Modified Ritz Method

Abstract

1. The Ritz variation method' enables one to approximate to the lowest energy level and wave function of that level in characteristic value problems of the form H+n = W4w WO < W1. W2... (1) where W5 is the energy level of the state 4t,, of the operator H. The An, are supposed to form a complete orthogonal set. It is based upon the fact that given any arbitary function t subject to the restriction fJ*dT = 1, then I, = fJH *dr) W0. Thus, the less we make I, by so choosing {, the closer is I, to Wo. MacDonald2 has shown that when applied to a linear combination of n independent functions, this method gives an upper bound to the n'th excited energy level. This method, however, suffers from the defect that only an upper bound is given. Thus, while it enables us to better the function, it gives us no information as to how accurate the function really is. We are going to show in what follows that if Wj is the j'th excited energy level to which I, lies closest, then 2-I + 11 Wj II-V12-where I, is as defined above, and I2 = f(Ht)2dT.