>>11686589I see the confusion. Let me know if this clears things up
part of the problem here is the understanding of energy vs. force and part of it is the confusion between classical and quantum systems. you're correct up until you mention a "repulsive force," because there is no physical force that takes over on length scales below N that repels the two electrons.
There is one force in this problem: electromagnetism. this is the force that attracts the two particles (in the case of the p/e) or repels them (in the case of ee). This force has an equivalent potential energy that depends on the separation, U q/r where the sign of q depends on the charge of the object being considered.
You want to minimize this energy, so if q>0 you would want r=\infinity, and if q<0 you would naively want r=0 (which would mean U=. However, there is also the contribution from Kinetic energy (which is a strictly positive quantity), and this is where the quantum vs classical differences come up. In classical mechanics, you are allowed to have 0 kinetic energy, which means your system will seek out the minimum potential only. In quantum mechanics, however, you have the uncertainty principle-if you choose to localize the electron at r=0, you end up with an infinite value for kinetic energy (see
>>11680983 (You) for a more intuitive understanding why this is the case). This is not an energy minimum, as is undefined.
Instead, you find that in the equation for the energy of the hydrogen atom, there is an interplay between the potential energy and the kinetic energy which comes from the same force, but it is not "repulsion" in the sense that you're talking about. Imagine a planet in a circular orbit. If you change the radius of this orbit while keeping the total energy (U+KE) the same, what happens to the orbit? It becomes highly elliptical. cont.
