Re: How is the Potential energy of electrons calculated ?

P.E.=mgh is just an approximation of the potential energy of an object
raised a certain distance h above the earth's surface.  Anytime you see g
used to represent the acceleration due to gravity, you can know that the
formula is only an approximation.  When dealing with an electron, gravity
is such a small force as to be nearly nonexistent compared to the other
forces involved.  Usually, when we speak of the "potential energy of an
electron," we are speaking of its electromagnetic potential energy.  This
potential energy is calculated using quantum mechanics.  For any atom
larger than Hydrogen, the calculations become extremely complex, but the
solutions for Hydrogen are well known.  These energy levels are described
quite well at http://hyperphysics.phy-astr.gsu.edu/hbase/hyde.html

The E=m0c^2 formula is for the relativistic rest energy of a particle, it
has nothing to do with the electromagnetic potential energy we are usually
talking about, nor is it related to the kinematic potential energy
approximated by P.E.=mgh.  The relativistic energy is the amount of energy
that would be released if the particle were completely annihilated by an
antiparticle (along with an equal amount of energy from the antiparticle).