$$H_c(t) = H_f + i\hbar\Big(i\frac{\omega_r}{2}\sigma_z\Big) = \frac{\hbar(\omega_0 - \omega_r)}{2}\sigma_z$$

Equation 131 · Evolutionary Physics

The Entry a Relabeling Cannot Write

This equation states an equality: the expressions on both sides have the same value under the article’s assumptions.

A single case makes the term concrete and ties it to a result nearly a century old. Let $H_f$ = ($\hbar$$\omega_0$/2)$\sigma_z$ , a spin precessing about a static field at its Larmor frequency, and let U(t) = $\exp(i\omega_r t\,\sigma_z/2)$ describe a frame rotating about the same axis at rate $\omega_r$ . Since U(t) is built entirely from $\sigma_z$ , it commutes with $H_f$ , so U$H_f$$U^\dagger$ = $H_f$ exactly and the bare conjugate is unchanged by the relabeling. But $\dot U$ $U^\dagger$ = i$\omega_r$$\sigma_z$/2 exactly, giving $H_c(t) = H_f + i\hbar\Big(i\frac{\omega_r}{2}\sigma_z\Big) = \frac{\hbar(\omega_0 - \omega_r)}{2}\sigma_z$. the detuned Hamiltonian that governs spin precession in a frame rotating with an applied field — the identical construction I. I. Rabi used to analyze spin behavior in a…

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