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ρ(x,x′,t)=ρ(x,x′,0) exp⁡ ⁣[−Λ(x−x′)2t]\rho(x, x', t) = \rho(x, x', 0)\, \exp\!\left[-\Lambda (x - x')^2 t\right]

Why this formula appears here

Joos and Zeh made this quantitative in 1985 by deriving a genuine, non-phenomenological master equation for the reduced density matrix of an object’s centre-of-mass position under repeated recoil-free scattering [ 7 ] . For the off-diagonal elements of that density matrix — the very quantity whose survival or destruction is the entire question of macroscopic superposition — their result, in the short-time, many-collisions regime, takes the form of a simple exponential decay in the separation between the two positions being superposed: ρ(x,x′,t)=ρ(x,x′,0) exp⁡ ⁣[−Λ(x−x′)2t]\rho(x, x', t) = \rho(x, x', 0)\, \exp\!\left[-\Lambda (x - x')^2 t\right]. Here Λ\Lambda is what Joos and Zeh call the localization rate: a single number, with units of inverse length squared per unit time, built…

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ρ(x,x′,t)=ρ(x,x′,0) exp⁡ ⁣[−Λ(x−x′)2t]\rho(x, x', t) = \rho(x, x', 0)\, \exp\!\left[-\Lambda (x - x')^2 t\right]

Equation 3 · Evolutionary Physics

Decoherence: The Quiet Selection That Makes the World Look Solid

This equation states an equality: the expressions on both sides have the same value under the article’s assumptions.

Joos and Zeh made this quantitative in 1985 by deriving a genuine, non-phenomenological master equation for the reduced density matrix of an object’s centre-of-mass position under repeated recoil-free scattering [ 7 ] . For the off-diagonal elements of that density matrix — the very quantity whose survival or destruction is the entire question of macroscopic superposition — their result, in the short-time, many-collisions regime, takes the form of a simple exponential decay in the separation between the two positions being superposed: ρ(x,x′,t)=ρ(x,x′,0) exp⁡ ⁣[−Λ(x−x′)2t]\rho(x, x', t) = \rho(x, x', 0)\, \exp\!\left[-\Lambda (x - x')^2 t\right]. Here Λ\Lambda is what Joos and Zeh call the localization rate: a single number, with units of inverse length squared per unit time, built…

Meanings in this article

  • Λ\Lambda: what Joos and Zeh call the localization rate: a single number.
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