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Equation 93 · Part 5 · The Clock That Comes Back Wrong by Exactly Its Mass

superscript

ℏkeff2T/m\hbar k_{\rm eff}^2T/m
superscript

What this part means

A raised number can be a power. When it is a label or bound, it selects a case or the upper limit of a sum; the formula’s structure distinguishes these uses.

Its job in the formula

A raised mark can be a power or an index. Its position and the surrounding notation determine which.

The passage around this formula

This is an exact analytic evaluation of the formula derived above for stated, idealized parameters — not a simulation, not a measurement, and not the output of any code run for this article. It is offered as illustrative, and it carries a built-in consistency check: because vrv_r=ℏ\hbar keffk_{\rm eff}/m by definition of a recoil kick, the same number equals ℏ\hbar keff2k_{\rm eff}^2T/m , the ordinary recoil phase used to extract ℏ\hbar/m in precision atom-interferometry measurements through the general path-integral formalism for atomic interferometry [ 14 ] . The two expressions are algebraically identical, so this is a check that the loop construction reproduces a quantity the field already measures by…

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Learn the underlying idea

An exponent tells how a base is used in multiplication. In x³, x is the base and 3 is the exponent: x³ = x × x × x.

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Sources cited in the surrounding passage

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