The Rayleigh Quotient

The central role of the Rayleigh quotient in many body physics is discussed. Various many body methods can be obtained from either an attempt to evaluate the Rayleigh Quotient directly or through various variational approximations. Rather than dwell on the technical details necessary to obtain the equations of the various many body methods, we concentrate on how they can be obtained from the Rayleigh Quotient, and some of the consequences of the approximations involved in their evaluation.