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Acta Physica Sinica

The equal probability comparison method of determining the equivalent-electron Young basis

HU Kun-ming, 商丘师范学院物理与信息工程系，商丘 476000

Published Jan 1, 2008• 0 citations
Publisher

Abstract

The concepts of the basic symmetric operator and complete symmetric operator of the equivalent-electron regular Young tableau T［λ］ig are presented, and the concepts of the root state and generative state generated by these symmetric operators acting on each Slater function i are also given. Based on the establishment rules of the vertical permutation operator A［λ］ig of the orthogonal normalization Young tableau T［λ］ie, the symmetric operators in A［λ］ie and the equiprobability comparison method for solving T［λ］ie are presented, which can avoid the complicated algebra involving many operators. Finally, a new method for sloving the Young basis of the electron system with a large N is presented.

DOI
10.7498/aps.57.6074

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The concepts of the basic symmetric operator and complete symmetric operator of the equivalent-electron regular Young tableau T［λ］ig are presented, and the concepts of the root state and generative state generated by these symmetric operators acting on each Slater function i are also given. Based on the establishment rules of the vertical permutation operator A［λ］ig of the orthogonal normalization Young tableau T［λ］ie, the symmetric operators in A［λ］ie and the equiprobability comparison method for solving T［λ］ie are presented, which can avoid the complicated algebra involving many operators. Finally, a new method for sloving the Young basis of the electron system with a large N is presented.
For n identical particles of electrons,the spatial configurations ( l f +l g - )with the given permutation symmetry are expressed in terms of n λ standard Young tableaux T [λ] i .The spatial wavefunction  1 of the n electron system related T [λ] 1 is Changed to the non perturbation function  1e .From the hypothesis of equal probability for spatial exchange between electrons with the opposite spins,the operator q +e related to T [λ] 1 is determined,the analytic forms of the non perturbation wavefunctions with the given permutation symmetry are derived,and the formulas for the symmetric spatial wavefunctions of identical electrons are obtained in terms of the Standard Young Tableaux.
We demonstrated in the previous chapter (Chapter 1) that the evaluation of matrix elements between states represented by the Slater determinant can be evaluated unambiguously using Slater’s rule [1, 28, 40]. While the determinantal representation and the evaluation matrix elements between the states are quite straightforward, this procedure can eventually be ineffective and cumbersome with increasing size of the system. The occupation number formalism or second quantization technique introduced by Dirac in quantum field theory, in this context has been found to be quite formidable and elegant. This formalism not only allows us to represent the states and the operators in a compact and convenient form but also enables us to deal with systems containing variable numbers of particles without explicitly specifying the number of particles. It implicitly assumes the existence of an unspecified number of functions in a single-particle basis. These one-particle functions, which constitute the Hilbert space for N particles, are generally chosen to be spinorbital because of their orthogonality. In the case of fermions, the many-particle basis is the anti-symmetric product of N one-particle basis functions (Slater determinant). The vector space consisting of all possible anti-symmetric Hilbert spaces is referred to as the Fock space, in which the operators and functions are represented without specifying the number of particles.