Stern-Gerlach Experiment

Prerequisites: Spin. Full path (19 pages)

Best explanations

Intuitive Spin and the Stern-Gerlach Experiment
Concrete Feynman Lectures on Physics, Vol. 3

Intuitive
Concrete
Abstract
Why is it interesting?
History

Intuitive

In 1922 Otto Stern and Walther Gerlach sent a beam of silver atoms through a magnet and looked at where the atoms landed on a glass plate. A silver atom behaves like a tiny bar magnet. The magnet in the experiment was shaped so that its field was stronger near one pole than near the other. In such a non-uniform field a small bar magnet is pushed up or down, depending on how it is tilted: one pointing "up" is pushed one way, one pointing "down" is pushed the other way, and one lying sideways is not pushed at all.

The atoms leave a hot oven with their little magnets pointing in random directions. Classically we therefore expect every possible amount of deflection between the two extremes, and the beam should spread into one continuous smear on the plate. Instead, Stern and Gerlach found two separate spots and nothing in between.

So the magnetic moment of the atom, measured along the direction of the field gradient, can only take two values. This is a direct sign that angular momentum in quantum mechanics is quantized in direction as well as in size. We now know that the magnetic moment of a silver atom comes from the spin of a single outer electron, and spin one-half allows exactly two outcomes.

The apparatus is also the simplest picture of a quantum measurement. If we take only the atoms from the upper spot and send them through a second magnet with the same orientation, all of them go up again. If the second magnet is turned by 90 degrees, they split half and half. And if we then send one of those halves through a magnet in the original orientation, it splits into two again. The second measurement erased the result of the first one.

Spin and the Stern-Gerlach Experiment by The Physics Mill

Concrete

A particle with magnetic moment $\vec\mu$ in a magnetic field $\vec B(\vec r)$ has potential energy $U = -\vec\mu\cdot\vec B$ and feels the force

$\vec F = \nabla(\vec\mu\cdot\vec B).$

In a homogeneous field the force vanishes and the moment only precesses. To get a deflection the field must be inhomogeneous. In the Stern-Gerlach magnet the field is large along $z$ , $\vec B \approx B_z(z)\,\hat z$ , with a strong gradient $\partial B_z/\partial z$ . Then

$F_z \approx \mu_z \frac{\partial B_z}{\partial z}.$

(Since $\nabla\cdot\vec B = 0$ , a gradient in $B_z$ along $z$ comes with gradients of the transverse components. Their effect averages out because the moment precesses rapidly about the large field along $z$ , so only $\mu_z$ matters.)

For silver the total orbital angular momentum of the electrons is zero and the magnetic moment is that of the single $5s$ electron,

$\vec\mu = -g_s \frac{\mu_B}{\hbar}\vec S, \qquad \mu_B = \frac{e\hbar}{2m_e}, \qquad g_s \approx 2.$

The spin component $S_z$ has the two eigenvalues $\pm\hbar/2$ , so $\mu_z \approx \mp\mu_B$ and the force is $F_z \approx \mp\mu_B\,\partial B_z/\partial z$ . An atom of mass $M$ and speed $v$ that spends the time $l/v$ in a magnet of length $l$ is displaced by

$\Delta z = \frac{1}{2}\frac{F_z}{M}\left(\frac{l}{v}\right)^2 = \mp\frac{\mu_B}{2M}\frac{\partial B_z}{\partial z}\frac{l^2}{v^2}$

at the exit of the magnet, plus further separation during the flight to the detector. Classically $\mu_z$ would vary continuously between $-|\vec\mu|$ and $+|\vec\mu|$ and we would see a continuous band. Two spots mean two eigenvalues.

Sequential measurements. Write $|{+}z\rangle$ , $|{-}z\rangle$ for the eigenstates of $S_z$ . The eigenstates of the spin component along a unit vector $\hat n$ with polar angle $\theta$ and azimuth $\phi$ are

$|{+}\hat n\rangle = \cos\frac{\theta}{2}\,|{+}z\rangle + e^{i\phi}\sin\frac{\theta}{2}\,|{-}z\rangle, \qquad |{-}\hat n\rangle = \sin\frac{\theta}{2}\,|{+}z\rangle - e^{i\phi}\cos\frac{\theta}{2}\,|{-}z\rangle.$

An atom selected in $|{+}z\rangle$ and sent into a magnet oriented along $\hat n$ exits in the upper beam with probability $|\langle{+}\hat n|{+}z\rangle|^2 = \cos^2(\theta/2)$ . For $\hat n = \hat x$ ( $\theta = \pi/2$ ) this is $1/2$ . An atom selected as $|{+}x\rangle = (|{+}z\rangle + |{-}z\rangle)/\sqrt{2}$ then gives $\pm\hbar/2$ along $z$ with probability $1/2$ each, although it came from the $|{+}z\rangle$ beam. This is because $S_x$ and $S_z$ do not commute, $[S_z, S_x] = i\hbar S_y$ , and have no common eigenstates.

The best description can be found in the Feynman Lectures Vol. 3
No-Nonsense Quantum Mechanics by Jakob Schwichtenberg, section "Spin Measurements"

Abstract

The internal state of a spin- $\tfrac{1}{2}$ particle lives in $\mathcal{H}_{\text{spin}} = \mathbb{C}^2$ , which carries the fundamental representation of $SU(2)$ , the double cover of the rotation group $SO(3)$ . The spin operators are $\vec S = \frac{\hbar}{2}\vec\sigma$ with the Pauli matrices $\sigma_i$ , satisfying $[S_i, S_j] = i\hbar\,\epsilon_{ijk}S_k$ . A spin- $j$ particle carries the $(2j+1)$ -dimensional irreducible representation, and a Stern-Gerlach magnet splits a beam of such particles into $2j+1$ components.

Pure spin states are rays in $\mathbb{C}^2$ , that is, points of $\mathbb{CP}^1 \cong S^2$ (the Bloch sphere). The general state is a density matrix

$\rho = \frac{1}{2}\left(\mathbb{1} + \vec r\cdot\vec\sigma\right), \qquad |\vec r| \le 1,$

with $|\vec r| = 1$ for pure states. A Stern-Gerlach magnet oriented along $\hat n$ realises the projection-valued measurement

$P_\pm(\hat n) = \frac{1}{2}\left(\mathbb{1} \pm \hat n\cdot\vec\sigma\right), \qquad \text{Prob}(\pm) = \mathrm{Tr}\left(\rho\, P_\pm(\hat n)\right) = \frac{1}{2}\left(1 \pm \hat n\cdot\vec r\right).$

The unpolarised beam from the oven is $\rho = \frac{1}{2}\mathbb{1}$ ( $\vec r = 0$ ), which gives $1/2$ for every orientation.

The device itself is a measurement in the sense of von Neumann: a unitary interaction that correlates the observable with a pointer variable, here the centre-of-mass position. The Hamiltonian is $H = \frac{\vec p^{\,2}}{2M} - \vec\mu\cdot\vec B(\vec r)$ acting on $L^2(\mathbb{R}^3)\otimes\mathbb{C}^2$ . Keeping only $B_z(z) \approx B_0 + b\,z$ with $b = \partial B_z/\partial z$ , the coupling is $-\vec\mu\cdot\vec B \approx g_s\frac{\mu_B}{\hbar}(B_0 + b\,z)\, S_z$ , which is diagonal in $S_z$ and acts on each component as a linear potential. The evolution therefore takes the form

$\left(\alpha\,|{+}z\rangle + \beta\,|{-}z\rangle\right)\otimes|\psi_0\rangle \;\longmapsto\; \alpha\,|{+}z\rangle\otimes|\psi_+\rangle + \beta\,|{-}z\rangle\otimes|\psi_-\rangle,$

where $|\psi_\pm\rangle$ are wave packets whose mean momenta along $z$ differ by $2\mu_B b\,\tau$ after a time $\tau$ in the field (for $g_s = 2$ ). Once $\langle\psi_+|\psi_-\rangle \approx 0$ , the position of the atom on the screen reveals $S_z$ with the Born probabilities $|\alpha|^2$ and $|\beta|^2$ . Before detection the process is reversible in principle: recombining the two beams coherently, with phases under control, restores the original spin state. Sequential Stern-Gerlach devices along non-commuting directions thus realise the non-commutativity of the projections $P_\pm(\hat n)$ , $P_\pm(\hat m)$ for $\hat n \neq \pm\hat m$ .

Why is it interesting?

The Stern-Gerlach experiment (1922) showed that the magnetic moment of silver atoms in a magnetic field takes only discrete orientations. The result was later understood as a consequence of an unusual internal property of the electron called spin.

History

Stern and Gerlach: How a bad cigar helped reorient atomic physics