does an electron have a continous range of energies? I think the answer is no. am i right?
and also... what does the solution of the schrodinger's equation provide?
I think you have a severe misunderstanding of how something can acquire energy. The energy of any electron (or any particle for that matter) depends on where it is, or what situation it is in. The energy of the emitted electrons out of a photo electric effect experiment is certainly NOT discrete. Look at any photoemission experiment. This is because the conduction band is, by definition of a "band", is continuous.