moment) along the field direction took only two values, exactly opposite each other.
The Implication
Angular momentum is quantized. For a single electron's contribution, only two values are possible. The "two-valuedness" was the empirical input that would soon be identified as spin ½ [2 ]. The discrete spots vindicated Sommerfeld's earlier theoretical hint about quantized angular momentum but went further by showing the values are discrete to a much finer level than anyone had suspected.
Misidentification
Initially Stern and Gerlach thought they were measuring orbital angular momentum of the electron in its silver atom. Bohr's old quantum theory predicted angular momentum quantization, and the result was taken as confirmation. Only later (after 1925-26) was it understood that what they had actually measured was electron spin — the angular momentum L for silver's outer electron is zero; the splitting came from spin alone.
Modern Interpretation
Today we understand the experiment in spin language. The silver atom's outer electron has spin ½ with magnetic moment μ = gμB S/ℏ, where g ≈ 2 (electron g-factor) and μB is the Bohr magneton.
Two Spin States
For spin ½, Sz = ±ℏ/2. The atom in the field has two possible energies E = −μ·B, leading to two forces and two beams. The splitting in the photographic plate is proportional to the gradient strength, the magnetic moment, and the time of flight through the field.
The Quantum Picture
The beam emerging from the furnace contains a statistical mixture of spin orientations (random unpolarized initial state). After passing through the Stern-Gerlach analyzer, the beam splits into two — the analyzer projects the spin state along the field direction. The two outgoing beams are eigenstates of Sz with eigenvalues +ℏ/2 and −ℏ/2.
Magnetic Moment Quantization
The experiment also established that the magnetic moment μB = eℏ/(2me ) is a fundamental constant of nature. The deflection magnitude pinned down its numerical value to within experimental error.
Sequential Measurements
If you place two Stern-Gerlach analyzers in sequence with different orientations, the results illustrate quantum measurement theory beautifully [3 ].
Same Direction Twice
Send atoms through a z-oriented analyzer, select only the "spin up" beam, then send through a second z-oriented analyzer. All atoms come out "spin up" — the second measurement gives the same result as the first, as expected for measuring the same observable on the same state.
Perpendicular Direction
Send atoms through z-oriented, select "spin up z," then send through an x-oriented analyzer. The atoms split equally into "spin up x" and "spin down x" — 50/50. The z eigenstate is an equal superposition of x eigenstates. Knowing Sz precisely makes Sx completely undetermined.
Then Back to Z
Now take the "spin up x" beam and send it through another z-oriented analyzer. You get 50/50 split into "spin up z" and "spin down z." The x measurement has "erased" the previous z measurement — measurement disturbs the system in a structured way determined by the commutation relations.
What This Demonstrates
The Stern-Gerlach experiment combined with sequential measurements illustrates:
Quantization of measurement outcomes.
The collapse of the wave function upon measurement.
Non-commutativity of conjugate observables.
Heisenberg-like uncertainty for angular momentum components.
Modern Variants
Atom Interferometers
Modern atomic-physics experiments use Stern-Gerlach-style separation as a building block of atom interferometers. Cold-atom beams are split, recombined, and used for precision measurements of gravity, rotation, and tests of fundamental physics [4 ].
Coherent Stern-Gerlach
If you remove the photographic plate and recombine the beams coherently, you reconstruct the original spin superposition. The "splitting" is reversible; the spin state has not collapsed because no measurement was made. This is fundamental for quantum information experiments.
Cesium Fountain Clocks
Modern atomic clocks use Stern-Gerlach-type magnetic state selection to prepare atoms in specific hyperfine states before microwave interrogation. The technique is essential for precision spectroscopy.
Spin-Polarized Beam Sources
Stern-Gerlach polarizers are used to prepare beams of spin-polarized atoms or electrons for scattering experiments. Modern variants achieve >99% polarization in laboratory settings.
Quantum Computing Demonstrations
Stern-Gerlach-style spin manipulation is used in trapped-ion and neutral-atom quantum computers for state preparation and readout.
Common Misconceptions
"Stern and Gerlach measured spin directly"
They measured a two-valued angular momentum but didn't yet have the concept of spin. The interpretation came later. Their 1922 paper called it "directional quantization."
"The two spots show quantization of the magnetic moment magnitude"
No — they show quantization of the projection along the field direction. The magnitude of the spin angular momentum is √(s(s+1))ℏ = √3/2 ℏ for spin ½, not ±ℏ/2. Only the projections quantize to ±ℏ/2.
"Stern-Gerlach is a complete description of spin"
It shows projection quantization but not the full structure (commutation relations, complex superpositions, spinor transformations under rotations). Those came from Pauli's and Dirac's theoretical work.
"Silver atoms with L = 0 should not be magnetic"
Right — by orbital angular momentum alone, they shouldn't be. The magnetic moment in the experiment comes entirely from electron spin, which was unknown at the time. The unexpected magnetic moment was a clue to spin's existence.
"The experiment can be done with one atom"
The 1922 experiment used a beam of many atoms (a statistical ensemble). Single-atom Stern-Gerlach-like measurements are now done routinely in trapped-ion and atomic physics labs.
"Stern and Gerlach won the Nobel Prize for this"
Stern won the 1943 Nobel Prize for "his contribution to the development of the molecular ray method and his discovery of the magnetic moment of the proton." The Stern-Gerlach experiment is mentioned in the citation but the prize is broader.
FAQ
Why silver and not hydrogen?
Silver was practical: it has a single valence electron and atomic beams can be produced from heated silver. Hydrogen, with its single electron, would have been ideal in principle but is harder to handle experimentally.
How precise was the original experiment?
The 1922 measurement determined the Bohr magneton to within about 10% of its modern value. Subsequent refinements improved this dramatically.
Can the experiment work with photons?
No — photons don't interact with magnetic field gradients in the same way. Their analog would be polarizing beam splitters separating different polarization states.
What if the magnetic field is misaligned?
The atoms still split into two beams, but along the new field direction. The "spin direction" is set by the orientation of the field, not by any preferred direction of the atom.
Does the experiment violate the uncertainty principle?
No. It measures the spin projection along one axis precisely (one of two values), leaving spins along perpendicular axes completely undetermined. This is exactly what the uncertainty principle for angular momentum predicts.
Why did it take so long to identify spin?
The 1922 result was interpreted in the framework of Bohr's old quantum theory, where angular momentum quantization was expected (though not in the discrete two-value form observed). Only after the development of modern quantum mechanics (1925-26) did the spin interpretation crystallize.
Sources
Gerlach, W., Stern, O. (1922). "Der experimentelle Nachweis der Richtungsquantelung im Magnetfeld." Zeitschrift für Physik , 9(1), 349–352.
Uhlenbeck, G., Goudsmit, S. (1925). "Ersetzung der Hypothese vom unmechanischen Zwang." Naturwissenschaften , 13, 953–954.
Sakurai, J. J., Napolitano, J. (2017). Modern Quantum Mechanics , 2nd ed.
Cronin, A. D., Schmiedmayer, J., Pritchard, D. E. (2009). "Optics and interferometry with atoms and molecules." Reviews of Modern Physics , 81(3), 1051–1129.
Nobel Foundation (1943). "The Nobel Prize in Physics 1943 — Otto Stern."
Friedrich, B., Herschbach, D. (2003). "Stern and Gerlach: How a bad cigar helped reorient atomic physics." Physics Today , 56(12), 53–59.