الفهرس 
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Abstract
The present thesis is devoted to the calculation of nearest neighbour level spacing distribution (NNS) for a chaotic system which is not invariant under time reversal. It consists of four chapters:- Chapter (1): The introduction, gives a review of some aspects of chaos in quantum mechanics, and classical mechanics. According to quantum mechanics, the behaviour of a system can be described by Hamiltonian given in the matrix from. Statistical theories of energy levels makes the hypothesis that characterize energies of chaotic systems, which behave locally as if they were eigenvalues of a matrix with randomly distributed elements, its diagonal elements are equal zeros. Chapter 1 contains a summary of the main results available in the literature, as well as brief description of kicked rotator model which will be used in the present thesis to test the derived NNS distribution. In Chapter (2): The nearest neighbour spacing distribution (NMS) for the energy levels of Guassian unitary ensemble of rank 2 matrices is derived. The resulting expression is compared with outcomes of previous numerical analysis of spectrum of a kicked rotator in a magnetic field which violates time reversal invariance. To calculate the level distribution for random Hamiltonian of rank 2 we proped a generalization of the method used by Shabin [1975]to the case in which time reversal invariance is broken. In Chapter (3): Two analytical expressions for nearest neighbour spacing distribution of eigenvalues of a time reversal symmetric Hamiltonian in the chaotic regime are compared with results of a previous numerical calculation of the quasi energy spacing of kicked rotator in magnetic field. The above-mentioned two expressions are compared with the predictions of these equations with the results of a numerical calculation of quasi-energy level spacing distribution of the generalized kicked rotator as an example for quintal time-reversal symmetric system which is chaoticin the classical limits. In Chapter (4): It is proposed to use the moment generating functions in comparing between the nearest neighbouring spacing distribution of the energy levels of quantum sysrems satisfying various invariant properties and the results of numerical experiments of simple quantum chaotic systems.