From: goldenfieldquaternions@gmail.com   
      
   On Saturday, August 19, 2017 at 2:44:43 AM UTC-5, Jos Bergervoet wrote:   
   > In solving the hydrogen atom we assume a 1/r electric potential.   
   > But since the electron wave function squared is a source for the   
   > electric field, this should be a screened potential. But if we   
   > do that, then it will be quite different from 1/r. At large   
   > distances it will vanish, and at distances around the Bohr radius   
   > the potential will already be significantly reduced. The paradox:   
   >=20   
   >    1) Energy levels will become quite different from known hydrogen   
   > levels with this changed potetial.   
   >    2) But without the change, we do not get the vanishing of the   
   > field outside the atom, so the atom is not neutral!   
   >=20   
   > How to easily resolve this? Can one use a 2-particle wave function   
   >    Psi_i_mu(x_e, x_p)   
   > with the position coordinates x_e and x_p of the electron and the   
   > photon respectively, and both their spin-indices?   
   >=20   
   > Or is another simplification starting from full field theory   
   > applicable? I can hardly imagine that an elaborate numerical   
   > lattice approach is needed just to solve hydrogen without   
   > inconsistent behavior for points 1) and 2) mentioned above..   
   >=20   
   > --=20   
   > Jos   
      
   The radial part of the wave function is given by Laguerre polynomials.   
   These are an asymptotic expansion. Unfortunately the format here   
   is to limited to give an expository on this. This is fairly classical   
   19th century mathematics, and it is what is used to avoid the radial   
   divergence and it also gives the radial eigenstates of the hydrogen   
   atom.   
      
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