Classical behavior emerges from quantum mechanics as you enter the classical domain. This can be explained through phase decoherence. As temperature rises, ran

Classical behavior emerges from quantum mechanics as you enter the classical domain. This can be explained through phase decoherence. As temperature rises, random phase shifts are introduced, which effectively removes the quantum effect. You can show mathematically how this works. Consider the young experiment: https://imgur.com/FqogDJj https://imgur.com/FqogDJj For a plane wave ψ ~ e^(ipx/ħ-iωt), the wave function at X is the sum of two components <X|ψ> = <X|P> + <X|Q> Where for some path-independent normalization function ψ(X,t), and using the small angle assumption (QX-PX = 2Xa/L), the components are: 1 ipXa/ħL <X|P>= ψ(X,t)- e 2 1 -ipXa/ħL <X|Q>= ψ(X,t) - e 2 And the probability of finding the particle at X is 2 2 2 pXa |<X|ψ>| = |ψ(X,t)| cos ----- ħL That is what you'd expect from the Young experiment. If we introduce a constant phase shift ϕ between P and Q, you get this average instead: 2 2 2 pXa |<X|ψ>| = |ψ(X,t)| cos (--- + ϕ) ħL If this phase shift is instead random, the formula becomes 2 1 ^ pXa 2 |<X|ψ>| - (1 + | dϕ P(ϕ)cos(2 --- + 2ϕ)) |ψ(X,t)| 2 v ħL Where P(ϕ) is a probability function for the phase shift. If the probability function is flat, the integral is zero since you're integrating the cosine across its domain. What you get is the classical result! 2 2 |<X|ψ>| = |ψ(X,t)| You can even re-phrase random phase shifts into a diffusion equation, and find that given α as the diffusion coefficient 2 2 1 -αt 2 pXa |<X|ψ>| = |ψ(X,t)| - (1 + e cos (--- + ϕ) ) 2 ħL i.e. the transition behavior from quantum to classical dependent on a direct measure of the decoherence! α small => quantum result, α = large, classical result.