Hamiltonian systems random matrix system

The random matrix was first introduced by E. P. Wigner as a model to mimic unknown interactions in nuclei, and it has been studied to describe statistical natures of spectral fluctuations in quantum chaos systems [17]. Here, we introduce a random matrix system driven by a time-dependent external field E(t), which is considered as a model of highly excited atoms or molecules under an electromagnetic field. We write the Hamiltonian... 

Traditional theoretical approaches to quantiun size elfects (QSE) in metal particles are based on random matrix theory (RMT), which was first established by Wigner and Dyson to describe the spectrum of heavy nuclei. It is assumed that the (random) Hamiltonian of the system is a random N x N Hermitian matrix, with a Gaussian probability distribution of the form ... 

In the case of an equilibrium system the Hamiltonian is the same as that of an ensemble of conservative systems in statistical equilibrium. If the energy of the system is measured to lie between Ek and EK + AE, then the representative ensemble is also restricted to the energy shell [AE K. From the hypotheses of equal a priori probabilities and random a priori phases it then follows that the diagonal elements of the density matrix lying inside [AE]k are all equal and that all others vanish. The density matrix of the quantum statistical microcanonical ensemble is thereby determined as... 

Anderson s simple model to describe the electrons in a random potential shows that localization is a typical phenomenon whose nature can be understood only taking into account the degree of randomness of the system. Using a tight-binding Hamiltonian with constant hopping matrix elements V between adjacent sites and orbital energies uniformly distributed between — W/2 and W/2, Anderson studied the modifications of the electronic diffusion in the random crystal in terms of the stability of localized states with respect to the ratio W/V. 

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