Quantum Stern-Gerlach Experiment

The apparatus of the last lesson was built to measure a continuous quantity and it returned exactly two answers — the same two answers, no matter which way you turned it. This lesson takes that result seriously enough to follow it through five cascaded experiments, and comes out the other side holding a two-dimensional complex vector space and the Born rule.

Learning Objectives

After this lesson you will be able to:

1. State what the 1922 plate actually showed, and identify precisely which classical assumption from the previous lesson each feature of the result violates.
2. Use the Stern-Gerlach analyzer abstraction — one input, two labelled outputs — to specify and reason about cascaded measurements.
3. Predict the outcome of the five canonical cascade experiments: repeated, reversed, perpendicular, three-stage, and rotated by an arbitrary angle.
4. Represent the atom's magnetic state as a vector in $\mathbb{C}^2$ and write down $\lvert{\pm z}\rangle$, $\lvert{\pm x}\rangle$, and $\lvert{+\hat{n}(\theta)}\rangle$.
5. Apply the Born rule to reproduce the empirical law $P(+,\theta) = \cos^2(\theta/2)$, and explain why the angle is halved.
6. Explain what it means for a measurement to prepare a state, and why the results cannot be explained by atoms that arrive with a pre-existing orientation.

Intuition

Run the apparatus of Lesson 1 and you should get a filled band: uniform intensity from one edge to the other, densest nowhere, populated everywhere. What Gerlach saw in February 1922 was two spots — one deflected up, one deflected down, by equal amounts, with the entire middle of the band empty.

Take a moment to appreciate how strange this is, because familiarity dulls it. The classical band is not merely wrong in detail; the observed pattern is its exact photographic negative in the region that matters. The only atoms that arrive are the ones at the two extreme values $\mu_z = \pm\mu$, which classically are the rarest and least special — they require the moment to point exactly along or exactly against the field, a set of measure zero on the sphere. Every atom in the beam behaves as though it had been pre-aligned, and pre-aligned both ways at once.

Now the sentence that turns a surprise into a crisis. The splitting is the same for every orientation of the magnet. Turn the apparatus on its side and you get two spots left and right, same separation. Tilt it 30° and you get two spots along that tilt, same separation. If atoms left the oven with fixed little arrows, some axis would have to be perpendicular to those arrows, and measuring along it would give zero deflection — one undeflected spot. That never happens. There is no arrangement of classical arrows, however cleverly chosen, that yields two equal and opposite projections along every axis, and that impossibility is what Freericks (after Styer) calls the conundrum of projections [Fre, §1.2]. Lesson 4 makes the argument airtight; this lesson gathers the experimental facts that make it necessary.

Theory

What the experiment showed, and what it did not prove

Stern and Gerlach sent a collimated beam of silver atoms through the strongest field gradient they could build, and collected the atoms on a glass plate. The beam speed was around 660 m/s, the magnet about 5 cm long, the expected splitting a tenth of a millimetre — small enough that alignment defeated them many times before the run that worked.

The immediate interpretation was that the result confirmed space quantization, the Bohr-Sommerfeld doctrine that an atom's orbit can only take discrete orientations relative to a field. It was celebrated as a triumph of the old quantum theory, and this is one of the great ironies of physics: the theory was wrong, the atom's two-valuedness has nothing to do with orbital quantization, and the correct explanation — the electron's spin — was not proposed until 1925 and not applied to this experiment until 1927. The experiment was right; the reason given for it was not.

That the old explanation fails is worth seeing concretely, because it rules out an entire class of patches. Orbital angular momentum with quantum number $\ell$ gives $2\ell + 1$ orientations, and $2\ell + 1$ is always odd — one, three, five — never two. Worse, silver's 47 electrons form closed shells plus a single valence electron in an $\ell = 0$ state, so silver's orbital moment is zero and the beam should not split at all. Phipps and Taylor closed the argument in 1927 by running the experiment on ground-state hydrogen, which has one electron and unambiguously zero orbital angular momentum, and still saw two spots.

The Stern-Gerlach analyzer

Since every orientation gives exactly two outcomes, we can stop drawing magnets and start drawing boxes. A Stern-Gerlach analyzer is an idealized device with one input port and two output ports, labelled $+$ and $-$, characterized entirely by the axis $\hat{n}$ its field defines: an atom entering leaves by exactly one of the two ports, and the port it leaves by is what we mean by "the value of the atom's magnetic projection along $\hat{n}$." Internally there are guide tubes that restore the two deflected beams to parallel travel; externally it is a black box that sorts.

Fix a coordinate system once and use it for the rest of this course: the beam travels along $y$, vertical is $z$, horizontal-transverse is $x$. An analyzer oriented along $\hat{z}$ we call SG $z$; along $\hat{x}$, SG $x$; and one rotated by $\theta$ from vertical in the $x$–$z$ plane, SG $\theta$.

Because the analyzer's output ports are spatially separated beams, we can chain them: feed one output port into the input of the next analyzer, and block the other. Five such cascades exhaust what there is to learn [Fre, §1.2].

Experiment 1 — repeatability

flowchart LR
    SRC["Oven<br/>(unpolarized)"] --> A["SG z"]
    A -->|"plus"| B["SG z"]
    A -->|"minus"| BLK["blocked"]
    B -->|"plus"| D1["100% of input"]
    B -->|"minus"| D2["0%"]

Take the $+$ output of an SG $z$ analyzer and send it into a second SG $z$ analyzer. Every atom leaves by the second analyzer's $+$ port; the $-$ port stays dark. The same holds with $-$ in place of $+$ throughout.

This is the single most important result in the set, and it is the reassuring one: measurement is repeatable. It means the first analyzer did not merely reveal a random number — it left the atom in a condition that the second analyzer reliably confirms. We say the first analyzer has performed state preparation: whatever "having a positive projection along $z$" means, an atom emerging from the $+$ port has it, definitely, and will keep having it until something else is done to it. Without this fact there would be no such thing as a quantum state.

Experiment 2 — reversing the analyzer

Now take the $-$ output of SG $z$ and feed it into an analyzer whose field points along $-\hat{z}$. Everything comes out that analyzer's $+$ port. This is bookkeeping rather than physics: reversing the axis exchanges the meaning of the labels, so an atom with a negative projection on $\hat{z}$ has a positive projection on $-\hat{z}$. Results along a reversed axis are just as definite as along the original one. Parallel and antiparallel analyzers are the two cases where the outcome is certain.

Experiment 3 — perpendicular analyzers

flowchart LR
    SRC["Oven<br/>5000 atoms"] --> A["SG z"]
    A -->|"plus (2500)"| B["SG x"]
    A -->|"minus (2500)"| BLK["blocked"]
    B -->|"plus"| D1["about 1250<br/>(25% of source)"]
    B -->|"minus"| D2["about 1250<br/>(25% of source)"]

Feed the $+$ output of SG $z$ into SG $x$. The atoms split 50–50. Each individual atom's choice is unpredictable; the proportions are utterly reliable.

This is the first appearance of genuine quantum randomness, and it is worth being precise about what kind of randomness it is. Every atom entering the second analyzer is in an identically prepared, completely known condition — they all came out the same port of the same analyzer. There is no hidden spread of initial conditions left to blame, as there was in the classical experiment where the randomness came entirely from the oven's isotropic source. The randomness here is not ignorance about which atom we have; it is irreducible. And "random" here does not mean "unpredictable": we cannot say where the next atom goes, but we can predict the frequencies to arbitrary precision by running long enough. Quantum mechanics is a theory of exactly this kind of prediction.

Experiment 4 — three analyzers, and the memory test

Chain three: SG $z$, then SG $x$ on the $+$ port, then SG $z$ again on the $+$ port.

flowchart LR
    A["SG z"] -->|"plus"| B["SG x"]
    B -->|"plus"| C["SG z"]
    C -->|"plus"| D1["50%"]
    C -->|"minus"| D2["50%"]

Two plausible lines of reasoning give opposite answers, and it is worth having argued both before seeing the result.

Reasoning A. The second analyzer prepared the atom along $x$. The third asks about $z$, which is perpendicular to $x$, so by Experiment 3 we should get 50–50.

Reasoning B. The first analyzer certified the atom as $+z$. The second certified it as $+x$. So entering the third, the atom is both $+z$ and $+x$; the third analyzer should confirm $+z$ with certainty.

The answer is A: a 50–50 split. Half the atoms emerge with a negative projection along $z$ — even though every one of them was certified $+z$ two stages earlier, and nothing in between touched the $z$ axis except the act of measuring $x$. Reasoning B fails because its premise is false: the atom is never "both $+z$ and $+x$". Measuring $x$ destroyed the $z$ information outright.

Confirm the diagnosis by making the third analyzer SG $x$ instead: then 100% emerge from its $+$ port, exactly as Experiment 1 would say. So the atom does have a perfectly definite $x$ projection — it simply no longer has a $z$ one. Freericks summarizes this memorably: atoms remember only the last axis they were measured on, and nothing before it [Fre, §1.2].

Bookkeeping for the whole chain: of the original beam, half is blocked at stage one, half of the remainder at stage two, and the survivors split evenly at stage three, so each final port receives $\tfrac12\times\tfrac12\times\tfrac12 = \tfrac18$ of the source — or $\tfrac14$ of the atoms that entered the second analyzer.

Experiment 5 — the $\cos^2(\theta/2)$ law

Finally, keep the first analyzer vertical and rotate the second one by an angle $\theta$, recording the fraction that emerges from its $+$ port. Three points are already known: $\theta = 0°$ gives 1 (Experiment 1), $\theta = 90°$ gives $\tfrac12$ (Experiment 3), $\theta = 180°$ gives 0 (Experiment 2). The measured curve through them is not the straight line joining them. At $45°$, about 85 atoms in 100 emerge from the $+$ port, not 75. The empirical law is

with $\theta$ the angle between the two analyzers, and the two probabilities summing to 1 as they must. Check it: $\cos^2(22.5°) = 0.854$, so 85 out of 100 is right on the curve. Check the special cases: $\cos^2 0 = 1$, $\cos^2 45° = \tfrac12$, $\cos^2 90° = 0$. Those three are worth memorizing; the rest is a calculator.

Note the half angle. It is not a typo, and it is not the classical $\cos\theta$ projection law of Lesson 1 in disguise. We are about to see where it comes from.

Two outcomes force a two-dimensional space

Let us build the smallest mathematical object that reproduces all five experiments.

Along any axis there are exactly two outcomes, and Experiment 1 says each of them is a condition an atom can definitely be in. Assign a state to each: $\lvert{+z}\rangle$ and $\lvert{-z}\rangle$ for the two exits of SG $z$. Experiment 2 says these are mutually exclusive — an atom certified $+z$ never comes out the $-$ port — so we take them orthogonal and normalized:

Any state of the atom's magnetic degree of freedom is a complex superposition $\alpha\lvert{+z}\rangle + \beta\lvert{-z}\rangle$ with $|\alpha|^2 + |\beta|^2 = 1$: the state space is $\mathbb{C}^2$. Probabilities come from the Born rule — the probability of finding outcome $\lvert{\phi}\rangle$ when the atom is in state $\lvert{\psi}\rangle$ is the squared modulus of their overlap:

Now fit the data. The states of the $x$ analyzer must give $\tfrac12$ against both $z$ states (Experiment 3) while being orthogonal to each other (Experiment 1 applied to SG $x$). The choice

does both: $|\langle{\pm x}|{+z}\rangle|^2 = |1/\sqrt2|^2 = \tfrac12$ and $\langle{+x}|{-x}\rangle = \tfrac12(1 - 1) = 0$. And it reproduces Experiment 4 with no further input, since $|\langle{\pm z}|{+x}\rangle|^2 = \tfrac12$ — the $z$ information really is gone, as a matter of arithmetic rather than of decree.

For a general angle $\theta$ in the $x$–$z$ plane, the state selected by SG $\theta$'s $+$ port is

whose overlap with $\lvert{+z}\rangle$ is $\cos(\theta/2)$ — giving exactly the measured law $P = \cos^2(\theta/2)$. At $\theta = 90°$ it reduces to $\lvert{+x}\rangle$; at $\theta = 180°$ it becomes $\lvert{-z}\rangle$, as Experiment 2 demands. Five experiments, one two-component vector, no free parameters left over.

Why the angle is halved

The half-angle is the signature of a spin-$\tfrac12$ system, and it encodes a real geometric fact: rotating the apparatus by $\theta$ rotates the state by $\theta/2$. Turn the analyzer through a full $360°$ and it is manifestly the same apparatus, but the state has only been taken half way around:

The state comes back with a minus sign. No measurement detects it — the Born rule squares moduli, so a global phase is invisible — but it means the correspondence between rotations of space and transformations of the state is two-to-one. This is the group $SU(2)$ double-covering $SO(3)$, and it is why the natural picture of a qubit is a sphere on which physical angles appear halved (1.2.2 The Bloch Sphere). The relative phase between $\lvert{+z}\rangle$ and $\lvert{-z}\rangle$, by contrast, is not invisible at all: it is what distinguishes $\lvert{+x}\rangle$ from $\lvert{-x}\rangle$, and Lesson 3 will make it do work.

What is actually precessing

Lesson 1 argued that the apparatus only works for objects carrying angular momentum, so the atom's moment must come with one. It is not orbital — silver and hydrogen both give two spots with $\ell = 0$. It is spin: an intrinsic angular momentum of magnitude $S_z = \pm\hbar/2$, with the associated moment $\boldsymbol{\mu} = -g_s\,(e/2m_e)\,\mathbf{S}$ and $g_s \approx 2$, so that each spot deflects as if carrying one full Bohr magneton — quantitatively what was measured.

Resist the picture of a spinning ball. Demanding that a sphere of the classical electron radius carry $\hbar/2$ requires an equatorial speed of order $100c$, and no experiment has ever resolved any spatial extent for the electron at all. Spin is a two-valued degree of freedom with the algebra of angular momentum and no mechanical picture underneath — one of several places where the quantum world supplies structure that classical language cannot narrate. The state space we just built, $\mathbb{C}^2$ with an orthonormal basis and the Born rule, is the qubit (1.2.1 The Qubit); rename $\lvert{+z}\rangle \to \lvert{0}\rangle$ and $\lvert{-z}\rangle \to \lvert{1}\rangle$ and every subsequent term of this program is about systems like the one on Gerlach's plate.

Hands-on (Python)

Simulate the cascades and recover the $\cos^2(\theta/2)$ curve from shot noise.

import numpy as np

rng = np.random.default_rng(7)

def state(theta):
    """+ output state of an analyzer rotated by theta (radians) from vertical."""
    return np.array([np.cos(theta / 2), np.sin(theta / 2)], dtype=complex)

def analyze(psi, theta, rng):
    """One analyzer: return (+1 or -1, collapsed state) via the Born rule."""
    plus, minus = state(theta), state(theta + np.pi)
    p_plus = abs(np.vdot(plus, psi)) ** 2
    if rng.random() < p_plus:
        return +1, plus
    return -1, minus

# --- Experiment 4: SG z (+) -> SG x (+) -> SG z ---------------------------
N, n_pass_x, n_final_plus = 100_000, 0, 0
for _ in range(N):
    s, psi = analyze(state(0.0), np.pi / 2, rng)        # z-prepared into SG x
    if s > 0:
        n_pass_x += 1
        s2, _ = analyze(psi, 0.0, rng)                  # back into SG z
        n_final_plus += (s2 > 0)

print(f"P(+x | +z)      = {n_pass_x / N:.3f}")            # ~0.500
print(f"P(+z | +x)      = {n_final_plus / n_pass_x:.3f}") # ~0.500  <- z memory gone
print(f"P(+x then +z)   = {n_final_plus / N:.3f}")        # ~0.250

# --- Experiment 5: rotate the second analyzer ----------------------------
print("\n theta   measured   cos^2(theta/2)")
for deg in (0, 45, 90, 135, 180):
    th = np.deg2rad(deg)
    hits = sum(analyze(state(0.0), th, rng)[0] > 0 for _ in range(2000))
    print(f"  {deg:3d}°    {hits / 2000:.3f}      {np.cos(th / 2) ** 2:.3f}")

Run it and watch Reasoning B die: the final $z$ measurement in Experiment 4 comes out 50–50 every time, no matter how definite the atom's $z$ projection was two stages earlier.

Exercises

E1 (easy). Atoms leaving the $+$ port of an SG $z$ analyzer enter an analyzer oriented along $-\hat{z}$. Which port do they leave by, and with what probability? Reconcile your answer with the $\cos^2(\theta/2)$ law.

E2 (easy). Atoms prepared $+z$ pass through an analyzer at $60°$. What fraction emerges from each port? If 4000 atoms enter, how many are expected at each?

E3 (medium). Take the two analyzers of the [Fre, §1.3.1] arrangement: the first oriented at $+45°$ from vertical, the second at $315° = -45°$. Atoms from the $+$ port of the first enter the second. Compute both output probabilities, and verify by explicit inner product with the state vectors.

E4 (medium). In Experiment 4, replace the middle SG $x$ analyzer by one at angle $\theta$, keeping the outer two vertical. What fraction of the atoms entering the middle analyzer emerges from the final $+$ port (summing over both middle ports would be a different experiment — here the $-$ port of the middle analyzer is blocked)? For which $\theta$ is it largest, and what happens at $\theta \to 0$?

E5 (hard). A run of 100 shots at $\theta = 45°$ yields 85 atoms in the $+$ port. (a) Is this consistent with $P = \cos^2(\theta/2)$? Quantify. (b) A skeptic proposes instead the linear law $P_{\text{lin}} = 1 - \theta/180°$, which agrees at $0°$, $90°$, and $180°$. How many shots at $45°$ are needed to distinguish the two laws at the $3\sigma$ level?

Checkpoint

1. What are the two features of the observed pattern that classical physics cannot reproduce — and which of them requires more than one orientation of the apparatus to notice?
2. Why does orbital angular momentum fail to explain two spots, and what did the 1927 hydrogen experiment add to the argument?
3. State the outcome of each of the five cascade experiments in one sentence apiece.
4. Write $\lvert{\pm x}\rangle$ in the $z$ basis and verify that they are orthonormal and give 50–50 against $\lvert{+z}\rangle$.
5. Where does the factor of $\tfrac12$ in $\cos^2(\theta/2)$ come from, and what happens to the state when the analyzer is rotated by a full turn?

Further Reading

• [Fre] J. K. Freericks, Quantum Mechanics Done Right, §1.2 — the five cascade experiments in the framing this lesson follows, including the analyzer-as-black-box device.
• [Sak] Sakurai & Napolitano, §1.1 — sequential Stern-Gerlach experiments used as the definition of quantum mechanics; the canonical treatment.
• [NC] Nielsen & Chuang, §1.5.1 — the same experiment told from the quantum-information side, as the prototype qubit.
• [ER] Eisberg & Resnick, Ch. 8 — the experimental history, including Phipps and Taylor.
• 1.2.1 The Qubit and 1.2.2 The Bloch Sphere — where the $\mathbb{C}^2$ built here becomes the program's central object.
• P.6.3 Magnetic Moments, Stern–Gerlach & Spin — the same cascades reached from the angular-momentum ladder.

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