Invariance hypothesis and higher correlations of Hamiltonian matrix elements

Abstract

A method is developed for calculating the averages of the components of a randomly oriented unit vector in an N dimensional orthogonal, unitary and symplectic space. The method is extended to calculate the averages of the components of two orthogonal N dimensional randomly oriented unit vectors. Using the invariance hypothesis it is shown that the off-diagonal matrix elements are distributed symmetrically about zero mean and that there are no correlations between an odd power of the off-diagonal matrix element and any power of diagonal or another off-diagonal matrix element. An expression is given for the correlation coefficient of the squares of diagonal matrix elements.