What's astonishing: many laws of physics emerge from statistical approximations of quantum mechanics.
One day, if I really get into quantum mechanics, I will try to understand how they rebuilt maxwell equations from QED.
One day, if I really get into quantum mechanics, I will try to understand how they rebuilt maxwell equations from QED.
This derivation is in the context of classical field theory, but QED is only a short hop away through path integrals.
It’s quite remarkable how the complexity of Maxwell’s equations can be reduced to a single term in the Lagrangian - (F_uv)(F^uv), assuming no charges. That’s really it!
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:
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
2 2 2 pXa
|<X|ψ>| = |ψ(X,t)| cos -----
ħL
2 2 2 pXa
|<X|ψ>| = |ψ(X,t)| cos (--- + ϕ)
ħL
2 1 ^ pXa 2
|<X|ψ>| - (1 + | dϕ P(ϕ)cos(2 --- + 2ϕ)) |ψ(X,t)|
2 v ħL
2 2
|<X|ψ>| = |ψ(X,t)|
2 2 1 -αt 2 pXa
|<X|ψ>| = |ψ(X,t)| - (1 + e cos (--- + ϕ) )
2 ħL
α small => quantum result, α = large, classical result.