7.6 Spin states and hyperfine

Source: Ideas from Computational Physics (Landau, Páez, Bordeianu), §7.6. Rewritten in original wording. Continues 7.5b String-problem labs. Labs: 7.6b Hyperfine labs.

Pauli matrices and a contact interaction

Spin-$\tfrac12$ operators are packaged as the Pauli matrices $\boldsymbol\sigma=(\sigma_x,\sigma_y,\sigma_z)$. For an electron–proton pair, a contact (Fermi) hyperfine interaction can be written $V=W\,\boldsymbol\sigma_e\!\cdot\!\boldsymbol\sigma_p$. Expand in the product basis $|\alpha\alpha\rangle,|\alpha\beta\rangle,|\beta\alpha\rangle,|\beta\beta\rangle$, read the spectrum, and form the splitting — pick each equation among lookalikes.

Labeling note. Some textbook passages swap which multiplet sits where. The algebra of this matrix is unambiguous: the triplet is the threefold $+W$ level and the singlet is $-3W$. In hydrogen that gap is the 21 cm / 1420 MHz line once $W$ is restored to laboratory units — labs often keep $W=1$ so $\Delta E/W=4$.

Why this is a matrix chapter

Building $V$, extracting the middle block's closed-form eigenvalues, and sorting the full spectrum are linear-algebra drills with a physics payoff: degeneracy counting and a measurable frequency. You do not need NumPy — a $2\times 2$ characteristic polynomial and a sorted list of four numbers suffice.

Quiz

Spin states and hyperfine

1 / 4

In the product basis used here, the interaction matrix V has…

Pick an answer to see if you are right.

Next: 7.6b Hyperfine labs.