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In Quantum mechanics we first built Hamiltonian and then find its eigenvalues and eigenvectors. How I can relate this with random matrix theory?

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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.

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  • $\begingroup$ But what it tells us about electron transport? After applying random matrix theory can we exactly know the value of each energy level? Is there any thing special in the off diagonal term of random matrices? $\endgroup$ – herry Oct 15 '18 at 9:16
  • $\begingroup$ @herry You seem to have a much more spesific question in mind than the one you originally posted. I would suggest you think carfully about what it is you actually want to know and post another question asking about that $\endgroup$ – By Symmetry Oct 15 '18 at 9:22
  • $\begingroup$ Actually I want to know does after applying Random matrix theory we exactly know where the energy level of the system reside?Because we argue that in complex nuclei we don't know what kind of interaction take place b/w particles so we replace hamiltonian with random matrix. Does this hamiltonian provies us exact energy level information like in Quantum mechanics? $\endgroup$ – herry Oct 15 '18 at 9:34
  • $\begingroup$ I think you misunderstand the purpose of random matrix theory and how you arrive at hamiltonians for specific quantum systems: either you list all the particles, all interactions and add up the corresponding terms to a quantum Hamiltonian. Random matrix theory starts from the other end where you propose an ensemble of matrices that statistically shares some properties of the complex, physical quantum system. With random matrices, you are not interested in the energy levels of one specific random matrix, but in eigenvalue statistics of the ensemble. $\endgroup$ – Max Lein Oct 16 '18 at 9:40

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