After reading about the hydrogen atom and understanding how Schrodinger's equation explains most part of the atomic spectrum of an hydrogen atom, and also came to know that, it explains most of the chemical reactions and a huge tool in chemistry.
I am now almost convinced, that it is wise to accept the Schrodinger equation as a law that govern's the motion of subatomic particles like electrons at quantum scales. Now I am a little curious about one problem. How does an electron (a distribution of charge) move under the influence of its own electrostatic Coulomb's field. I am interested only in strictly theoretical sense, but also like to know if there is any practical importance to it.
I'd like to consider this problem first in a 1-D setup, purely due to my lack of acquaintance with partial differential equations. So lets consider a 1-D electron, as a linear charge distribution of constant density $\rho$ and distributed over a length $2r_e$.
Now I am interested to setup the Schrodinger equation for it, in the case where there is no external field. I'd appreciate some help/comments on setting it up, solution and analysis/interpretation of the resulting wave function and what it actually means, at diefferent energy levels(very high, very low, etc).