How to relate random matrix theory with Quantum mechanics approach In Quantum mechanics we first built Hamiltonian and then find its eigenvalues and eigenvectors. How I can relate this with random matrix theory?
 A: Random matrices were suggested as Hamiltonians in very complex systems from nuclear physics. Here, you had a lot of different energy levels appearing in your spectra, but unlike with, say, the hydrogen atom, it was no longer clear how to label them in a systematic way. So rather than trying to derive or propose an explicit (and presumably practically unsolvable) model, Wigner proposed to use random matrices which reproduce the correct eigenvalue statistics. 
More precisely, Wigner suggested that the entries of the random matrix are i. i. d. Gaußian variables under the constraint that the matrix be hermitian, and to normalize the matrices so that the eigenvalues fall into a pre-determined window. You call the set of such matrices the Gaußian orthogonal ensemble (GOE). Due to the normalization, you can study e. g. the eigenvalue distribution and level spacing distribution as the size of these matrices becomes large. For GOE you get the “semicircle law” whee the eigenvalue density approaches the function $P(x) = \frac{2}{\pi} \, \sqrt{1 - x^2}$. 
This has been refined to include e. g. symmetries such as even and odd time-reversal symmetries, which leads to the Wigner-Dyson classes. (These are also relevant for the study of topological insulators.) Of course, the presence of these symmetries has an effect on the eigenvalue statistics. Or you could look at “band matrices” where only entries close to the diagonal can be non-zero. 
In any case, it is important to keep in mind that within the framework of random matrix theory you do not study individual matrices, but the statistics of distributions of matrices. 
