In 1922 Stern and Gerlach sent silver atoms, each carrying the magnetic moment of one unpaired electron, through a magnetic field that grows stronger in one direction. A magnetic moment in such a field feels a force proportional to its component along the field. Classically the moments would point every way and the beam would smear into a line. It split into two spots. Along any axis the spin has exactly two values, and sending one of the beams through a second apparatus turned another way shows what measurement does to a state.

A sieve that lets through only two sizes of grain, with nothing in between, would tell you that the grains come in two sizes. The Stern–Gerlach magnet is a sieve for the component of spin, and it finds two sizes.

Silver atoms enter from the left and pass between the poles of a magnet whose field grows stronger upwards. Each is pushed up or down according to its spin component along the field, and the beam lands on the screen, on the right, in two spots. Classically it would have smeared into a line.

A magnetic moment in a field that varies in space feels a force. In a field along $z$ that grows with $z$, the force is proportional to $\mu_{z}$, so the atoms are sorted by their spin component along $z$.

• Two spots for silver: its one unpaired electron has $s=\tfrac12$, and the rest of the atom has no angular momentum.
• A beam that left the $+z$ exit and meets a second $z$ apparatus all leaves by again: the measurement prepared the state .

A magnetic moment in a field has energy $-\vec{\mu}\cdot\mathbf{B}$, and a force is minus the gradient of an energy. Near the axis of the magnet the field points along $z$ and changes with $$, so only the component of the moment feels a force, in proportion to the field’s gradient.

The energy of a magnetic moment in a field.

A force is minus the gradient of the energy; $\vec{\mu}$ does not depend on position.

Near the axis of the magnet the field points along $z$.

Differentiate with respect to $z$.

Two values of $\mu_{z}$ give two forces, equal and opposite: two spots.

Classically the magnetic moments of the atoms would point in every direction, and the beam would spread into a continuous line. For silver it splits into exactly two spots.

A particle of spin $s$ gives $2s+1$ spots in a Stern–Gerlach apparatus, one for each value of $m$.

Atoms that left the upper exit of a $z$ magnet all leave the upper exit of a second $z$ magnet: the first measurement prepared the state spin up along $z$.

Spin up along $z$, measured along $x$, gives $\pm\tfrac12\hbar$ with equal probability.

Great work! You now know how to:

• derive the force on a magnetic moment in a varying field
• read the two spots as spin $\tfrac12$
• predict what chains of Stern–Gerlach magnets do