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Evaluation of quantum mechanical perturbative sums in terms of quadratic surds and their use in the approximation of ζ(3)/π 3
Author(s) -
Bracken Paul,
Čížek Jiri
Publication year - 2002
Publication title -
international journal of quantum chemistry
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.484
H-Index - 105
eISSN - 1097-461X
pISSN - 0020-7608
DOI - 10.1002/qua.1803
Subject(s) - ground state , perturbation theory (quantum mechanics) , algebraic number , mathematics , quadratic equation , euler's formula , quantum , euclidean geometry , quantum mechanics , perturbation (astronomy) , non perturbative , physics , mathematical physics , mathematical analysis , geometry
The first correction to the energy using Rayleigh–Schrödinger perturbation theory is calculated for the ground state of the one‐dimensional 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 state of the one‐dimensional 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 ζ(3) can be approximated by a product of nested radicals and rational constants, multiplied by π 3 , much in the same way that ζ(2) was expressed by Euler as a product of rational numbers and π 2 . It is discussed how this analysis can be continued to higher orders in perturbation theory such that ζ(2 m +1) is obtained as a multiple of π 2 m +1 and nested radicals. © 2002 Wiley Periodicals, Inc. Int J Quantum Chem, 2002