The experiment that defines spin and cannot be done on it

The experiment that established that spin exists, and that it takes only two values, is a beam through an inhomogeneous magnetic field. A magnetic moment $\mu$ in a field gradient feels a force $\mu\,\partial B/\partial z$; the two orientations feel opposite forces; and the beam splits in two.

It was done in 1922 on silver atoms, and it was done to test something else entirely. Stern and Gerlach were looking for the space quantisation the old quantum theory predicted for orbital angular momentum, expecting either a continuous smear if the classical picture held or an odd number of components if the quantisation did. They got two, which neither theory allowed, and the interpretation as electron spin came four years later from other people.

That is worth keeping beside the result. The experiment is remembered as the discovery of spin and it was designed as a test of something that turned out not to be what it measured — and the reason it succeeded at all is the property nobody in it was thinking about. Silver was chosen for a chemical reason — one unpaired outer electron, so the atom’s moment is that electron’s, by the filling order that leaves it alone — and for a practical one: it is easy to evaporate and it makes a visible deposit on glass.

There is a third reason, which nobody was looking for at the time and which turns out to be the only one that matters. A silver atom is neutral.

The obstruction, which is one line of vector calculus #

A magnetic field has no divergence — its lines have no ends, which is the same statement. So a field whose $z$-component varies with $z$ must have some other component varying in some other direction:

$$\frac{\partial B_z}{\partial z} + \frac{\partial B_x}{\partial x} + \frac{\partial B_y}{\partial y} = 0.$$

A gradient in $B_z$ therefore comes with a transverse field $B_x$ that varies across the beam — zero on the axis, growing linearly with the distance off it, and of the same order as the gradient times that distance.

A neutral atom does not care. A charged particle does: moving along the beam with speed $v$ through a transverse field $B_x$, it feels a Lorentz force $qvB_x$, which is along $z$ — the same direction the spin splitting is in. That force does no work and it does plenty of harm.

So the same magnet that separates the two spin states also deflects the beam by an amount that depends on where in the beam a particle happens to be. One is a splitting and the other is a blurring, and they are produced by the same field.

The magnet cancels out #

The arithmetic is short and its ending is the reason the argument is a serious one rather than a caution.

The spin force is $\mu\,\partial B/\partial z$ and gives a splitting proportional to it. The Lorentz force is $qv\,(\partial B/\partial z)\,w$ for a particle at the edge of a beam of width $w$, and gives a blurring proportional to the same gradient. Both act for the same time, over the same magnet. Dividing,

$$\frac{\text{splitting}}{\text{blurring}} \sim \frac{\mu}{qvw} = \frac{e\hbar/2m}{evw} = \frac{\hbar}{2mvw} = \frac{\bar{\lambda}_{\text{dB}}}{2w}.$$

The field gradient has gone. The magnet’s length has gone. What is left is the electron’s own de Broglie wavelength divided by the width of the beam — and the only reason the Bohr magneton and the charge combined that way is that the magneton is $e\hbar/2m$.

That cancellation is what makes the argument about the electron rather than about an apparatus. There is no magnet to improve, because the magnet is not in the answer.

The one escape, closed by diffraction #

The ratio improves as the beam is narrowed, and it reaches one when the beam is as narrow as the de Broglie wavelength. That is the escape, and it is closed by the other thing a narrow beam does.

A beam of width $w$ spreads by diffraction through an angle of order $\bar{\lambda}/w$, so over a magnet of length $L$ it spreads by $L\bar{\lambda}/w$. Staying collimated requires that to be less than $w$, which requires $w > \sqrt{L\bar{\lambda}}$ — and with $L$ a tenth of a metre and $\bar{\lambda}$ twenty picometres, that is about a micrometre and a half.

At that width the ratio is $\bar{\lambda}/2w$, about a part in a hundred thousand. Narrowing further improves the ratio and destroys the beam faster than it improves it, and the two requirements never both hold. The first figure draws all three lengths together and there is no window anywhere in it.

Nothing about this is a limit of technology. The three quantities are fixed by the electron’s charge, its mass and Planck’s constant, and the only free parameter — the beam’s width — makes one worse whichever way it is moved.

The same three lengths at a different setting #

The two comparisons above change the particle and leave the apparatus alone. The obvious objection runs the other way: the electron’s numbers are hopeless at one particular magnet and one particular beam energy, and an apparatus is a thing an experimenter may redesign. A stronger gradient is available, a longer pole piece is available, and a slower beam is available. The right way to test the objection is to grant all three at once and look at what moves.

What moves is every length, and by the same factor. The splitting is proportional to the gradient and to the square of the time in the field; so, term for term, is the Lorentz blurring, because the force that deflects the spin states apart and the force that deflects a charge sideways are both proportional to the field the magnet produces and both act over the same flight. Their ratio is built out of the electron’s charge, its mass and $\hbar$ and out of nothing the designer controls. The diffraction length is the only one of the three that ignores the magnet entirely, and it is set by the beam’s width, which is the quantity the two magnetic lengths have already fixed.

That is the practical content of the cancellation. An experimenter improving the apparatus moves every curve on the figure together, and the only quantity that changes the ordering is the beam’s width — which is the one quantity that cannot be pushed, because pushing it invokes diffraction.

It is worth being clear about which of the three lengths is quantum. The splitting is quantum, because the Bohr magneton contains Planck’s constant. The blurring is entirely classical. And the diffraction is quantum. So the argument as a whole is a comparison of two quantum quantities against one classical one, and the reason it comes out so decisively is that the two quantum quantities carry $\hbar$ to different powers.

What Bohr took from it, and what he did not #

Bohr made this argument in the 1920s, and Pauli reported and refined it; the conclusion drawn at the time was stronger than the argument supports, and the difference is worth stating because it recurs.

What the argument shows is that a particular method — separating states by a force in a field gradient — cannot be applied to a free electron. That is a definite and correct result, and the figures compute it.

What was often said is that the spin of a free electron is therefore not measurable in principle, and that spin is essentially non-classical in a way that orbital angular momentum is not. That is a larger claim and it does not follow from the argument. It has been contested since the 1990s by several proposals for arrangements that evade the specific obstruction — using time-varying fields, using the longitudinal rather than the transverse geometry, or measuring a correlation rather than a deflection — and the question of whether any of them works has not been closed. What is agreed is that no one has done it.

And what is certainly false is that the electron’s moment cannot be measured. It is the most precisely measured quantity in physics: about twelve significant figures, from a single electron in a trap, where the arrangement is nothing like a beam. The spin’s precession frequency is compared with the cyclotron frequency of the same electron in the same field, the ratio is the gyromagnetic factor — the two that is not exactly two — and the field cancels out of the ratio exactly as the gradient cancels out above — for the same reason, and to the opposite effect.

That is the useful lesson rather than the philosophical one. A quantity that resists one method is not thereby inaccessible; it usually means the method is measuring the wrong ratio. The beam experiment asks for a deflection and gets it swamped by a deflection; the trap asks for a ratio of frequencies and gets an answer to twelve figures, because a frequency ratio has no length in it to be blurred.

The trap that does measure it #

The arrangement that succeeds where the beam fails deserves more than the sentence above, because the contrast is instructive.

A single electron is held in a Penning trap: a strong uniform magnetic field to confine it radially, and a quadrupole electric field to confine it along the axis. It circles the field lines at the cyclotron frequency $\omega_c = eB/m$, and its spin precesses at $\omega_s = g\,eB/2m$. Both frequencies are proportional to the same field, so the ratio

$$\frac{\omega_s}{\omega_c} = \frac{g}{2}$$

contains no field at all. What is measured in practice is the small difference $\omega_s - \omega_c$, which is $(g-2)/2$ times $\omega_c$ — a difference of about a part in a thousand of a frequency, measured directly, which is why the anomaly is known far better than the moment itself.

The spin state is read out by a method that is itself a Stern–Gerlach effect in disguise. A small magnetic bottle — a deliberate inhomogeneity in the trap’s field — makes the electron’s axial oscillation frequency depend on its spin, by a few parts in a billion, and that shift is watched in the current the electron induces in the trap’s electrodes. The electron is not deflected anywhere; its spin is read off a frequency.

Two features of that make it work where a beam cannot. There is one electron, so there is no beam width for a Lorentz force to act across. And the observable is a frequency, which has no length in it, so nothing about the electron’s position enters the answer. The obstruction in the beam experiment is entirely about a displacement being compared with another displacement.

The same shift in method shows up throughout precision measurement: whenever a quantity resists measurement as a displacement, the thing to look for is a frequency ratio in which the troublesome factor cancels. That is the same trick a ringdown uses on a black hole, where a pitch and a decay rate between them determine two properties of an object nobody can approach.

An order-of-magnitude estimate, with trajectories standing in for a wavepacket #

The estimate is an order-of-magnitude one. The transverse field is taken as the gradient times the offset, the offset as half the beam width, and the deflections as those forces acting for the transit time. A careful treatment with a real pole geometry changes the numbers by factors of a few and does not change the cancellation, which is where the argument’s force is.

The beam is treated as a collection of trajectories. A proper treatment follows a wavepacket through the field, in which the spin and the position become entangled and the “two beams” are two branches of one state. The conclusion is the same and the language is different, and the distinction matters for the proposals that claim to evade the obstruction, since several of them turn on exactly that structure.

The Lorentz term can be reduced and not removed. Arrangements with the beam along the gradient rather than across it, or with compensating fields, change the geometry and the numbers. What survives every such rearrangement is that the magneton is $e\hbar/2m$, so any force that separates spins is the same size as a Lorentz force on a charge displaced by a de Broglie wavelength.

And it is specific to a charge. A neutron has a magnetic moment and no charge, and Stern–Gerlach separation of neutron spins is routine. So is the separation of neutral atoms, which is how atomic beams are polarised, and how a beam of them is cooled and trapped. The obstruction is about charge and nothing else.

Why the coincidence in the arithmetic is not one #

The cancellation that makes the argument work looks like luck and is not, and saying why turns a calculation into a reason.

The two forces compared are $\mu\,\partial B/\partial z$ and $qv\,(\partial B/\partial z)\,w$. They have the same gradient, so the ratio is $\mu/qvw$. That ratio is small because $\mu$ is small — and $\mu$ is small because the Bohr magneton is $e\hbar/2m$, in which the charge appears once in the numerator and once, through $v$ and $m$, in the denominator of the comparison.

Put differently: the magnetic moment of an elementary charge is exactly the moment a charge would have if it circulated with one unit of angular momentum. That is not an accident of the electron; it is what $g \approx 2$ and $\mu = g\,(e/2m)\,S$ amount to with $S = \hbar/2$. So the spin force is the same size as the Lorentz force on a charge displaced by the quantum of angular momentum divided by its momentum — which is the de Broglie wavelength.

The comparison is therefore between a length that belongs to the particle and a length that belongs to the beam, and no apparatus supplies either. That is the structural reason the magnet cancels, and it says immediately which particles the obstruction applies to: any particle whose moment is of order its own charge times its own de Broglie wavelength, which is every charged elementary particle there is.

A hypothetical particle with a much larger moment for its charge would escape it — and the only particles with anomalously large moments for their charge are composite ones, whose moments come from constituents, and among those the useful ones are neutral.

Two beams that are one superposition until something looks #

They cannot show that the two spin states are not two beams until something has looked. A wavepacket passing through the magnet becomes a superposition of a piece displaced upward with spin up and a piece displaced downward with spin down, and it stays a superposition — the two pieces can be brought back together and made to interfere, which they do, showing that nothing was measured by the deflection alone. The figure’s two dots are what a screen records and not what exists in the gap.

Nor can they show the beam’s own charge. A beam of electrons dense enough to see repels itself, spreading far faster than diffraction requires, and that space-charge spreading is a further obstruction not drawn here because it depends on the current rather than on anything fundamental. It can be reduced by using fewer electrons, and diffraction cannot.

And they cannot show what happens at one electron at a time. The argument is about a beam with a width, and a single electron has no width — it has a wavepacket with a spread, which is the same quantity by another name, and the argument goes through unchanged with $w$ read as the packet’s spread. That equivalence is worth stating because it is the point at which an uncertainty relation enters an argument that is otherwise classical.

Still open: whether the obstruction can be evaded at all #

Proposals to separate free-electron spins have appeared regularly since the 1990s and the field has not converged. They fall into three groups: geometries in which the beam travels along the gradient rather than across it, so that the Lorentz force is transverse rather than parallel to the splitting; arrangements using time-dependent fields, where the argument’s steady-state assumption fails; and schemes that measure a spin-dependent correlation between position and something else rather than a bulk deflection.

Analyses of each have been published on both sides, and the disagreements are usually about whether an approximation retains the term that does the blocking. No experiment has been attempted at the precision that would settle it, partly because the payoff is modest: a polarised electron source is available by other means, and the electron’s moment is known far better than any beam experiment could measure it.

So the question is a question about principle, and its interest is in what kind of statement it is. It is not a statement about what can be known, and it is not a statement about what apparatus exists. It is a statement that one particular measurement scheme has an obstruction whose size is fixed by three constants of nature — and whether a scheme exists that does not is a question nobody has answered by construction rather than by argument.

The habit worth carrying away is about impossibility arguments. An argument that a measurement cannot be made is only as strong as the class of measurements it quantifies over, and the class is usually much narrower than the conclusion drawn from it. Here the class is deflection in a static gradient, the obstruction inside it is real and computable, and the quantity said to be unmeasurable is in fact the best-measured number in physics.