Spin-½ as a Two-Level System

A spin-½ particle — an electron, a proton, a neutron, or a silver atom's valence electron — carries an intrinsic angular momentum whose measured component along any axis takes only two values: $+\tfrac{\hbar}{2}$ or $-\tfrac{\hbar}{2}$ . There is no third outcome. This is the experimental heart of the Stern–Gerlach experiment, and it is precisely what makes spin-½ the physical archetype of a qubit.

Two states, two basis vectors

Because a measurement along a chosen axis (conventionally $z$ ) yields exactly two results, the quantum state lives in a two-dimensional complex Hilbert space. We label the two eigenstates of the $z$ -component of spin

S_z\,|{\uparrow}\rangle = +\tfrac{\hbar}{2}\,|{\uparrow}\rangle, \qquad S_z\,|{\downarrow}\rangle = -\tfrac{\hbar}{2}\,|{\downarrow}\rangle,

and we identify them with the computational basis of a qubit:

|{\uparrow}\rangle \equiv |0\rangle, \qquad |{\downarrow}\rangle \equiv |1\rangle.

The most general spin-½ state is a normalized complex superposition

|\psi\rangle = \alpha\,|0\rangle + \beta\,|1\rangle, \qquad |\alpha|^2 + |\beta|^2 = 1.

This is exactly the qubit state vector. Spin-½ and the qubit are the same mathematical object: a unit vector in $\mathbb{C}^2$ .

Spin along other axes

Measuring spin along the $x$ -axis also gives only two outcomes, $\pm\tfrac{\hbar}{2}$ , with eigenstates

|{+x}\rangle = \frac{1}{\sqrt{2}}\bigl(|0\rangle + |1\rangle\bigr), \qquad |{-x}\rangle = \frac{1}{\sqrt{2}}\bigl(|0\rangle - |1\rangle\bigr).

The state $|{+x}\rangle$ is what emerges from a Stern–Gerlach magnet oriented along $x$ when we keep the deflected-upward beam. In qubit language $|{+x}\rangle$ is the familiar $|{+}\rangle$ state, and it is produced by acting with the Hadamard gate on $|0\rangle$ :

H|0\rangle = \frac{1}{\sqrt{2}}\bigl(|0\rangle + |1\rangle\bigr) = |{+x}\rangle.

Why the mapping matters

Identifying spin-½ with a qubit lets us reuse every tool from quantum computing — gates, the Bloch sphere, measurement statistics — to reason about a real physical system, and conversely lets the intuition built on spins (precession, Stern–Gerlach filtering, magnetic resonance) illuminate abstract qubit manipulations. Throughout this module we will keep both pictures in view at once.

Try it

Prepare the spin-up-along- $x$ state $|{+x}\rangle = (|0\rangle + |1\rangle)/\sqrt{2}$ on a single qubit starting from $|0\rangle$ . The Bloch panel should show the arrow pointing along $+x$ after you run it.

Run your code to see the quantum state.