I am currently in my second semester of undergraduate quantum mechanics. We have recently starting discussing two particle systems, usually in relation to spin interactions. In all of our calculations, we have assumed entirely non-interacting particles. This does not seem physical, especially if we are considering charged particles as most particles are.
Say we have two electrons in an infinite square potential well. Classically, there should be potential between the charged electrons. But, since the electrons are defined by wave functions rather than positions, we cannot use the standard $kqq/r$ formula for potential.
A quick Wikipedia search told me that an electron's charge density is proportional to its wave function. Can you use this to define some sort of probability distribution for the potential due to the uncertainty in the position of the electron? How would an uncertainty in potential manifest itself in the Schrodinger Equation? What would the probability density of two electrons in an infinite square potential well look like assuming the two electrons have some sort of potential between them?