7.3.15. Stern-Gerlach Experiment

Table of Contents

Revealing Space Quantization

The Stern-Gerlach experiment is one of the most famous experiments in quantum physics because it showed that angular momentum does not behave like a continuously orientable classical vector when measured. Instead, certain components can take only specific discrete values. This result gave direct evidence for quantization and later became central to the idea of spin.

In the experiment, a beam of atoms is sent through a nonuniform magnetic field. If magnetic moments inside the atoms could point in any direction continuously, the beam would spread into a continuous band on the detector. Instead, under the right conditions, the beam splits into distinct parts. This means the measured magnetic property has only certain allowed outcomes.

Basic Setup

A typical Stern-Gerlach apparatus has an oven that produces neutral atoms, a collimator that narrows the beam, a region with a strongly nonuniform magnetic field, and a screen or detector where the atoms land. Neutral atoms are used so that electric forces do not dominate the motion, while the atoms can still carry a magnetic moment.

Rendering scientific sketch…

Basic Stern-Gerlach apparatus

Why a Magnetic Field Splits the Beam

An atom with magnetic moment $\boldsymbol{\mu}$ in a magnetic field $\mathbf{B}$ has potential energy

$$
U = -\boldsymbol{\mu}\cdot\mathbf{B}.
$$

If the magnetic field is uniform, the atom may experience a torque, but there is no net translational force. To separate the beam, the field must vary from place to place. Then the atom feels a force related to how the energy changes with position.

If the magnetic field varies mainly along the $z$ direction, the force is approximately

$$
F_z = \mu_z \frac{dB_z}{dz}.
$$

So the vertical force depends on the component $\mu_z$ of the magnetic moment. Different values of $\mu_z$ produce different deflections.

A Stern-Gerlach beam splitter requires a nonuniform magnetic field.
If the field is uniform, there is no spatial separation of different magnetic moment states.

Classical Expectation and Quantum Result

Classically, one might imagine atomic magnetic moments pointing in arbitrary directions. Then $\mu_z$ could have any value in a continuous range, and the detector would show a smeared vertical distribution.

Quantum mechanics predicts something very different. For a system with angular momentum, a component such as $L_z$ or spin component $S_z$ can take only discrete values. Therefore the magnetic moment component along the measurement axis is also quantized, and the beam separates into distinct spots.

This is called space quantization. It does not mean the atom can point only in a few directions in ordinary space at all times. It means that when the component along the chosen axis is measured, only specific values can be obtained.

The Silver Atom Case

The historic experiment used silver atoms. Silver is especially useful because its outermost electron determines the relevant magnetic behavior. In the simplest description, the observed splitting is due to the electron spin magnetic moment.

For a spin-$\frac{1}{2}$ system, the measured spin component along the chosen axis can take only two values:

$$
S_z = +\frac{\hbar}{2}, \qquad S_z = -\frac{\hbar}{2}.
$$

Since the magnetic moment is tied to spin, the beam splits into two parts, one deflected upward and one downward.

For a spin-$\frac{1}{2}$ particle, a measurement of spin along any chosen axis gives only two possible outcomes:
$$
+\frac{\hbar}{2} \quad \text{or} \quad -\frac{\hbar}{2}.
$$

Meaning of the Measurement

The Stern-Gerlach apparatus measures the component of angular momentum, or magnetic moment, along the direction defined by the magnetic field gradient. If the apparatus is aligned vertically, it measures the $z$ component. If it is rotated, it measures the component along a different axis.

This is important because quantum states can be prepared and analyzed by passing particles through successive Stern-Gerlach devices. For example, if a beam is first separated into spin-up and spin-down along $z$, and then the spin-up beam is sent into another apparatus also measuring along $z$, it remains spin-up and does not split again. But if that same beam is sent into an apparatus measuring along $x$, it splits into two parts again.

This shows that spin components along different axes are not simultaneously definite in the same way.

Sequential Stern-Gerlach Experiments

A sequence of Stern-Gerlach devices reveals a uniquely quantum behavior. Suppose the first apparatus selects atoms in the state spin-up along $z$, written as $|+z\rangle$. If these atoms are then passed through a second apparatus measuring along $x$, the beam splits into two equal parts, corresponding to $|+x\rangle$ and $|-x\rangle$.

If one of those beams, say $|+x\rangle$, is then sent into a third apparatus measuring along $z$, it again splits into both $|+z\rangle$ and $|-z\rangle$. The earlier information about definite $z$ spin has been lost after the $x$ measurement.

Rendering scientific sketch…

Sequential Stern-Gerlach measurement

Relation to Quantum States

The Stern-Gerlach experiment gives a concrete way to think about quantum states. A spin state is not just a hidden classical direction. Instead, it is a quantum state that produces certain probabilities for measurement outcomes along different axes.

For spin-$\frac{1}{2}$ particles, the standard basis states for measurement along the $z$ axis are

$$
|+z\rangle, \qquad |-z\rangle.
$$

A state prepared as $|+x\rangle$ is a superposition of the $z$ basis states. In simple form,

$$
|+x\rangle = \frac{1}{\sqrt{2}}\left(|+z\rangle + |-z\rangle\right).
$$

This is why a measurement along $z$ after preparation along $x$ gives two possible outcomes.

What the Experiment Proved

The Stern-Gerlach experiment provided strong evidence for several key quantum ideas.

Idea	What the experiment shows
Quantization	Measurement outcomes are discrete, not continuous
Measurement axis matters	Different orientations measure different components
State preparation	A selected output beam is a well-defined quantum state
Incompatibility of components	Measuring along one axis changes predictions along another axis

It also played an important role in establishing the concept of intrinsic spin, which has no exact classical analog.

The Stern-Gerlach experiment demonstrates that angular momentum components are quantized and that measurement outcomes depend on the chosen axis.
A particle can be definite in one spin component, such as $S_z$, without being definite in another, such as $S_x$.

Limits of the Classical Picture

It is tempting to imagine the atom as a tiny spinning charged ball pointing either up or down. This picture is not correct. Quantum spin is an intrinsic property, not simply literal mechanical spinning in the classical sense. The Stern-Gerlach experiment forces us to move beyond classical intuition because the results of sequential measurements cannot be explained by a hidden continuous orientation alone.

Summary Formula

The essential physical mechanism of the apparatus is the force on a magnetic moment in a field gradient:

$$
F_z = \mu_z \frac{dB_z}{dz}.
$$

Because $\mu_z$ can take only discrete allowed values, the beam separates into discrete spots rather than forming a continuous band.

Core result of the Stern-Gerlach experiment:
A nonuniform magnetic field converts quantized magnetic moment components into spatially separated beams.

Historical Importance

The experiment was first performed by Otto Stern and Walther Gerlach in 1922. It became one of the earliest direct demonstrations of quantum discreteness in microscopic systems. Today it remains a standard example for understanding measurement, spin, and the probabilistic structure of quantum mechanics.

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