Spin-½ and the Stern-Gerlach experiment

Intrinsic angular momentum

Quantum particles can have intrinsic angular momentum called spin. Spin is not literally a tiny ball rotating about its axis. Electrons, for example, appear pointlike in modern experiments, yet they possess intrinsic angular momentum and magnetic moment.

For a spin-1/2 particle, the magnitude of spin angular momentum is determined by

$S^2=s(s+1)hbar^2$

with

s=rac{1}{2}

so

S^2=rac{3}{4}hbar^2

A chosen component, such as $S_z$, has possible values

S_z=pmrac{hbar}{2}

Stern-Gerlach experiment

The Stern-Gerlach experiment sends atoms through a nonuniform magnetic field. A magnetic dipole in a field gradient experiences a force depending on its magnetic moment orientation. Classically, one might expect a continuous smear of deflections because dipoles could have any orientation.

Instead, the beam splits into discrete spots. For spin-1/2 systems, there are two outcomes. This reveals quantization of angular momentum components.

Spin states

For spin along the z-axis, the two basis states are often written

angle$$ and $$|-z angle$$ A general spin-1/2 state can be written as $$|psi angle=alpha|+z angle+eta|-z angle$$ with normalization $$|alpha|^2+|eta|^2=1$$ If $S_z$ is measured, the probability of getting $+hbar/2$ is $|alpha|^2$, and the probability of getting $-hbar/2$ is $|eta|^2$. ## Sequential measurements If a particle is measured to be $|+z angle$ and immediately measured again along z, the result is $+hbar/2$ with certainty. But if it is then measured along x, the result is probabilistic. A spin state definite along z is not definite along x. This shows that spin components along different axes are incompatible observables. ## Spin and magnetic moment Spin is associated with a magnetic moment. For an electron, $$ec{mu}=-grac{e}{2m_e}ec{S}$$ where $gapprox2$ for electron spin. The minus sign appears because the electron has negative charge. The interaction energy in a magnetic field is $$U=-ec{mu}cdotec{B}$$ This coupling explains why magnetic field gradients can separate spin states. ## Two-state systems Spin-1/2 is the simplest nontrivial quantum system: a two-dimensional Hilbert space. It provides a clean setting for superposition, incompatible measurements, basis changes, and quantum information. A quantum bit, or qubit, is mathematically a two-state quantum system: $$|psi angle=alpha|0 angle+eta|1 angle$$ Spin-1/2 particles are physical examples of qubits. ## The big idea Spin is intrinsic quantum angular momentum. The Stern-Gerlach experiment reveals that spin components are quantized, giving two outcomes for spin-1/2 particles. Spin states form a two-dimensional Hilbert space, making them central to both foundational quantum mechanics and modern quantum information.