The answer that was not there before

Send a beam of silver atoms through a non-uniform magnetic field and it does not spread into a smear. It splits into exactly two, one deflected up and one down, with nothing in between.

The splitting alone is a result, and it is the same kind of result a spectrum is: a continuous quantity turning out to take only certain values. A classical magnetic moment pointing in a random direction would be deflected by an amount proportional to its component along the field, giving a continuous band. Two spots means the component along the field takes only two values.

The chain is a much sharper result, and it is what this essay is about.

Two spots, whatever direction is chosen #

The first strange feature is that the two values do not depend on which direction the analyser points.

Turn the apparatus to any orientation and the beam still splits in two — not into a band as a distribution of speeds would give, along that orientation, with the same separation. There is no direction in which the beam fails to split, and no direction in which it splits into three. Whatever is being measured has exactly two possible answers along every axis.

That is already impossible for a vector. A classical arrow has a definite component along every direction at once, and those components are related — knowing the components along three perpendicular axes fixes the arrow completely, and fixes its component along every other axis too. Here, measuring along z gives ±1, and measuring the same atom afterwards along x gives ±1 with no memory of the z answer constraining it.

The chain, and what it removes #

Take the first analyser’s “up” beam and nothing else. Every atom in it has been measured along z and found to be up. Send it through a second analyser also along z, and everything comes out up: the measurement is repeatable, which is what makes it a measurement of something rather than a randomiser.

Now put the second analyser at ninety degrees. The beam splits into equal halves along x, which is unsurprising — the atoms were prepared with a definite z answer and nothing was said about x.

Take the “up along x” half and send it through a third analyser back along z. Classically the answer is obvious: every atom in this beam was found to be up along z by the first analyser, so all of them should come out up.

Half of them come out down.

So the middle analyser did not filter the beam. It replaced what the beam had an answer to. Measuring the x component destroyed the z answer that had been established, and destroyed it completely rather than partially.

The rule the intensities follow #

The numbers are not arbitrary. With an analyser at angle θ to the one that prepared the beam, the fraction going into each output is

$$P_+ = \cos^2\!\frac{\theta}{2}, \qquad P_- = \sin^2\!\frac{\theta}{2},$$

which is Malus’s law with a half-angle in it.

The half is the signature of spin-½ and it has a consequence worth stating: an analyser turned through 180° — pointing exactly the opposite way — gives cos²90° = 0, so what was “up” along z is never “up” along −z, which is right. But a full 360° rotation of the state rather than the analyser returns the amplitude with a minus sign, and it takes 720° to come back. That is not measurable in this experiment and it is measurable in a neutron interferometer, where a rotated beam is interfered with an unrotated one.

What a spin is, and what it is not #

Some care is owed to the word, because the mechanical picture it invites is not merely inaccurate — it is impossible.

An electron with the measured magnetic moment, treated as a spinning charged ball of any radius consistent with scattering experiments, would need a surface speed far above the speed of light to produce it. The calculation was done in 1925 by Lorentz and used to talk Uhlenbeck and Goudsmit out of publishing; they published anyway, and were right about the phenomenon and wrong about the mechanism.

What spin is, operationally, is an intrinsic angular momentum that is not associated with any motion through space. It contributes to the total angular momentum, it produces a magnetic moment, it obeys the conservation law — and it does not correspond to anything rotating. The electron is, as far as every experiment can tell, a point.

That has a consequence for reading the figure. The two outputs of an analyser are labelled “+” and “−” rather than “up” and “down” because there is no arrow to point up. What the labels name is which of two possible outcomes a measurement along the chosen axis produced, and nothing more.

Why hidden values cannot rescue it #

The natural repair is to suppose each atom carries definite answers for every direction, fixed at the source, and the analyser merely reads one of them off. The beam is then a mixture of atoms with different pre-assigned answer sets, and the statistics come from the mixture.

That works for the three-analyser chain. It does not work in general, and the reason is Bell’s, in 1964.

Consider two atoms prepared together so that their spins are opposite, sent to two distant analysers whose orientations are chosen independently. If each atom carries a full set of pre-assigned answers, then the correlations between the two analysers’ outcomes, averaged over any distribution of answer sets whatever, must satisfy an inequality. Quantum mechanics predicts correlations that violate it, at certain angle combinations, by a wide margin.

The experiments have been done, refined for fifty years, and closed the loopholes one at a time — detection efficiency, locality, and finally the freedom-of-choice loophole in experiments using astronomical sources to set the analyser angles. The inequality is violated. Pre-assigned local answers are not merely unnecessary; they are ruled out.

What survives, and what does not #

It is worth being careful about what that does and does not license, because the popular readings overshoot in both directions.

What does not survive is the picture — the one the double slit also excludes — of a particle carrying definite values for every property, with measurement as a passive reading. That picture is excluded by experiment, not by interpretation.

What does survive is that nothing observable travels faster than light. The outcomes at one analyser are random whatever the other one does; the correlation only appears when the two records are brought together and compared, which requires an ordinary signal. The chain above cannot be used to send a message, and no arrangement of it can.

What is not settled is what replaces the excluded picture. Several accounts fit every experiment — the state collapses, or it does not and the branches decohere, or the values are real but non-local. Choosing between them is not a matter this page can decide, and the honest position is that the rule is exact and the story is contested.

The rule stated as an operation #

The most useful way to hold on to the result is procedural, and it is the one that generalises to everything else in the field.

A measurement has a set of possible outcomes. A state is a combination of the corresponding possibilities with complex weights. The probability of an outcome is the squared magnitude of its weight, and after the measurement the state is the one corresponding to the outcome obtained.

Applied to the chain: “up along x” is an equal combination of “up along z” and “down along z”. So a beam prepared up along x has half its weight on each z answer, and a z analyser splits it evenly. The middle analyser did not disturb anything mechanically — it left the beam in a state that, expressed in the z basis, is an equal mixture.

That polariser demonstration is worth doing, because it makes the essential point with no quantum mechanics at all: the middle element does not select from what is there, it re-expresses the state in a new basis, and the new basis has components along directions the old one had removed.

The same rule stated as an operation is what makes it useful. A measurement is a projection: the state is resolved onto the basis the apparatus asks about, and the probability of each outcome is the square of the corresponding component. Nothing about that is special to spin — a polariser does it to light, and the cosine-squared law an optician knows as Malus’s is the identical statement about a two-dimensional space. What is peculiar is not the arithmetic but that the projection leaves the system in the state it projected onto, so the second measurement of the same quantity agrees and a measurement of a different one does not.

Where the analogy with light stops #

The parallel with polarisation is exact for the arithmetic and misleading about the mechanism, and the difference is instructive.

A light wave’s polarisation is a direction in a plane, and Malus’s law has cos²θ rather than cos²(θ/2) — a polariser turned through 90° blocks everything, and one turned through 180° blocks nothing, because a polarisation direction and its opposite are the same direction. Spin-½ has the half-angle, so the opposite orientation is a different state and 180° is complete extinction.

The deeper difference is that a light beam is many photons and a polariser’s action on a beam is describable classically. The chain above works one atom at a time, and each atom either goes up or down — there is no fraction of an atom in either channel. The classical wave picture reproduces the intensities and cannot reproduce the discreteness, which is the same split the double slit shows in a different variable.

Made into a measuring device #

Two uses have come out of the arrangement, and both depend on the property that makes it philosophically awkward.

Magnetometry and atomic clocks. The separation of the two beams is proportional to the field gradient and to the moment, so the apparatus measures either given the other. More importantly, the ability to prepare a beam in a known state and then interrogate it is the basis of the atomic beam method: Rabi’s technique of state-selecting, applying a radio-frequency field, and state-selecting again is how hyperfine transitions are measured to sixteen figures and how the second is defined.

Quantum key distribution. If a state cannot be read without disturbing it, then an eavesdropper reading a transmission leaves evidence. The BB84 protocol has the sender encode bits in one of two randomly chosen bases and the receiver measure in one of two randomly chosen bases; where the bases agree the bits agree, and an eavesdropper who guessed wrong introduces errors at a rate the two parties can measure by comparing a sample. The security argument is precisely the third analyser on this page: measuring in the wrong basis does not fail quietly, it randomises.

That second application is worth pausing on. The feature that makes the experiment hard to interpret — that a measurement in one basis destroys information about another — is not a defect being worked around. It is the resource, and there is no classical channel with the same property.

Putting the beams back together #

There is one variant of the chain that settles what the middle analyser is actually doing, and it is the sharpest experiment in the subject.

In the arrangement above, the middle analyser splits the beam and one output is blocked. Suppose instead that neither is blocked: reverse the field gradient downstream, bring the two paths back together, and recombine them with equal path lengths so that no record survives of which one an atom took.

The emerging beam is the beam that went in. Send it through a third analyser along $z$ and every atom comes out up, and none comes out down. The middle analyser, operated this way, has done nothing whatever.

That is decisive about the mechanism, because the magnetic field, the gradient and the forces on the atoms were identical in both versions. What differed is only whether the two branches were kept separate at the end. So the destruction of the $z$ answer in the original chain was not caused by the apparatus jostling the atoms; it was caused by discarding a branch, which is to say by making it possible in principle to know which way each atom went.

The general statement is the one this collection keeps arriving at from different directions. What destroys interference is the availability of which-path information, not the violence of the interaction. A gentle measurement that records which branch is taken destroys the coherence completely; a fierce interaction that records nothing does not destroy it at all.

Feynman built the whole of his treatment of quantum mechanics around exactly this apparatus for that reason, calling it the improved Stern–Gerlach filter and using it to introduce amplitudes before anything else. It is a superb teaching device and it was, for ninety years, a thought experiment: recombining two atomic beams so completely that no trace of the separation remains requires matching their positions, velocities and phases to a precision that was not available.

It has now been done. Full-loop Stern–Gerlach interferometers built on atom chips, with the splitting and recombination performed by magnetic gradients from wires micrometres away, have recovered the coherence and demonstrated the recombination directly. The result is the one Feynman assumed, and the ninety-year gap between the argument and the experiment is a fair measure of how hard it is to un-split a beam of atoms.

The cigar that developed the plate #

The experiment that started all of this was performed in Frankfurt in February 1922 with hand-built apparatus, and it very nearly produced nothing.

Silver was evaporated in an oven, collimated into a beam a few hundredths of a millimetre wide, passed through a magnet with a knife-edge pole piece to make the field gradient as sharp as possible, and deposited on a glass plate. The expected separation of the two traces was about a fifth of a millimetre. The apparatus was unstable enough that Gerlach worked at night, when the trams had stopped.

After one long exposure Gerlach took out the plate and saw nothing on it at all. The deposited silver film was a few atoms thick and completely transparent. He showed it to Stern, and as the two of them examined it the trace slowly darkened and became visible.

The reason was Stern’s cigars. As a poorly paid junior academic he smoked the cheapest available, which were high in sulphur, and his breath over the plate converted the invisible silver film to black silver sulphide. The experiment that established spatial quantisation was developed, chemically, by a bad cigar.

They sent Bohr a postcard with the photograph attached and a note congratulating him on the confirmation of his theory. As the section above records, it was not a confirmation of his theory. It was a confirmation of a prediction his theory happened to share with the correct one, arrived at from an atom whose orbital angular momentum is zero, three years before the quantity actually being measured had been proposed.

What the picture cannot show #

The figure draws beams with intensities on them, which is a description of an ensemble. A single atom does not have an intensity; it goes one way or the other, and the numbers on the drawing are the fractions obtained after many atoms.

It also draws the blocked outputs as stubs, which suggests the atoms in them are somewhere. They are — in a beam stop — and the point of drawing them is that they exist and are discarded. What cannot be drawn is that discarding them is what makes the surviving beam a prepared state rather than a filtered one.

And nothing on the figure shows the magnet. The apparatus is a field gradient that couples to the atom’s magnetic moment, and the whole reason the experiment splits a beam at all is that the moment is quantised — a fact the figure assumes and the original experiment discovered.

The experiment that was right for the wrong reason #

Stern and Gerlach ran it in 1922 to test Bohr and Sommerfeld’s quantisation of orbital angular momentum, which predicted that silver’s outer electron in an l = 1 state would give three spots. They got two, and took it as confirmation.

Silver’s outer electron is in an l = 0 state. It has no orbital angular momentum at all, and the two spots come from spin — which had not been proposed in 1922 and was not established until 1925. The experiment confirmed a theory that was wrong, by measuring a quantity the theory did not contain, and the number of spots happened to differ from the classical prediction in the same direction.

The episode is a standing warning about confirmations. A prediction of “not a continuous band” was confirmed; a prediction of “three spots” was refuted, and the refutation went unremarked for three years because the qualitative half was what everyone was looking at.

Where the ladder goes next #

The rungs from here: entanglement and the Bell inequalities, derived rather than described; the quantum Zeno effect, where repeated measurement freezes an evolution; weak measurement, which extracts partial information at the cost of partial disturbance; decoherence, which explains why large objects never show this behaviour without invoking any collapse; and the applications — quantum key distribution, whose security rests directly on the impossibility of reading a state without disturbing it.

The claim to carry forward is what the third analyser demonstrates. A measurement is not a reading. The first analyser produced a beam with no “down” component along z, and the third found one — so the property the first measured was not a property the atoms had and kept, but an answer that existed only in relation to the question that was asked.