In Wavefunction in quantum mechanics and locality, wavefunction is constrained by H=√m2−ℏ2∇2, and taylor-expanding H results in:
H=⋯=m√1−ℏ2/m2⋅∇2=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)=∫+∞−∞d3→k(2π)3F(→k)ei→k⋅→z for suitable transform function F(→k). Then:
f(i∇)ψ(x)=∫+∞−∞d3→k(2π)3F(→k)ei→k⋅(i∇)ψ(→x)=∫+∞−∞d3→k(2π)3F(→k)e−k⋅∇ψ(→x)=∫+∞−∞d3→k(2π)3F(→k)ψ(→x−→k)
This is the superposition of values of g calculated at points different than →x. that is the non-locality.
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