What are the benefits and reasons behind considering a probability density distribution for electrons as opposed to a mere density distribution? Before now, I had never questioned this matter. Why must we assume a probability density distribution (pdd) as opposed to a general density distribution (gdd)? Perhaps I have a misunderstanding.
My current understanding is that a pdd for an electron would suggest that the electron is never in one place, but given a coordinate $x$,$y$,$z$ we could assign a prob value of the electron being found here. Today it dawned on me that a gdd for an electron may be different. A gdd would suggest the electron is, for a coordinate $x$,$y$,$z$ actually there, only in part as opposed to a pdd where the electron would be potentially there in full. Is there a difference? Perhaps not the best question, surely not the worst.
 A: Interesting question.  And though, of course, one must always be cognizant of definitions, I'll provide something of an answer, though your exact definition of a 'gdd', I may not fully grasp.
In condensed matter physics, a full many-body solution of the Schrodinger equation, which would produce a wave function from which you could then calculate a probability density, cannot really be solved.
But the problem can be approached in a different way, using what is know as density functional theory.  The link here gives a readable comparison between the two methods.  To quote one section:

Nevertheless, we have been able to carry out this program of calculations [solution of the ground state electronic energy of a solid] because there is an alternative theoretical formulation for determining the electronic structure. In two important theorems (Hohenberg and Kohn 1964, Kohn and Sham 1965, Dreitzler and Gross 1990), it has been shown that the total energy of a solid (or atom) may be expressed uniquely as a functional of the electron density.

So indeed there is great benefit to working with equations of the electron density distribution of electrons as opposed to a probability density distribution.  Hopefully this is somewhere in the ballpark of what you are asking.
