Monday, December 31, 2018

quantum mechanics - Delocalization in the square root version of Klein-Gordon equation


In this Wikipedia article a relativistic wave equation is derived using the Hamiltonian H=p2c2+m2c4

Substituting this into the Schrödinger equation gives the square root version of the Klein-Gordon equation: ((i)2c2+m2c4)ψ=itψ
Then the article says:



Another problem, less obvious and more severe, is that it can be shown to be nonlocal and can even violate causality: if the particle is initially localized at a point r0 so that ψ(r0,t=0) is finite and zero elsewhere, then at any later time the equation predicts delocalization ψ(r,t)0 everywhere, even for r>ct which means the particle could arrive at a point before a pulse of light could.



What is this solution explicitly? I have read also this Phys.SE question but there is no clue for my question.




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