Abstract
This chapter discusses the Gaussian unitary and symplectic ensembles. It presents the properties of the eigenvalues of random real symmetric matrices chosen from an orthogonal ensemble. It is of some interest to compare the results obtained there with the corresponding results for the eigenvalues of random Hermitian matrices chosen from a unitary ensemble. The chapter reviews a model for the energy levels of a complex system without invariance under time reversal. The chapter presents the expression of the product of differences as the Vandermonde determinant and also presents an introduction to the normalized oscillator wave functions. On squaring and integrating, all cross terms drop out and each square term take the same coefficient (N — n)!.
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