Spin Operators Sᵢ = (ħ/2)σᵢ

The Pauli matrices are dimensionless. To turn them into physical observables — operators whose eigenvalues carry units of angular momentum — we attach the natural quantum of action $\hbar$ . The spin operators are defined as

S_x = \frac{\hbar}{2}\,\sigma_x, \qquad S_y = \frac{\hbar}{2}\,\sigma_y, \qquad S_z = \frac{\hbar}{2}\,\sigma_z.

The factor of $\tfrac{1}{2}$ is exactly what makes this a spin-½ particle.

Eigenvalues are ±ħ/2

Because $\sigma_z$ has eigenvalues $\pm 1$ , the operator $S_z = \tfrac{\hbar}{2}\sigma_z$ has eigenvalues

S_z\,|0\rangle = +\frac{\hbar}{2}\,|0\rangle, \qquad S_z\,|1\rangle = -\frac{\hbar}{2}\,|1\rangle.

A measurement of the $z$ -component of spin can only return $+\tfrac{\hbar}{2}$ ("spin up") or $-\tfrac{\hbar}{2}$ ("spin down"). The same is true along any axis, because $S_x$ and $S_y$ are likewise $\tfrac{\hbar}{2}$ times an involution.

The angular-momentum commutators

Scaling the Pauli commutators by $(\hbar/2)^2$ and using $[\sigma_j,\sigma_k] = 2i\varepsilon_{jkl}\sigma_l$ gives the canonical angular-momentum algebra

[S_x, S_y] = i\hbar\,S_z, \qquad [S_y, S_z] = i\hbar\,S_x, \qquad [S_z, S_x] = i\hbar\,S_y.

This is the same algebra obeyed by orbital angular momentum $\mathbf{L} = \mathbf{r}\times\mathbf{p}$ . Spin is genuine angular momentum, even though no particle is literally spinning.

The total spin and its magnitude

The squared total spin operator is

S^2 = S_x^2 + S_y^2 + S_z^2 = 3\left(\frac{\hbar}{2}\right)^2 I = \frac{3}{4}\hbar^2\,I,

using $\sigma_k^2 = I$ . This matches the general angular-momentum formula $S^2 = s(s+1)\hbar^2$ with $s = \tfrac{1}{2}$ :

s(s+1)\hbar^2 = \tfrac{1}{2}\cdot\tfrac{3}{2}\,\hbar^2 = \tfrac{3}{4}\hbar^2.

Every spin-½ state is an eigenstate of $S^2$ with the same eigenvalue $\tfrac{3}{4}\hbar^2$ — the magnitude of the spin is fixed, only its direction is quantum.

Try it

Working in units where $\hbar = 1$ , compute the spin-up eigenvalue of $S_z$ (the larger of the two). It should equal $\tfrac{\hbar}{2} = 0.5$ .

Run your code to see the quantum state.