Recall, from Chapter 29, that an orbiting electron generates a microscopic magnetic moment $\vec{\mu }$. (In this chapter we will use $\vec{\mu }$ for the magnetic dipole moment rather than the $\vec{m}$ of Chapter 29 to avoid confusion with quantum numbers $m$ and $m_s$.) FIGURE 41.9 reminds you that a magnetic moment, like a compass needle, has north and south poles. Consequently, a magnetic moment in an external magnetic field experiences forces and torques. In the early 1920s, the German physicists Otto Stern and Walter Gerlach developed a technique to measure the magnetic moments of atoms. Their apparatus, shown in FIGURE 41.10, prepares an atomic beam by evaporating atoms out of a hole in an “oven.” These atoms, traveling in a vacuum, pass through a nonuniform magnetic field. Reducing the size of the upper pole tip makes the field stronger toward the top of the magnet, weaker toward the bottom.

FIGURE 41.9 An orbiting electron generates a magnetic moment.

FIGURE 41.10 The Stern-Gerlach experiment.

A magnetic moment experiences a net force in the nonuniform magnetic field because the field exerts forces of different strengths on the moment’s north and south poles. If we define a $z$-axis to point upward, then an atom whose magnetic moment vector $\vec{\mu }$ is tilted upward $({\mu }_z>0)$ has an upward force on its north pole that is larger than the downward force on its south pole. As the figure shows, this atom is deflected upward as it passes through the magnet. A downward-tilted magnetic moment $({\mu }_z<0)$ experiences a net downward force and is deflected downward. A magnetic moment perpendicular to the field $({\mu }_z=0)$ feels no net force and passes through the magnet without deflection. In other words, an atom’s deflection as it passes through the magnet is proportional to ${\mu }_z$, the $z$-component of its magnetic moment.

It’s not hard to show that an atom’s magnetic moment is proportional to the electron’s orbital angular momentum: $\vec{\mu }\propto \vec{L}$. Because the deflection of an atom depends on ${\mu }_z$, measuring the deflections in a nonuniform field provides information about the $L_z$ values of the atoms. The measurements are made by allowing the atoms to stick on a collector plate at the end of the apparatus. After the experiment, the collector plate is removed and examined to learn how the atoms were deflected.

With the magnet off, the atoms pass through without deflection and land along a narrow line at the center, as shown in FIGURE 41.11a. If the orbiting electrons are classical particles, they should have a continuous range of angular momenta. Turning on the magnet should produce a continuous range of vertical deflections, and the distribution of atoms collected on the plate should look like FIGURE 41.11b. But if angular momentum is quantized, as Bohr had suggested several years earlier, the atoms should be deflected to discrete positions on the collector plate.

FIGURE 41.11 Distribution of the atoms on the collector plate.

For example, an atom with $l=1$ has three distinct values of $L_z$ corresponding to quantum numbers $m=-1\text{,}0$, and 1. This leads to a prediction of the three distinct groups of atoms shown in FIGURE 41.11c. There should always be an odd number of groups because there are $2l+1$ values of $L_z$.