# 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?

• – Qmechanic Oct 15 '18 at 9:30

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 $$P(x) = \frac{2}{\pi} \, \sqrt{1 - x^2}$$.