Equation 130 · Part 7 · The Entry a Relabeling Cannot Write

subscript

\dot U U^\dagger = i\omega_r\sigma_z/2

subscript

What this part means

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.

Its job in the formula

A subscript distinguishes a version, component, step, or member of a quantity. It does not automatically mean multiplication.

Full expressionsubscriptArticle meaning

The passage around this formula

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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Learn the underlying idea

A subscript is a label attached below a symbol. It often selects a time step, component, category, or member of a sequence.

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Sources cited in the article section

[14] Space Quantization in a Gyrating Magnetic Field ↗

These citations provide research context; check each source for the exact claim it supports.