From: g.scholten@nospam.gmx.de   
      
   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.   
      
   You mean, you interpret the wave function of the electron that way that   
   there would be a classical charge cloud with charge density   
      
   rho(x) = -e |psi(x)|^2  (1)   
      
   This interpretation is wrong, though. According to quantum mechanics,   
   one should rather imagine the electron as a point charge with an   
   uncertain position. Instead of a certain classical charge density, there   
   is a charge density operator   
      
   \op{rho}(x) = -e delta(x - \op{x0})   
      
   with \op{x0} being the position operator of the electron. Calculating   
   the average value  =  yields   
      
    = -e |psi(x)|^2   
      
   what is similar to (1), though, but this is only an average value, not a   
   classical certain value.   
      
   Therefore, there is no such shielding effect as you described it. There   
   is no portion of the electron charge inside the Bohr radius that would   
   shield the charge of the proton and reduce the potential outside the   
   Bohr radius. The electron always feels the full charge of the proton,   
   independently from the distance.   
      
   You also can see this by considering the two-particles wave function   
   Psi(xe, xp) of electron and proton. In the corresponding Schroedinger   
   equation, a two-particles potential operator   
      
   \op{W} = -e^2 / (4pi eps0 |\op{xe} - \op{xp}|)    (2)   
      
      
   occurs, with \op{xe} and \op{xp} being the position operators of   
   electron and positron. This is due to the rule for the transition from   
   classical theory to quantized theory: if in the classical theory a   
   classical two-particles potential W(x1,x2) occurs in the Hamilton   
   function, in the quantized theory a corresponding two-particles   
   potential operator like (2) occurs in the Hamilton operator.   
      
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