Equation 128 · The Entry a Relabeling Cannot Write

What does this equation mean?

H_f

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H_f

Symbol H_f

$H_f$ is a part of this expression. Its role is fixed by the surrounding article and by the operations shown in the formula.

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subscript

The lower label selects a particular version, component, or indexed member of the quantity. For example, x₀ and xₜ can be values at different positions.

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How to interpret it

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What the article says around this equation

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

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Sources cited in the article section

[14] Space Quantization in a Gyrating Magnetic Field ↗

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