What is Spin one-half? — Physics

Definition 12.2 University Physics — Year 3 · Chapter 12 — Spin and Two-Level Systems

The electron (and the proton, the neutron, the quarks) carries an intrinsic angular momentum with $j = \tfrac12$ : a two-dimensional state space spanned by $\ket\uparrow, \ket\downarrow$ (eigenstates of $\hat S_z$ with $\pm\hbar/2$ ), on which

\hat{\vect S} = \frac\hbar2\,\hat{\vect\sigma} , \qquad \sigma_x = \begin{pmatrix}0&1\\1&0\end{pmatrix} ,\; \sigma_y = \begin{pmatrix}0&-\iu\\\iu&0\end{pmatrix} ,\; \sigma_z = \begin{pmatrix}1&0\\0&-1\end{pmatrix}

— the Pauli matrices, realising the angular momentum algebra in its smallest possible home. A general state $\alpha\ket\uparrow + \beta\ket\downarrow$ is a spinor. The electron’s magnetic moment is

\hat{\vect\mu} = -g\,\frac{\mu_{\text{B}}}{\hbar}\,\hat{\vect S} , \qquad g \approx 2 :

twice the orbital rate per unit angular momentum — the anomaly Einstein–de Haas had measured (Exercise 10.12) and the Dirac equation would later predict. Spin is not a rotation of anything: no radius, no “spinning ball” survives scrutiny — it is intrinsic, like charge.

Examples

Example 12.3 (Chained Stern–Gerlach filters)

Select the $S_z = +\hbar/2$ beam, and measure $S_z$ again: all atoms answer $+$ . Measure $S_x$ instead: half and half — the state $\ket\uparrow$ is the superposition $(\ket{+x} + \ket{-x})/\sqrt2$ . Now keep the $S_x = +$ beam and measure $S_z$ once more: half and half again — the $S_x$ measurement erased the previously sharp $S_z$ . Three magnets suffice to exhibit incompatibility, collapse and Born’s rule; this chain is Example 8.7 performed with atoms.

Example 12.1 (Stern–Gerlach)

A magnetic moment $\vect\mu$ in an inhomogeneous field feels the force $F_z = \mu_z\,\partial B_z/\partial z$ : the deflection measures $\mu_z$ . Classical expectation: moments oriented at random, a continuous fan. Quantum expectation for orbital momenta: $2l + 1$ spots: one, three, five — always odd. Observed for silver (and for hydrogen): two spots, symmetric, nothing between. The measured component takes exactly two values — the signature of $j = \tfrac12$ , forbidden to orbits, and the direct display of quantisation: the apparatus is a measuring device for one spin component, and the beam splits into its two eigenvalues.

Example 12.7 (Hyperfine structure and the 21 cm line)

In hydrogen’s ground state the electron’s and proton’s spins interact through their magnetic moments: the four spin states split into a triplet and a singlet separated by only $\Delta E = 5.9\,\text{µ}\mathrm{eV}$ — the hyperfine splitting, $\nu = 1420\,\mathrm{MHz}$ , $\lambda = 21\,\mathrm{cm}$ . The transition is absurdly slow (one flip per ten million years), but the Galaxy holds $10^{66}$ hydrogen atoms: the 21 cm line is bright enough to have mapped the spiral arms, the warp of the disc, and — through its Doppler shifts — the flat rotation curves that argue for dark matter. The same physics, in caesium’s ground state, splits levels by exactly $9\,192\,631\,770\,\mathrm{Hz}$ : since 1967, the definition of the second is a hyperfine spin flip counted out (Exercise 12.12).

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