Evaluation of quantum mechanical perturbative sums in terms of quadratic surds and their use inthe approximation of zeta(3) /pi~3

摘要

The first correction to the energy using Rayleigh-Schrodinger perturbation theory is claculated for the ground state of the one-dimensinal Hubbard model.An algebraic technique is developed which can be used to evaluate these perturbative sums such that the final result is expressed as a finite linear combination of the elements which occur in the individual terms of the sum.The first nontrivial correction to the energy for the ground stateof the one-dimensinal Hubbard model in closed form is evaluated.A number theoretic application of this calculation will be given,which is related to the Euclidean construction of regular polygons inside the circle.It is shown that (zeta(3) can ve approximated by a product of nest3d radicals and ratinal constants,multiplied by pi~3,much in the same way the zeta(2) was expressed b Euler as a product of rational numbers and pi~2.It is discussed how this analysis can be continued tohigher ordersin perturbation theory such that zeta(2m+1) is obtained as a multiple of pi~(2m+1) and nested radicals.