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Equation 102 · Part 3 · The Entry a Relabeling Cannot Write

=

ω/2π=5 GHz\omega/2\pi = 5\ \mathrm{GHz}
=

What this part means

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

Its job in the formula

The equals sign connects the complete expression on the left with the complete expression on the right. Both sides must have compatible units.

The passage around this formula

Putting numbers to the swing keeps it honest rather than decorative. Take ω\omega/2π\pi = 5\ GHz\mathrm{GHz} and g/2π\pi = 10\ MHz\mathrm{MHz} , both squarely inside the range of qubit and resonator frequencies and coupling strengths reported across circuit quantum electrodynamics architectures, which span a few gigahertz in frequency and from a fraction of a megahertz up to several hundred megahertz in coupling depending on the regime [ 12 ] . The swap time is t∗t^\ast = π\pi/(2g) = 1/(4×\times10\ MHz\mathrm{MHz}) = 25.0\ ns\mathrm{ns} , and the ledger’s swing at that instant is ℏ\hbarω\omega/2 ≈\approx 1.66×\times10^{-24}\ J\mathrm{J} ≈\approx 10.3\ μ\mueV\mathrm{eV} — an exact analytic evaluation under the stated…

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Learn the underlying idea

An equals sign says that the expression on its left and the expression on its right have the same value under the stated definitions and assumptions.

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Sources cited in the surrounding passage

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