Skip to content
Quantum Mechanics

Stern–Gerlach Revisited

A Stern–Gerlach analyzer is the laboratory prototype of a spin-component measurement. In the modern spin-1/21/2 idealization, orienting the magnet along a unit vector n^\hat{\mathbf n} corresponds to measuring

Sn^=ℏ2n^⋅σ,S_{\hat{\mathbf n}} = \frac{\hbar}{2}\hat{\mathbf n}\cdot\boldsymbol\sigma,

with two possible spin eigenvalues,

±ℏ2.\pm\frac{\hbar}{2}.

The apparatus turns an internal spin distinction into spatially separated output beams. This page revisits Stern–Gerlach using the spin-state formalism. The original 1922 silver-atom experiment, its historical context, and its experimental caveats live in Stern–Gerlach Experiment.

Stern–Gerlach apparatus splitting a neutral atom beam into two output paths

An inhomogeneous magnetic field correlates magnetic-moment projection with spatial path. The modern idealized analyzer is described by spin-component projectors along the magnet axis.

Magnetic Force and the Ideal Analyzer

Section titled “Magnetic Force and the Ideal Analyzer”

In a region where a bias field fixes an approximately constant quantization axis and transverse spin flips are negligible, write B≃B(z)n^\mathbf B\simeq B(z)\hat{\mathbf n}. From Hint=−γSn^B(z)H_{\rm int}=-\gamma S_{\hat n}B(z), the transverse force on the two spin eigenstates is approximately

Fz(±)=±γℏ2dBdz.F_z^{(\pm)}=\pm\frac{\gamma\hbar}{2}\frac{dB}{dz}.

A uniform field changes spin phases but does not separate a beam. Real magnet fields obey Maxwell’s equations and are not exactly a single-component gradient everywhere; the expression describes the dominant force near the selected beam trajectory. Resolved paths, negligible spin flips, and a calibrated magnetic-moment sign are needed to approximate a projective spin analyzer.

For spin-1/21/2, define the operator

σn^=n^⋅σ.\sigma_{\hat n} = \hat{\mathbf n}\cdot\boldsymbol\sigma.

It has eigenstates

σn^∣+n^⟩=∣+n^⟩,σn^∣−n^⟩=−∣−n^⟩.\sigma_{\hat n}|+\hat{\mathbf n}\rangle = |+\hat{\mathbf n}\rangle, \qquad \sigma_{\hat n}|-\hat{\mathbf n}\rangle = -|-\hat{\mathbf n}\rangle.

Equivalently,

Sn^∣±n^⟩=±ℏ2∣±n^⟩.S_{\hat{\mathbf n}}|\pm\hat{\mathbf n}\rangle = \pm\frac{\hbar}{2}|\pm\hat{\mathbf n}\rangle.

An ideal Stern–Gerlach analyzer oriented along n^\hat{\mathbf n} correlates these eigenstates with different spatial paths:

∣+n^⟩∣in⟩⟶∣+n^⟩∣path+⟩,|+\hat{\mathbf n}\rangle|\mathrm{in}\rangle \longrightarrow |+\hat{\mathbf n}\rangle|\mathrm{path}_+\rangle,

and

∣−n^⟩∣in⟩⟶∣−n^⟩∣path−⟩.|-\hat{\mathbf n}\rangle|\mathrm{in}\rangle \longrightarrow |-\hat{\mathbf n}\rangle|\mathrm{path}_-\rangle.

For a superposition,

∣ψ⟩=α∣+n^⟩+β∣−n^⟩,|\psi\rangle = \alpha|+\hat{\mathbf n}\rangle +\beta|-\hat{\mathbf n}\rangle,

the coherent pre-detection idealization is

∣ψ⟩∣in⟩⟶α∣+n^⟩∣path+⟩+β∣−n^⟩∣path−⟩.|\psi\rangle|\mathrm{in}\rangle \longrightarrow \alpha|+\hat{\mathbf n}\rangle|\mathrm{path}_+\rangle + \beta|-\hat{\mathbf n}\rangle|\mathrm{path}_-\rangle.

If the paths are detected, blocked, or become environmentally distinguishable, the apparatus acts as a measurement or preparation device rather than merely a reversible beam splitter.

Projectors and Probabilities

Section titled “Projectors and Probabilities”

The projectors for an analyzer oriented along n^\hat{\mathbf n} are

P±(n^)=12(I±n^⋅σ).P_\pm^{(\hat n)} = \frac12 \left( I\pm\hat{\mathbf n}\cdot\boldsymbol\sigma \right).

For an incoming pure state ∣ψ⟩|\psi\rangle,

p±=⟨ψ∣P±(n^)∣ψ⟩.p_\pm = \langle\psi|P_\pm^{(\hat n)}|\psi\rangle.

If the incoming state has Bloch vector r\mathbf r, so that

ρ=12(I+r⋅σ),\rho = \frac12 \left( I+\mathbf r\cdot\boldsymbol\sigma \right),

then

p±=Tr⁡(ρP±(n^))=12(1±r⋅n^).p_\pm = \operatorname{Tr} \left( \rho P_\pm^{(\hat n)} \right) = \frac12 \left( 1\pm\mathbf r\cdot\hat{\mathbf n} \right).

For a pure state prepared as ∣+a^⟩|+\hat{\mathbf a}\rangle, where γ\gamma is the angle between a^\hat{\mathbf a} and n^\hat{\mathbf n},

p+=12(1+a^⋅n^)=cos⁡2γ2,p_+ = \frac12(1+\hat{\mathbf a}\cdot\hat{\mathbf n}) = \cos^2\frac{\gamma}{2},

and

p−=sin⁡2γ2.p_- = \sin^2\frac{\gamma}{2}.

This half-angle law is one of the cleanest operational signatures of spin-1/21/2 geometry.

Orientation Is the Observable

Section titled “Orientation Is the Observable”

A Stern–Gerlach magnet does not measure “spin” without qualification. It measures a component selected by the apparatus orientation:

n^⟼Sn^=ℏ2n^⋅σ.\hat{\mathbf n} \quad\longmapsto\quad S_{\hat{\mathbf n}} = \frac{\hbar}{2}\hat{\mathbf n}\cdot\boldsymbol\sigma.

Rotating the magnet changes the observable. A state with a definite SzS_z value is not generally a state with a definite SxS_x value. For example,

∣+z⟩=12(∣+x⟩+∣−x⟩),|+z\rangle = \frac{1}{\sqrt2} \left( |+x\rangle+|-x\rangle \right),

so an xx-oriented analyzer sends an incoming ∣+z⟩|+z\rangle beam into two equal-intensity outputs.

The physical labels “upper” and “lower” are apparatus-calibration labels. They depend on field gradients, magnetic-moment sign conventions, and beam geometry. The formal labels ∣+n^⟩|+\hat{\mathbf n}\rangle and ∣−n^⟩|-\hat{\mathbf n}\rangle refer to eigenstates of the chosen spin component.

Sequential Analyzers

Section titled “Sequential Analyzers”

Sequential Stern–Gerlach analyzers make noncommutativity visible.

Start with a zz analyzer and select the +z+z output:

∣ψ⟩=∣+z⟩.|\psi\rangle=|+z\rangle.

Send that beam into an xx analyzer. Since

∣+z⟩=12(∣+x⟩+∣−x⟩),|+z\rangle = \frac{1}{\sqrt2} \left( |+x\rangle+|-x\rangle \right),

the xx analyzer produces

P(+x)=P(−x)=12.P(+x)=P(-x)=\frac12.

If the +x+x output is selected, the state is updated to ∣+x⟩|+x\rangle. A final zz analyzer then gives

∣+x⟩=12(∣+z⟩+∣−z⟩),|+x\rangle = \frac{1}{\sqrt2} \left( |+z\rangle+|-z\rangle \right),

so

P(+z)=P(−z)=12.P(+z)=P(-z)=\frac12.

The selected xx output no longer has the original certainty for a later zz result. In the quantum model this follows from the state update and the noncommuting observables:

[Sx,Sz]=−iℏSy.[S_x,S_z] = -i\hbar S_y.

Selection Versus Recombination

Section titled “Selection Versus Recombination”

There are two different idealizations that are often conflated.

In a selective measurement, one output beam is kept and events in the other output are excluded. Merely ignoring the outcome while retaining both outputs is a nonselective measurement and does not prepare a pure spin eigenstate. Keeping the +n^+\hat{\mathbf n} output prepares

∣+n^⟩.|+\hat{\mathbf n}\rangle.

In a fully coherent beam-splitting model, the two paths could in principle be recombined before path information is lost. Then the device is closer to a unitary entangler between spin and spatial degrees of freedom than to an irreversible measurement.

Real Stern–Gerlach apparatuses are macroscopic, imperfect, and path-distinguishing. For most textbook spin examples, the projective measurement model is the appropriate abstraction. For foundational discussions, it matters that measurement is not merely a symbol attached to the magnet; it involves spatial separation, amplification, and loss or use of path information.

Relation to the Original Silver Experiment

Section titled “Relation to the Original Silver Experiment”

The original Stern–Gerlach experiment used neutral silver atoms. It did not send a beam of isolated free electrons through an ideal spin analyzer. Its ground electronic state has L=0L=0 and J=S=1/2J=S=1/2: closed electron shells cancel, and the unpaired valence electron carries the dominant magnetic moment. This explains the leading two-beam pattern. Nuclear spin and hyperfine structure require a more detailed model when resolved; the two-state description concerns the dominant electronic response.

This historical fact is why the same experiment can be both:

  • an early experimental landmark for space quantization;
  • a modern pedagogical model for spin-1/21/2 measurement.

Keeping those statements separate prevents two common errors: treating the original experiment as if it already used modern spin language, and treating the modern spinor analyzer as if all atomic-structure details had no role in the actual apparatus.

Common Mistakes

Section titled “Common Mistakes”
  • Saying a Stern–Gerlach device measures “spin” without specifying the axis.
  • Treating the two output paths as proof of pre-existing classical arrows along all axes.
  • Forgetting that selecting an output prepares a new state.
  • Assuming a zz analyzer followed by an xx analyzer should preserve the original zz certainty.
  • Confusing physical beam direction with the sign of the spin eigenvalue; magnetic-moment signs and apparatus geometry matter.
  • Describing the original silver-atom experiment as a direct free-electron spin measurement.
  • Ignoring the distinction between coherent path separation and an actual detected measurement.

Exercises

Section titled “Exercises”
  1. A spin-1/21/2 particle is prepared in ∣+z⟩|+z\rangle. What are the probabilities for an xx-oriented Stern–Gerlach analyzer?
Solution

Use

∣+z⟩=12(∣+x⟩+∣−x⟩).|+z\rangle = \frac{1}{\sqrt2} \left( |+x\rangle+|-x\rangle \right).

Therefore

P(+x)=12,P(−x)=12.P(+x)=\frac12, \qquad P(-x)=\frac12.
  1. Use the projector formula to find the probability that ∣+a^⟩|+\hat{\mathbf a}\rangle gives the +n^+\hat{\mathbf n} output.
Solution

For a pure state ∣+a^⟩|+\hat{\mathbf a}\rangle, the Bloch vector is a^\hat{\mathbf a}. The +n^+\hat{\mathbf n} projector is

P+(n^)=12(I+n^⋅σ).P_+^{(\hat n)} = \frac12 \left( I+\hat{\mathbf n}\cdot\boldsymbol\sigma \right).

Thus

p+=12(1+a^⋅n^).p_+ = \frac12 \left( 1+\hat{\mathbf a}\cdot\hat{\mathbf n} \right).

If γ\gamma is the angle between the axes, a^⋅n^=cos⁡γ\hat{\mathbf a}\cdot\hat{\mathbf n}=\cos\gamma, so

p+=12(1+cos⁡γ)=cos⁡2γ2.p_+ = \frac12(1+\cos\gamma) = \cos^2\frac{\gamma}{2}.
  1. Explain why a zz analyzer, followed by an xx analyzer with +x+x selected, followed by another zz analyzer gives equal final zz probabilities.
Solution

After the +x+x output is selected, the state is ∣+x⟩|+x\rangle. In the zz basis,

∣+x⟩=12(∣+z⟩+∣−z⟩).|+x\rangle = \frac{1}{\sqrt2} \left( |+z\rangle+|-z\rangle \right).

Therefore a final zz analyzer gives

P(+z)=P(−z)=12.P(+z)=P(-z)=\frac12.

The intermediate xx selection prepares a new state rather than revealing a pre-existing definite zz value.

  1. Why is the original Stern–Gerlach experiment not literally a free-electron spin measurement?
Solution

The original experiment used neutral silver atoms. Neutral atoms avoid the dominant Lorentz-force deflection that a charged beam would experience. The effective two-state magnetic degree of freedom in silver is closely tied to the unpaired valence electron, but the actual beam particles were atoms with atomic structure. The modern free spin-1/21/2 analyzer is an idealization distilled from that physical setting.

References

Section titled “References”