Two spots

A beam of silver atoms through a magnet, and a photographic plate that stayed blank until a man with a cheap cigar breathed on it. They were testing the wrong theory and got the right answer for a reason nobody would understand for three more years.

1922 – 192718 min

By the end you should be able to

Say what the classical prediction was and why two spots refuted it
Explain the cos²(θ/2) law and why the half-angle matters
Describe what the sequential experiment shows that a single measurement cannot

By 1922 there was a peculiar claim in circulation called space quantisation. Sommerfeld had extended Bohr's model so that an atom's angular momentum could not point in arbitrary directions: relative to an external magnetic field, only certain discrete orientations were allowed. Not merely a quantised magnitude — a quantised direction. It sounds like nonsense, and Otto Stern thought so. He is reported to have said that if this business turned out to be right, he would give up physics. He then designed the experiment that would settle it, which is the correct response.

A commemorative plaque in Frankfurt showing the Stern–Gerlach apparatus and a reproduction of the postcard sent to Bohr. — The memorial at Frankfurt, where the experiment was done. It reproduces the postcard Gerlach sent to Bohr showing the split beam — the deposit is a few hundredths of a millimetre across, and it only became visible because Stern smoked cheap cigars: the sulphur in the smoke turned the invisible silver film into black silver sulphide.

Why the field has to be non-uniform

The apparatus turns on one piece of physics that is easy to miss. A magnet sitting in a uniform field feels no net force — the pull on its north pole and the push on its south pole cancel exactly. It only twists. A magnet in a field that varies across space does feel a net force, because one end sits in a stronger field than the other. And the size and direction of that force depend on how the magnet is oriented. So: shape the pole pieces so the field is deliberately lopsided, send atoms through, and each one is deflected by an amount that reports the orientation of its magnetic moment. Then let them land on a glass plate and see where they went. It is about as direct as an experiment can be. The apparatus asks a question and the plate records the answer.

Silver was chosen carefully, and the reasons are worth stating. It has 47 electrons, of which 46 pair up and cancel, leaving exactly one unpaired in the outer 5s shell. So the entire atom carries the magnetic moment of a single electron and behaves as one — no complicated internal bookkeeping. It evaporates at a manageable temperature, giving a usable beam from a simple oven. And it blackens on a photographic plate, so you can see where it landed. That last property is about to matter more than they expected.

Compare the classical band against what actually lands on the plate.

Loading the magnets…

Thermal atoms leave an oven pointing in every direction, and a classical magnetic moment is deflected in proportion to cos θ — so the beam should arrive as a continuous vertical smear, brightest in the middle. Two spots with a gap between them is not something that picture can produce at all.

The experiment

Silver atoms through a lopsided magnet

Otto Stern and Walther Gerlach · February 1922 · Physikalischer Verein, Frankfurt

The question: Can an atomic magnetic moment point in any direction relative to an applied field, as classical physics requires — or only in certain discrete orientations, as Sommerfeld’s space quantisation claims?
The apparatus: Silver evaporated in an oven at about 1300 K, collimated by slits into a beam roughly 0.03 mm wide, passed through a 3.5 cm magnet with one knife-edge pole and one grooved pole to produce a strongly non-uniform field, then flown about 10 cm to a glass plate. The whole apparatus ran under vacuum and required exposures of around eight hours; the resulting traces are about 0.2 mm apart and were measured under a microscope.

Theory predicted

Classically, a continuous band: thermal atoms emerge at every orientation, deflection varies smoothly with orientation, so the beam simply broadens with the greatest density at the centre. Sommerfeld’s theory predicted a discrete pattern — for the state silver was believed to be in, three lines.

They measured

Two distinct traces, separated by about 0.2 mm, with no trace of atoms in between. Not a band, not three lines. Two spots and a clean gap where the undeflected beam would have been.

How sure could they be? The separation is a fifth of a millimetre from a beam initially 0.03 mm wide, which is why the result is unambiguous rather than marginal — the gap is far wider than the beam. The initial plate appeared blank, and the image emerged only when Stern leaned in to look at it.

Why it mattered

The classical picture is refuted outright: no distribution of orientations produces a gap in the middle. But the experiment also refuted the theory it was testing. Silver’s ground state has zero orbital angular momentum, so there was nothing there for space quantisation to quantise, and two is not a possible number of orientations for any orbital angular momentum — those always give an odd number. What Stern and Gerlach had measured was electron spin, which would not be proposed until 1925.

You might think

Stern and Gerlach set out to test space quantisation, and confirmed it.

Actually

They confirmed something, and it was not the theory under test. Silver’s ground state has zero orbital angular momentum — there was nothing there for space quantisation to act on, and the observed splitting should not have happened at all on Sommerfeld’s account. Worse, orbital angular momentum ℓ always gives 2ℓ+1 orientations, which is an odd number: 1, 3, 5. Two is not on the list. Sommerfeld predicted three lines for the state he assumed silver occupied; two lines is a result his theory cannot produce. What the plate recorded was electron spin, a property with two orientations that nobody proposed until Uhlenbeck and Goudsmit in 1925, three years later. The experiment is rightly regarded as one of the most important of the century, and essentially nobody involved understood what it had shown until well after the fact.

Sweep the angle. Only ever two outcomes — and watch the split follow cos²(θ/2), not cos θ.

Loading the magnets…

Rotating the analyser never produces an intermediate deflection. What varies is how the atoms divide between two fixed outcomes, and the weighting follows the half-angle — 50/50 at 90°, and certainly down at 180°, where a classical projection would give cos 180° = −1.

Step through z→z, then z→x, then z→x→z. Then check the control, z→z→z.

Loading the magnets…

The chain is where this becomes more than a curiosity. z→z passes everything, so the first measurement really does leave a persisting definite state. z→x splits evenly, which is unremarkable. z→x→z splits evenly again — and that is not.

Measuring x destroys z

Follow the chain carefully, because the conclusion is forced. z → z. Every atom passes. So the first magnet did leave each atom in a definite state with respect to z, and that state is stable — it is still there when you ask again. z → x. Splits 50/50. Unremarkable on its own: x was never determined, and you could still say each atom simply had an x value you had not looked at. z → x → z. Splits 50/50 again. But every atom entering that third magnet was measured up-along-z two stages ago, and nothing has touched its z component since. It should still be up. z → z → z. Passes intact, 100%. So it is not the act of measuring that scrambles anything, and not the magnets being clumsy — you can measure z as often as you like without harm. The only difference is what was asked in between. Measuring x destroys z. And notice what this rules out: the idea that the atom quietly carries definite values for both, waiting to be read. If it did, the third magnet would find the z value still sitting there.

1916Sommerfeld proposes space quantisation. Stern says he will quit physics if it is right.
1922Two spots. The plate is blank until a cheap cigar develops it.
1925Uhlenbeck and Goudsmit propose electron spin — the thing actually measured three years earlier.
1927Phipps and Taylor repeat it with hydrogen, ruling out any residual orbital contribution.
1943Stern receives the Nobel Prize. Gerlach, who had worked on the German nuclear programme, does not.
1975Neutron interferometry detects the 720° sign flip directly.

Where this leaves us

This is the cleanest system in physics for thinking about measurement: two outcomes, no continuum, no complications, and a chain of magnets you can lengthen at will. And it makes the question that has been building for four lessons unavoidable. Before the x-magnet, the atom was definitely up along z — the z→z test proves it. After the x-magnet, it definitely is not. Something happened. But Schrödinger's equation is smooth, continuous and completely deterministic. Feed it a state and it hands you the state at any later time, with no probabilities and no jumps anywhere in it. It contains nothing that looks like "and then one of the two outcomes occurred". So the theory has two rules that do not fit together: one for how things evolve when nobody looks, and another for what happens when somebody does. Nothing in the theory says which is which, or what counts as looking. That gap opened in 1926. It has not closed.