how do you solve quantum equation for example prove that for any operator A, E|<i|A|j>|^=Tr(AA^T), where Tr denotes the trace help would be appreciated
As far as how to solve the equations that is going to depend heavily on what the equation is. For example, the Navier-Stokes equations are miserably difficult to solve even under simple circumstances, whereas it can be hard not to find solutions to the Poisson equation. (At least solutions that might be likely for a charge distribution, for example.)
Your question is really too broad to even try to answer. Do you have any specific interests?
this page. Let me put it to you this way: I decided I wanted to study Quantum theory because 1) I stink at experimentation and 2) I didn't EVER want to try to solve something as nasty as Navier-Stokes. (Oddly enough fluid mechanics is more complicated, at least Mathematically, than QM.) (Then of course, I ran headlong into General Relativity and found out that there are certain similarities between an incompressible fluid flow and quantum probability densities. Ah well. )
Despite my interests I try to steer clear of the truly nasty stuff.
Keep at it. You're young enough to have lots of time to run into all of this stuff. Give it a few more years and you'll hit the more advanced techniques that your are learning the basics of now.