Equation 132 · The Entry a Relabeling Cannot Write

What does this equation mean?

\omega_r = 0

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Result or conditionomega_r

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\omega_r

Symbol omega_r

omeg $a_r$ is part of the quantity the equation computes from the expression on the right.

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=

The expressions on both sides represent the same quantity under the stated assumptions.

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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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What the article says around this equation

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 gyrating magnetic field in 1937 [ 14 ] . Two checks confirm the recovery is exact rather than incidental: at $\omega_r$ = 0 , the non-rotating case, $\Delta_{\rm frame}$ $\equiv$ 0 and $H_c$ = $H_f$ , the already-verified baseline; at $\omega_r$ = $\omega_0$ , exact corotation, $H_c$ = 0 and the spin’s energy in this frame vanishes identically, which is the ordinary resonance condition, not a puzzle, because $\Delta_{\rm frame}$ = - $\hbar$ $\omega_0$ $\langle$ $\sigma_z$ $\rangle$ /2 exactly cancels the lab-frame value for a state that starts, and stays,…

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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 gyrating magnetic field in 1937 [ 14 ] . Two checks confirm the recovery is exact rather than incidental: at $\omega_r$ = 0 , the non-rotating case, $\Delta_{\rm frame}$ $\equiv$ 0 and $H_c$ = $H_f$ , the already-verified baseline; at $\omega_r$ = $\omega_0$ , exact corotation, $H_c$ = 0 and the spin’s energy in this frame vanishes identically, which is the ordinary resonance condition, not a puzzle, because $\Delta_{\rm frame}$ = - $\hbar$ $\omega_0$ $\langle$ $\sigma_z$ $\rangle$ /2 exactly cancels the lab-frame value for a state that starts, and stays, aligned with the field.

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Sources cited in the surrounding passage

[14] Space Quantization in a Gyrating Magnetic Field ↗

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