Monday, 22 January 2018

schroedinger equation - nabla and non-locality in simple relativistic model of quantum mechanics


In Wavefunction in quantum mechanics and locality, wavefunction is constrained by H=m222, and taylor-expanding H results in:


H==m12/m22=m(1)


While the person who asked this question accepted the answer, I was not able to understand fully.


Why would 200 in the taylor-expansion of H be so problematic (200 can be replaced by any arbitrary number) - resulting in non-locality? Isn't this just some exponentiation of dot product of gradient?




Answer



Non-locality comes from presence of infinite many terms in that expansion. To see that, lets assume we are applying the non-polynomial function f(z) of i on any function ψ(x): f(i)ψ(x). Assuming that f(z) is "nice enough" to have the Fourier representation f(z)=+d3k(2π)3F(k)eikz for suitable transform function F(k). Then:


f(i)ψ(x)=+d3k(2π)3F(k)eik(i)ψ(x)=+d3k(2π)3F(k)ekψ(x)=+d3k(2π)3F(k)ψ(xk)


This is the superposition of values of g calculated at points different than x. that is the non-locality.


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