Intiutive approach to Green's function for SE

gulsen

Griffiths develops an intelgral equation for Scrödinger equation in his QM book. As doing so, he requires Green's function for Helmholtz equation

[tex]
(k^2 + \nabla^2) G( \mathbf r) = \delta^3(\mathbf r)
[/tex]

A rigourious series of steps, including Fourier transforms and residue integrals follow immidiately. As it turns out, the Green's function is independent of direction of [itex]\mathbf r[/itex], i.e. [itex]G(\mathbf r) = G(r)[/itex]. Does anyone know a trick deducing this condition without actually finding the Green's function. The importance to prove is that, it turns out that there's a relatively simple solution if we assume [itex]G(\mathbf r) = G(r)[/itex])

dextercioby

This simpler form of the GF is caused by a space-invariance of the potential.