Spin & the Stern–Gerlach Experiment

Last session named a fourth quantum number, the spin number mₛ, without explaining it. Its story begins with magnetism. Circulating charge produces a magnetic moment, a measure of how strongly something behaves like a bar magnet, written μ and measured in joules per tesla. An electron with orbital angular momentum is a tiny current loop, so atoms should behave like miniature compass needles. The natural unit is the Bohr magneton, μB = eħ/2mₑ, about 9.27×10⁻²⁴ joules per tesla, the moment of an electron orbit carrying one unit, ħ, of angular momentum.

In a uniform magnetic field a compass needle feels a torque, a twist toward alignment, but no net push: its two poles are pulled equally hard in opposite directions. To push a magnet sideways you need a field gradient, a field that changes from place to place, so that one pole sits in a stronger field than the other. If the field strengthens along a direction we call z, the force is the moment's component along z, μz, times the gradient. A magnet pointing up the slope is pulled toward the stronger field, one pointing down is pushed away, and one lying sideways feels nothing. A beam of atomic magnets is sorted by μz.

That makes a decisive test. Atoms leaving a hot oven point every which way, so classically μz takes every value between −μ and +μ, and the beam should merely broaden into a single continuous smear, with no gap anywhere. The old quantum theory of Bohr and Sommerfeld disagreed: its rule of space quantization let an orbit take only certain orientations relative to a field, so the beam should break into separate stripes. In 1921 Otto Stern, then working at Max Born's institute in Frankfurt, saw that an atomic beam could decide between the two. The idea, he later recalled, came to him early one morning in a bed too cold to leave.

Walther Gerlach, an experimenter in the neighbouring institute, built the apparatus with Stern; it took more than a year to work. Silver vapour from an oven at about 1,000 degrees Celsius was trimmed into a narrow beam by slits 0.03 millimetres wide, then flew 3.5 centimetres between shaped magnet poles producing a field of about 0.1 tesla that changed by roughly 10 tesla per centimetre. The expected splitting was a fraction of a millimetre, so a misalignment of 0.01 millimetres could spoil a run, and the apparatus rarely ran more than a few hours between breakdowns.

The silver deposits were about one atom thick and invisible to the eye. In Stern's telling, an early plate showed nothing until he peered at it closely: his cheap cigars were rich in sulfur, his breath turned the silver into jet-black silver sulfide, and the trace emerged like a developing photograph. When Friedrich and Herschbach re-enacted the scene in 2002, breath alone did nothing but cigar smoke blackened the film, so the smoke probably did the work. Early in 1922 the two men met in Göttingen and agreed to give up. Then a railway strike stranded Gerlach on the way home; he spent the day rethinking every detail, improved the alignment, and soon the beam split. A postcard to Niels Bohr, dated 8 February 1922, carried a photograph of the result: 'Attached is the experimental proof of directional quantization. We congratulate you on the confirmation of your theory.'

The plate looked like a pair of lips. Where the gradient was strongest, the trace divided into two separate bands with an empty gap between them, merging only at the ends, where the gradient weakened. No atom arrived undeflected. Each was pushed up or down by an amount set by its speed, and from the size of the split Stern and Gerlach found the silver atom's moment to be one Bohr magneton, within about 10 percent. Not a smear but two stripes: a component of an atom's magnetism, and therefore of its angular momentum, takes only discrete values.

The answer came from spectroscopy. In late 1924 Wolfgang Pauli, puzzling over the anomalous Zeeman effect, in which spectral lines in a magnetic field split into more components than any orbit allowed, concluded that the electron carries a 'two-valuedness not describable classically'. Early in 1925 Ralph Kronig proposed that the electron rotates about its own axis, and Pauli's scorn kept him from publishing. That autumn in Leiden, George Uhlenbeck and Samuel Goudsmit had the same idea, and their mentor Paul Ehrenfest sent it to a journal. As Goudsmit remembered it, Ehrenfest told them the idea might be wrong, but with no reputation yet they had nothing to lose.

The spinning ball did not survive; the name did. Spin is intrinsic angular momentum: angular momentum that belongs to a particle itself, like its charge or mass, rather than to any motion through space. Electrons, protons, neutrons and quarks all have spin quantum number s = ½. Measured along any axis, the spin component is mₛħ with mₛ = ±½, that is, +ħ/2 or −ħ/2, about 5.27×10⁻³⁵ J·s. The rule that gives orbital motion 2l + 1 orientations gives spin 2s + 1 = 2, hence two stripes. In silver, 46 of the 47 electrons pair off, their spins and orbits cancelling for reasons that are next chapter's business, leaving one outer electron in the fifth shell's s orbital, with l = 0. Magnetically, a silver atom is a single electron spin.

Why not a spinning ball? Three fatal reasons. Size: the electron behaves as a point in every experiment so far, and even a ball as large as the classical electron radius, 2.8×10⁻¹⁵ metres, would need a surface speed above a hundred times the speed of light to carry ħ/2. Magnetism: a ball with charge and mass spread alike has a g-factor, the ratio of magnetism to angular momentum in natural units, of 1; electron spin has a g-factor of 2.0023, which is why a silver atom's moment is a full Bohr magneton. Counting: orbital angular momentum always has whole-number l and an odd number of orientations. Two stripes demand a half-integer.

Treat a Stern-Gerlach magnet as a question. A magnet whose gradient points along z asks 'up or down along z?', and its two exits are the two answers, the states |↑⟩ and |↓⟩. In the language of chapter 14 they are eigenstates of the z-spin operator, with eigenvalues +ħ/2 and −ħ/2. Turn the magnet 90 degrees about the beam and it asks the same question along x, with answers |+x⟩ and |−x⟩. Written in the z language these are equal-weight superpositions: |+x⟩ = (1/√2)(|↑⟩ + |↓⟩) and |−x⟩ = (1/√2)(|↑⟩ − |↓⟩). Block one exit with a stop and the magnet becomes a filter passing a single answer.

Now chain the magnets, in the idealized experiments that open Sakurai's textbook and anchor Feynman's Lectures. Z then Z: send the up atoms from one Z magnet through another, and every one exits up; asking again changes nothing. Z then X: send them into an X magnet instead, and they split fifty-fifty, because |↑⟩ = (1/√2)(|+x⟩ + |−x⟩) and each amplitude squared is one half. Perhaps each atom carries two hidden labels, one for z and one for x. Z then X then Z: send only the +x atoms into a final Z magnet. Hidden labels would bring every atom out up, since we filtered for up at the start. Instead, half come out down.

The X magnet did not read a pre-existing label. It measured, leaving each atom in |+x⟩, a state with no definite z value, so the final magnet tosses fresh quantum dice. As chapter 15 showed, z and x spin are incompatible observables: their operators do not commute, so no state has a definite value of both. A spinning top has a definite component along every axis at once. An atom has a definite answer to one spin question at a time, and a new question wipes out the old answer.

Tilt a magnet by an angle θ from z, in the plane across the beam, and it asks 'up or down along this axis?' Its up state is |↑θ⟩ = cos(θ/2)|↑⟩ + sin(θ/2)|↓⟩. Check: at θ = 0 it is |↑⟩; at θ = 90°, cos 45° = sin 45° = 1/√2, giving |+x⟩; at θ = 180° it is |↓⟩, since up along minus z is down along z. The Born rule of chapter 9 then gives the odds for an atom known to be up along z: P(up along θ) = cos²(θ/2). At 0, 60, 90, 120 and 180 degrees the chance of up is 1, three-quarters, one half, one quarter and 0.

Why half the angle? Up and down are opposite directions in space, yet they are perfectly distinguishable states, and perfectly distinguishable states must have zero overlap, like perpendicular arrows. A 180-degree turn in space must therefore be a 90-degree turn between state arrows, so every angle in space is halved in the space of states. Photon polarization, by contrast, obeys Malus's cos²θ with no halving, because its perfectly distinguishable pair, horizontal and vertical, is only 90 degrees apart.

This rule settles the myth that spin can point in any direction when measured. A spin can be prepared along any direction you like: atoms leaving a tilted magnet's up exit are in |↑θ⟩. But a measurement along any axis returns only +ħ/2 or −ħ/2, never an in-between value. What varies smoothly is the average: the expected component along the tilted axis is (ħ/2)(cos²(θ/2) − sin²(θ/2)) = (ħ/2)cos θ, exactly the projection a classical arrow would give. Single atoms answer in clicks, and only crowds trace out the smooth classical curve. Nor can the spin vector lie exactly along any axis, since its length, √(s(s + 1))·ħ ≈ 0.87ħ, exceeds its largest component, ħ/2.

The half angle hides a stranger consequence. Rotating a spin state through an angle α multiplies its amplitudes by cosines and sines of α/2, so a full 360-degree turn, which returns every arrow, top and teapot to its starting position, turns |↑⟩ into −|↑⟩; only after 720 degrees does the state come back to itself. Two-component objects with this property are called spinors; Pauli built them into quantum mechanics in 1927, with the matrices you met in chapter 14. A minus sign in front of a whole state changes no probability, so a lone rotated beam looks untouched. To see the sign you must compare the rotated state with an unrotated copy of itself.

Neutron interferometers do exactly that. A neutron is a spin-one-half particle with a small magnetic moment, and a perfect silicon crystal can split its wave into two paths that later recombine. In 1975 a team led by Helmut Rauch in Vienna, with a young Anton Zeilinger among the authors, and an American team led by Samuel Werner sent one path through a magnetic field that made the neutron's spin precess, wheeling round the field like a tilted top, a motion called Larmor precession. As the field grew, the interference signal rose and fell with a period of 720 degrees of precession, not 360; Rauch's team measured 704 ± 38 degrees. A 360-degree turn on one path turned bright into dark.

No rigid object behaves like this, yet the geometry of rotations permits it. Balance a cup on your flat palm and turn it through one full circle without letting go: your arm ends up twisted. Keep turning the same way through a second circle, sweeping the cup over your head, and your arm comes back untwisted. A belt fixed at one end agrees: after one full turn of the free end, no way of moving that end without turning it removes the twist, while after two full turns one does. Spin-one-half particles carry this two-turn memory of rotation, and in chapter 19 the same sign helps sort every particle into fermions and bosons.

Because spin carries a magnetic moment, a magnetic field B gives its two states different energies. For an electron the gap is ΔE = g·μB·B, and a photon of exactly that energy, frequency f = ΔE/h, can flip the spin. This is magnetic resonance. Free electrons resonate at about 28.0 GHz per tesla, so near 0.35 tesla the resonance sits at 9.8 GHz, in the microwave band. Yevgeny Zavoisky discovered electron spin resonance in Kazan in 1944. Only unpaired electrons respond, which makes it a sensitive detector of free radicals and crystal defects; its descendants include the diamond quantum sensors of chapter 24.

Protons have spin one-half too, but are about 658 times less magnetic than electrons, so they resonate at radio frequencies, 42.58 MHz per tesla. In 1938 Isidor Rabi, a veteran of Stern's Hamburg laboratory, detected nuclear magnetic resonance in molecular beams; in 1946 Felix Bloch and Edward Purcell found it in ordinary liquids and solids. Magnetic resonance imaging adds an idea Stern and Gerlach would recognize, a field gradient: when the field varies across the body, so does the resonance frequency, and a signal's frequency reveals where it came from. Paul Lauterbur proposed gradient imaging in 1973 and Peter Mansfield made it fast; they shared the 2003 Nobel Prize in Physiology or Medicine.

A 3-tesla hospital scanner therefore works at 127.7 MHz, just above the FM radio band. The energy gap, about 5×10⁻⁷ electron-volts, is tiny beside the thermal energy at body temperature, 0.027 eV, so the lower-energy orientation wins by only about 10 parts per million. But a millilitre of water holds 6.7×10²² hydrogen nuclei, so the imbalance is still about 7×10¹⁷ spins; tipped by a radio pulse, their combined magnetization precesses and induces a voltage in the scanner's coils. That signal is the smooth average of an enormous crowd, the (ħ/2)cos θ curve of the tilted magnets; each nucleus, measured alone, would still answer only up or down.