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Equation 91 · Part 11 · The Clock That Comes Back Wrong by Exactly Its Mass

Denominator: hbar

Φ[LG]=m vrbℏ≈6.29×104 rad.\Phi[\mathcal L_{\rm G}]=\frac{m\,v_rb}{\hbar}\approx6.29\times10^{4}\ \text{rad}.
ℏ\hbar

What this part means

The complete quantity below the fraction bar; it must be nonzero for this division.

Its job in the formula

hbar occurs below the fraction bar. The quantity above the bar is divided by this expression; zero is excluded as a denominator.

The passage around this formula

…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…

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Learn the underlying idea

A fraction a/b means a divided by b. The top number is the numerator; the bottom number is the denominator, and it cannot be zero.

Open the illustrated fractions: division written vertically guide →

Sources cited in the surrounding passage

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