Equation 15 · Decoherence: The Quiet Selection That Makes the World Look Solid
What does this equation mean?
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This equation states an equality: the expressions on both sides have the same value under the article’s assumptions. Read the equation part by part below; each part has a contextual explanation and a link to its mathematical background.
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Symbol l
l is part of the quantity the equation computes from the expression on the right.
Symbol t
t is an argument of the function-like quantity on the left; its role is set by that function’s stated inputs.
=
The expressions on both sides represent the same quantity under the stated assumptions.
See an illustrated explanation →superscript
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.
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What the article says around this equation
The same paper also tracks what happens to the coherence length itself, not just the decay rate: solving the full master equation for a free particle gives a coherence width l(t) = (8 t)^{-1/2} , meaning it shrinks as the inverse square root of time for as long as scattering dominates, and does so from whatever width the particle’s wave packet started with [ 7 ] . Joos and Zeh flag the counterintuitive consequence themselves: the everyday expectation that a free wave packet spreads out over time is not what this equation describes at all. “The much-discussed dispersion hardly ever shows up even for small dust particles or large molecules,” they write; instead “the coherence length…
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The same paper also tracks what happens to the coherence length itself, not just the decay rate: solving the full master equation for a free particle gives a coherence width l(t) = (8 t)^{-1/2} , meaning it shrinks as the inverse square root of time for as long as scattering dominates, and does so from whatever width the particle’s wave packet started with [ 7 ] . Joos and Zeh flag the counterintuitive consequence themselves: the everyday expectation that a free wave packet spreads out over time is not what this equation describes at all. “The much-discussed dispersion hardly ever shows up even for small dust particles or large molecules,” they write; instead “the coherence length decreases towards the thermal de Broglie wave length of the object, whereas the incoherent spread increases” [ 7 ] . Scattering does not let a macroscopic object’s superposition grow lazily wider with time, the way an isolated wave packet’s would; it clamps the coherent part down to a wavelength set by the temperature of whatever is doing the scattering, almost as fast as the two effects can compete.
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