The Stern-Gerlach Experiment

In a Frankfurt basement in 1922, two physicists named Otto Stern and Walther Gerlach built an apparatus that everyone expected would confirm the classical picture of atomic magnetism. A hot oven boiled silver atoms into a beam. The beam passed through the narrow gap of a strangely-shaped magnet — one pole was a sharp wedge, the other a broad flat face — so that the magnetic field between the poles was stronger near the wedge and weaker near the flat. Finally, the beam smacked into a glass plate at the far end.

If silver atoms behaved like the tiny spinning bar magnets that classical physics said they were, the plate should have shown a single vertical smear — a smooth distribution of atoms deflected by every amount between "maximally up" and "maximally down," depending on how each atom's internal magnet happened to be oriented when it entered the field. That is what Stern and Gerlach expected to see. That is what every classical calculation said they should see.

What they saw, after running the beam overnight, was two dots. A clean upper dot. A clean lower dot. Nothing in between.

The shock is hard to overstate. Silver atoms had refused to behave like continuous bar magnets. Their magnetic moment along the z axis — the direction of the field gradient — was not "some value between -\mu and +\mu." It was +\mu or -\mu, exactly, with nothing else on the menu. This is quantisation, in its most visceral experimental form, and it is the picture you should hold in your head for every later argument about why quantum is different from classical.

This chapter is about that picture. You will see how the apparatus works, why the two-dot pattern forces you to reject the classical model, how the spin-z eigenstates turn out to be the computational basis \{|0\rangle, |1\rangle\} of a qubit, and — the most beautiful part — what happens when you chain two Stern-Gerlach magnets in a row and ask them consecutive questions about the same atom. The answer to that last experiment is the reason every textbook says "quantum measurements do not commute," and it is the experimental ancestor of every qubit measurement you will ever do.

The apparatus

Three components, in a row:

1. An oven that heats silver metal to about 1000 °C until it is emitting a thermal beam of neutral silver atoms. A slit at the oven's mouth collimates the beam — only atoms heading forward in a narrow direction survive.
2. A non-uniform magnet with a wedge-shaped north pole sitting above a flat south pole. Between them, the magnetic field \vec B points roughly downward (toward the flat face), but — critically — its strength varies with height. Near the wedge the field is stronger; near the flat pole the field is weaker. There is a gradient \partial B_z / \partial z, and that gradient is what does the work.
3. A detector plate at the end of the flight path. Silver deposits a visible stain wherever atoms land.

The physics of the deflection is classical and simple. A magnetic moment \vec\mu sitting in a non-uniform magnetic field feels a force \vec F = \nabla(\vec\mu \cdot \vec B). If only the z-component of the gradient matters (the design is deliberate about this), the force reduces to F_z = \mu_z \,(\partial B_z / \partial z). An atom whose magnetic moment along z is \mu_z = +\mu feels an upward force; an atom with \mu_z = -\mu feels a downward force; an atom with \mu_z somewhere between feels a proportional force and lands somewhere in the middle.

That is the classical prediction. Keep it in your head, because the experimental result is about to contradict it.

What classical physics says — and what you actually see

A silver atom has 47 electrons. 46 of them form a neutral, non-magnetic inner core; the 47th is a single unpaired outer electron whose spin gives the atom its magnetic moment. (Silver was not a random choice — Stern and Gerlach picked it precisely because its magnetic moment comes from one lone electron.)

Classically, an atom's magnetic moment is a tiny bar magnet. When the beam leaves the oven, each atom's bar magnet is pointing in some arbitrary direction — the oven is hot and the orientations are random. Once the atom enters the non-uniform field, its \mu_z is the projection of its moment onto the z axis, which can take any value from -\mu to +\mu depending on orientation. So classically you expect every value of \mu_z to be present in the beam, and the detector plate should show a smooth, continuous vertical smear of deflections.

Stern and Gerlach saw two dots. Only two dots. Every silver atom emerged from the magnet either maximally deflected upward or maximally deflected downward — no atoms in between, ever, no matter how long they ran the oven. The internal magnetic moment of every silver atom, when its z-component was measured, turned out to be either +\mu_B or -\mu_B where \mu_B is the Bohr magneton. It was quantised — restricted to two discrete values.

Why this is so violent a shock: classical physics has no mechanism for this. A bar magnet pointed at 30° above the horizontal has a perfectly well-defined projection onto the vertical axis — call it \mu \cos(60°) = 0.5\mu. The experiment is saying that projection does not exist; every measurement of the vertical component returns either +\mu or -\mu, as if no intermediate orientation is permitted. This is the headline fact of the quantum world, delivered directly into a glass plate. Every later quantum-mechanical reconstruction of what the atom is "really doing" has to reproduce this two-dot outcome.

The reason, which it took another three years of physics to articulate (Pauli, Goudsmit and Uhlenbeck, Schrödinger, Heisenberg), is that the outer electron of the silver atom carries an intrinsic angular momentum called spin, and spin is a strictly two-valued quantum observable along any axis. Measure spin along the z axis and you get +\hbar/2 (spin up) or -\hbar/2 (spin down) — never anything else, never a partial value. The magnetic moment is proportional to the spin, so the two possible \mu_z values map directly onto the two dots on the plate.

From spin-z to the qubit basis

Now translate this into quantum-computing language.

The "spin up along z" state is a genuine state of the silver atom's outer electron — a unit vector in a two-dimensional complex Hilbert space, the "spin Hilbert space" of a spin-1/2 particle. Call it |{\uparrow}_z\rangle. The "spin down along z" state is a different unit vector, orthogonal to the first; call it |{\downarrow}_z\rangle. These two states are the two eigenstates of the observable S_z (spin along z), with eigenvalues +\hbar/2 and -\hbar/2 respectively:

Any state of the electron's spin can be written as a complex superposition of these two:

This should look familiar. It is the state of a qubit. You match the two languages letter by letter:

Why the identification is honest and not just an analogy: the electron spin's Hilbert space is a two-dimensional complex vector space with a preferred orthonormal basis, and every quantum-mechanical prediction you make about it is computed using exactly the formalism you have been using for qubits — Dirac notation, unit vectors, orthogonality, the Born rule. A single electron spin-1/2 is a qubit. Stern-Gerlach is not a "classical analogue" of qubit measurement; it is the literal experimental realisation of a projective measurement in the computational basis on a naturally-occurring qubit.

The computational basis \{|0\rangle, |1\rangle\} you have been writing down since chapter 1 is the spin-z basis of a spin-1/2 particle. The Stern-Gerlach magnet is a computational-basis measurement apparatus: it takes whatever state the atom is in, projects it onto |0\rangle or |1\rangle, and the probability of each outcome is what the Born rule predicts.

You could equally well build a Stern-Gerlach magnet with its gradient along the x axis (rotate the whole apparatus by 90°). The atoms would then land on two dots — but these are the eigenstates of S_x, which in the qubit language are |+\rangle and |-\rangle. Rotate to y, and the two dots are |+i\rangle and |-i\rangle. The axis you pick is the basis you measure in. This is the entire content of §7 of the projective-measurement article, made real with magnets and silver atoms.

Probabilities: what fraction of atoms hits which dot?

Given a beam of identically prepared silver atoms, each in the spin state |\psi\rangle = \alpha|0\rangle + \beta|1\rangle, what fraction lands on the upper dot and what fraction on the lower?

The Born rule answers this directly. The probability that any one atom is deflected upward (outcome spin-up-along-z, i.e., outcome 0 in qubit language) is

and similarly p(\downarrow_z) = |\beta|^2. Run N atoms through the apparatus and you expect about N|\alpha|^2 on the upper dot and N|\beta|^2 on the lower, with the usual \sqrt{N} statistical fluctuations.

If the beam is completely unpolarised — atoms emerge from the oven in random spin directions, a classical thermal mixture — then |\alpha|^2 = |\beta|^2 = 1/2 for every measurement direction and the two dots receive roughly equal counts. That is what Stern and Gerlach actually saw.

If instead you first prepare all atoms in |{\uparrow}_z\rangle (by, say, running them through a previous z-Stern-Gerlach and keeping only the upper beam), every atom will land on the upper dot in the next z-measurement with probability 1. This is the deterministic-outcome-on-an-eigenstate fact from the measurement chapter, now in a form you can see with your eyes on a piece of glass.

Sequential Stern-Gerlach — the best demonstration of non-commutation

Now for the experiment that gives Stern-Gerlach its philosophical punch. Imagine three Stern-Gerlach magnets arranged in a row. You will use them to ask a silver atom consecutive questions about its spin along different axes, and see what happens when the axes do not match.

The setup

Magnet 1 (oriented along z): the atom enters in a random state; the magnet splits the beam in two. Block the lower beam. Now you have atoms prepared in |{\uparrow}_z\rangle = |0\rangle.

Magnet 2 (oriented along x): take the upper beam from magnet 1 and feed it into a magnet oriented along the x axis. This magnet splits the beam in two — into |{\uparrow}_x\rangle = |+\rangle and |{\downarrow}_x\rangle = |-\rangle. Block the lower one (atoms with spin down along x). Now you have atoms in |+\rangle.

Magnet 3 (oriented along z again): take the |+\rangle beam from magnet 2 and feed it into a magnet oriented along z again. What do you expect to see at the detector?

The naive classical prediction

You might reason like this. Magnet 1 selected atoms that were spin-up along z. Magnet 2 then asked "what is the spin along x?" and we kept only those that came out spin-up along x. But those atoms were already spin-up along z — that fact has not changed. So when magnet 3 measures along z again, we should see every single atom emerge in the spin-up beam. A 100/0 split.

This is wrong. The experiment — done with real silver atoms, verifiable today in any undergraduate labotatory — gives a 50/50 split at magnet 3.

The quantum calculation

Apply the formalism step by step. After magnet 1, the state is |0\rangle. Magnet 2 is a measurement in the x basis, with projectors P_+ = |+\rangle\langle +| and P_- = |-\rangle\langle -|. Compute the probability of outcome + given input |0\rangle:

Why \langle + | 0\rangle = 1/\sqrt{2}: by definition |+\rangle = (|0\rangle + |1\rangle)/\sqrt{2}, so \langle + | = (\langle 0| + \langle 1|)/\sqrt{2}, and \langle + | 0\rangle = 1/\sqrt{2}. Half the atoms make it past magnet 2 into the + beam.

If outcome + occurred, the post-measurement state is |+\rangle (not |0\rangle any more). That is the collapse step. The atom that comes out of magnet 2 is no longer described by "spin up along z" — it is described by "spin up along x."

Now magnet 3 measures along z again on the state |+\rangle. Compute the probabilities:

The state |+\rangle = (|0\rangle + |1\rangle)/\sqrt{2} is an equal superposition in the z basis, so a z-measurement gives 0 or 1 with equal probability. Magnet 3 sees a 50/50 split, exactly as observed.

Why the earlier "fact that the atom is spin-up-along-z" has vanished: magnet 2 projected the state onto |+\rangle, which means whatever component of |0\rangle that was "in the x-measurement-compatible direction" survived, and the rest was discarded. After the projection, the state is |+\rangle — and |+\rangle contains equal amplitudes on |0\rangle and |1\rangle. The "spin up along z" information was never hiding somewhere inside |+\rangle; it was destroyed the instant magnet 2 performed its measurement.

The operators, in matrix form

The three Pauli matrices \sigma_x, \sigma_y, \sigma_z are the matrix representations of the spin observables S_x, S_y, S_z (up to the factor \hbar/2):

where

Why \sigma_z is diagonal: its eigenvectors are |0\rangle = \binom{1}{0} (eigenvalue +1) and |1\rangle = \binom{0}{1} (eigenvalue -1) — the computational basis is built to diagonalise \sigma_z. A Stern-Gerlach magnet aligned along z is implementing exactly the operator \sigma_z as a projective measurement.

Reading \sigma_z. The matrix has +1 at position (0,0) and -1 at position (1,1) — those are the two eigenvalues, with eigenstates |0\rangle and |1\rangle. A Stern-Gerlach measurement along z returns one of these two eigenvalues, with probabilities given by the Born rule.

The fact that \sigma_z and \sigma_x do not commute —

— is the algebraic expression of "measurements along different axes do not agree." Commuting observables can be simultaneously diagonalised and simultaneously measured; non-commuting ones cannot. The z\to x\to z cascade is the experimental face of this algebraic fact, and the Heisenberg uncertainty relation for spin components is its quantitative refinement.

Common confusions

• "Is spin really angular momentum?" Yes. Spin is a genuine angular momentum — it adds to orbital angular momentum when computing the total, it obeys the same commutation relations [S_x, S_y] = i\hbar S_z, it couples to magnetic fields the way other angular momentum does. The only thing it is not is angular momentum from the particle literally spinning like a top. The electron is point-like at every length scale we can probe; there is no tiny ball rotating. Spin is an intrinsic quantum degree of freedom that mathematically acts like angular momentum but has no classical rotational picture. The magnetic moment, the Zeeman splitting, the Stern-Gerlach deflection are all observable consequences.

• "Why does measuring S_x reset S_z?" Because a measurement of S_x projects the state onto an S_x eigenstate, and S_x eigenstates are superpositions of S_z eigenstates. Once the state has been projected onto |+\rangle, it is no longer an S_z eigenstate; it is equally |0\rangle and |1\rangle. The S_z-value is undefined — not "still +1 but hidden," not "now -1," but no value at all until you measure it. The next S_z measurement will give a random outcome weighted by the Born rule.

• "Is the dot position on the detector random?" Yes, at the level of the individual atom. You cannot predict which dot a given atom will hit — only the long-run statistics (p = |\alpha|^2 or |\beta|^2). This randomness is fundamental in the Born-rule sense: no hidden information about the atom would let you predict the outcome. Bell's theorem, refined by decades of experiments (Aspect 1982, Hensen et al. 2015), has ruled out every natural class of "hidden-variable" theories that tried to make individual outcomes secretly deterministic.

• "Could I use a Stern-Gerlach magnet to read an unknown spin state completely?" No. One measurement gives you one bit — up or down along one axis. The amplitudes \alpha and \beta are two complex numbers (minus the norm and global phase constraints, so two real parameters). You need measurements in multiple different bases on many copies of the state (quantum state tomography) to pin both down. A single Stern-Gerlach gives a single bit. This is the projective-measurement story, made experimental.

• "Silver has 47 electrons, but there are only two dots. Where did the other 46 electrons go?" They contribute nothing to the spin-1/2 observable. The 46 inner-shell electrons pair up into filled shells with zero net spin and zero magnetic moment. Only the 47th outer-shell electron has an unpaired spin, and only that electron's spin couples to the magnetic field gradient. Silver is the ideal atom for this experiment because it isolates a single-electron spin inside a neutral atomic carrier.

• "Is the Stern-Gerlach effect unique to silver?" No. Any atom with an unpaired electron spin shows the same two-dot pattern. Sodium works. Hydrogen works. Modern experiments use neutral potassium, rubidium, and caesium — all alkali metals with a single outer-shell electron. The 1922 choice of silver was dictated by oven engineering, not by physics.

Going deeper

The g-factor and the electron's magnetic moment

The classical formula for the magnetic moment of a current loop is \mu = IA (current times area). Applied to an electron orbiting a nucleus, this gives the magnetic moment as proportional to its orbital angular momentum: \mu_{\text{orbital}} = -(e/2m_e)\,L. By analogy, you might guess that the electron's spin magnetic moment is \mu_{\text{spin}} = -(e/2m_e)\,S. This guess is wrong by a factor of 2.

The actual relationship is \mu_{\text{spin}} = -g_s(e/2m_e)\,S where g_s \approx 2 is the electron's spin g-factor. Dirac's 1928 relativistic theory of the electron predicted g_s = 2 exactly. Modern QED (Schwinger's 1948 calculation and all refinements since) gives a tiny correction: g_s = 2.00231930436\ldots, and the agreement between this calculated value and the measured one is the most precise test of any physical theory ever performed (twelve significant figures). The Stern-Gerlach deflection magnitude in the 1922 experiment is set by g_s, and the numerical agreement was one of the early pieces of evidence for spin as a genuine quantum degree of freedom.

Stern-Gerlach as the inspiration for all of quantum measurement theory

The sequential Stern-Gerlach experiment is, historically, the first experiment to make "non-commuting observables" visible. Heisenberg's 1927 paper on the uncertainty principle cited the Stern-Gerlach setup as the thought-experiment motivation. Bohr's "complementarity" discussions of the 1930s used Stern-Gerlach as the paradigm of a measurement that forces a choice between incompatible properties. Every later treatment of projective measurement — von Neumann's 1932 axiomatisation, the measurement-problem literature, the Copenhagen-vs-Everett debate — is, in some sense, thinking through the consequences of what Stern and Gerlach saw on that glass plate.

India's connection to this era is substantial. C.V. Raman was doing his own landmark experiments on inelastic light scattering in Calcutta in the 1920s, published as the Raman effect in 1928 (Nobel Prize 1930). Meghnad Saha had already written his 1920 ionisation equation, applying quantum statistical mechanics to stellar atmospheres — one of the first large-scale uses of quantum quantisation outside the laboratory. Satyendra Nath Bose's 1924 paper, championed by Einstein, gave the statistics for identical bosons (including the photon). The same decade that produced the Stern-Gerlach two-dot pattern produced, from Indian physicists working with modest resources, three of the foundational applications of the quantum idea. The cross-pollination between German and Indian quantum physics in the interwar period is one of the quiet great stories of 20th-century science.

Commutators and the uncertainty principle for spin

The non-commutation of \sigma_x and \sigma_z has a precise quantitative consequence. For any two observables A and B, the Robertson-Heisenberg inequality states

where \Delta A is the standard deviation of A-outcomes on the state and \langle [A,B]\rangle is the expectation value of the commutator. For spin along x and z on a state where \langle \sigma_y\rangle \neq 0, this gives a strict lower bound on the product \Delta S_x \cdot \Delta S_z — you cannot prepare a state with arbitrarily well-defined values of both. Eigenstates of \sigma_z (like |0\rangle) have \Delta S_z = 0 and \Delta S_x = \hbar/2 (maximal); eigenstates of \sigma_x (like |+\rangle) have \Delta S_x = 0 and \Delta S_z = \hbar/2. There is no state where both are smaller than \hbar/2 at once. The sequential Stern-Gerlach experiment is exactly this bound in experimental form: the moment you sharpen S_x to a definite value by measuring it, S_z has to spread.

Contextuality — a preview

A stronger statement than "non-commutation" is contextuality: the outcome of a measurement can depend on what other measurements you choose to perform (or not perform) alongside it, even on the same shot. The Kochen-Specker theorem (1967) proves that no "non-contextual hidden-variable" theory can reproduce quantum predictions in Hilbert spaces of dimension \geq 3. For spin-1/2 specifically, the original Stern-Gerlach result is compatible with many hidden-variable stories — but once you enlarge to spin-1 systems, or bring in multiple qubits, contextuality becomes unavoidable. This is the conceptual heir of Stern-Gerlach, expanded into a full no-go theorem.

Modern variants

The original 1922 experiment used a hot thermal beam of silver atoms. Today's versions look nothing like it on the outside but share the same essence:

• Neutron Stern-Gerlach. Thermal neutrons have spin 1/2 and a small magnetic moment; they can be split into two beams by a strong field gradient.
• Optical Stern-Gerlach. Instead of a magnetic gradient on a magnetic moment, use a light-intensity gradient on an atom's electric polarisability. Modern cold-atom experiments can split clouds of rubidium or sodium into two "dots" this way, with exquisite resolution.
• Molecular Stern-Gerlach. The same principle works on polar molecules — their electric-dipole moments quantise in strong electric-field gradients.
• Solid-state analogues. The Quantum Hall effect can be read as a kind of Stern-Gerlach for electrons in a 2D gas; so can the spin-Hall effect.
• Ground-state atom interferometers. Splitting a cold atomic cloud into coherent "up" and "down" arms, then recombining, is how the most accurate atomic clocks and inertial sensors work. The apparatus is a direct descendant of Stern and Gerlach.

Most importantly for quantum computing: the projective measurement of a superconducting qubit, or a trapped-ion qubit, or a silicon spin qubit, is a miniaturised, engineered Stern-Gerlach. The principle is the same: couple the qubit to a measurement apparatus in the computational basis, read out one of two outcomes, and the post-measurement state is the corresponding basis state. The 1922 silver beam is the ancestor of every qubit readout pulse fired on every quantum computer today.

Where this leads next

• The Bloch sphere — the geometric picture that makes "spin along any axis" a point on a unit sphere, and a Stern-Gerlach measurement a projection onto an axis.
• Projective measurement — the general formalism of which Stern-Gerlach is the first physical realisation.
• Pauli X, Y, Z — the algebraic structure of the three spin observables, the commutation relations, and the canonical single-qubit gates that come from them.
• Measurement in arbitrary bases — how to implement an x-basis or y-basis measurement on hardware that only reads the z basis, via the software rotation trick.
• Unitary evolution — the other half of postulate 2; everything that is not measurement.

References

1. Wikipedia, Stern-Gerlach experiment — the original 1922 paper, historical context, and modern variants.
2. John Preskill, Lecture Notes on Quantum Computation, Ch. 2 (spin, measurement, and the Stern-Gerlach apparatus) — theory.caltech.edu/~preskill/ph229.
3. Nielsen and Chuang, Quantum Computation and Quantum Information (2010), §2.2.3 on projective measurement and §1.5 on historical context — Cambridge University Press.
4. Wikipedia, Spin (physics) — spin as an intrinsic angular momentum, the g-factor, and its role in quantum mechanics.
5. Friedrich and Herschbach, Stern and Gerlach: How a Bad Cigar Helped Reorient Atomic Physics (2003) — Physics Today 56(12), 53. The origin story in 1922 Frankfurt, including why the experiment was done at all.
6. Wikipedia, Satyendra Nath Bose — Bose's 1924 paper on photon statistics and the Indian side of the 1920s quantum revolution [6].