Introduction: A New Kind of Angular Momentum
In 1922, Stern and Gerlach discovered that silver atoms have an intrinsic angular momentum unrelated to their orbital motion. This spin angular momentum is purely quantum with no classical analog.
The Experiment
- Beam of silver atoms passes through inhomogeneous magnetic field
- Magnetic moment experiences force: \(F_z = \mu \frac{\partial B}{\partial z}\)
- Classically: continuous distribution of deflections expected
- Actually observed: beam splits into exactly two spots!
The Surprising Result
Two spots mean only two possible values of \(\mu_z\), hence two values of \(S_z\):
\[S_z = +\frac{\hbar}{2} \quad \text{(spin up)} \quad \text{or} \quad S_z = -\frac{\hbar}{2} \quad \text{(spin down)}\]This can't come from orbital angular momentum (\(l\) integer gives odd number of spots).
Sequential Measurements
Repeating with different orientations reveals quantum superposition:
- Z-up beam through X-oriented field → splits again!
- \(|+_z\rangle\) is a superposition of \(|+_x\rangle\) and \(|-_x\rangle\)
The Quantum Connection
Spin is the simplest quantum system: just two states. It perfectly demonstrates superposition, measurement, and non-commutativity. The Stern-Gerlach experiment is the prototypical quantum measurement, now used to introduce quantum mechanics to every physics student.