The Stern–Gerlach Experiment

In 1922, Otto Stern and Walther Gerlach sent a beam of silver atoms through a strongly inhomogeneous magnetic field and collected the deflected atoms on a glass plate. Classical physics predicted a continuous smear; the experiment produced two discrete spots. That result announced something entirely outside classical mechanics: angular momentum is quantized in space.

The classical expectation

A magnetic dipole moment $\boldsymbol{\mu}$ in an inhomogeneous field $\mathbf{B}$ experiences a force

$F_z = \mu_z \frac{\partial B_z}{\partial z},$

where $z$ is the direction of the strong field gradient, transverse to the beam's direction of travel, so the force deflects atoms along $z$ . Classically, a spinning charged object can have any orientation, so $\mu_z$ takes a continuous range of values and the deflected beam should spread into a continuous stripe. That is not what Stern and Gerlach observed.

What actually happened

Two and only two spots appeared on the collector plate, symmetric about the undeflected position. Silver has a single valence electron with zero orbital angular momentum ( $\ell = 0$ ), so the splitting cannot be attributed to orbital motion. The deflection arises from an intrinsic angular momentum — spin — of the electron.

For spin- $\tfrac{1}{2}$ , the spin quantum number is $s = \tfrac{1}{2}$ . The magnetic quantum number $m_s$ is restricted to the values

$m_s \in \left\{-\tfrac{1}{2},\, +\tfrac{1}{2}\right\},$

so there are exactly $2s + 1 = 2$ discrete projections of the spin onto any chosen axis. This restriction is spatial quantization: the spin angular momentum does not point in an arbitrary direction but instead "chooses" one of a finite set of projections.

The spin magnetic moment

The force on each atom is proportional to $\mu_z$ . For an electron the spin magnetic moment is

$\mu_z = -g_s \frac{e}{2m_e} m_s \hbar,$

where $g_s \approx 2$ is the electron $g$ -factor, $e$ is the elementary charge, $m_e$ is the electron mass, and $\hbar$ is the reduced Planck constant. Because $m_s = \pm\tfrac{1}{2}$ , the force takes exactly two values, equal in magnitude and opposite in sign, producing the two spots.

Spatial quantization

Spatial quantization is the general statement: for a particle with spin quantum number $s$ , the component of spin along any axis is restricted to

$m_s = -s,\; -s+1,\; \ldots,\; s-1,\; s,$

giving $2s + 1$ allowed values. For spin- $\tfrac{1}{2}$ this yields two values; for spin- $1$ it would yield three. The Stern–Gerlach experiment confirmed this discreteness directly and irreversibly, ruling out any classical continuous distribution.

Why the result is remarkable

Classical angular momentum is a continuous vector — you can tilt it to any angle. Quantum spin is fundamentally different: no matter which axis you choose to measure, you get only one of a discrete set of outcomes. Moreover, if you prepare an atom in spin-up along $z$ and then measure along $x$ , you again get only two outcomes, each with probability $\tfrac{1}{2}$ . Measurement along a new axis does not reveal a pre-existing value; it creates a new definite outcome. This irreducible randomness is a signature of the quantum world, and the Stern–Gerlach apparatus is the clearest window into it.