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arXiv:cond-mat/0311288 · Annals of Physics

Second quantized reduced Bloch equations and the exact solutions for pairing Hamiltonian

Liqiang Wei, Chia‐Chung Sun, Zong-Hao Zeng

Published May 13, 2004• 0 citations
arXivHTMLPDF
DOI
10.1016/j.aop.2004.04.007

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Hamiltonian Approach to the Second Quantization

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The Hamiltonian for an arbitrary Coulomb three-body system is written in closed form in terms of nine generators of three independent o(2,1) algebras and its six physical parameters (i.e., three masses and three charges). Moreover, it is shown that such a Hamiltonian can also be written in the second quantized form by using six pairs of creation and annihilation (boson) operators. The obtained finite-term representation of the Coulomb three-body Hamiltonian in the second quantized form is exact, i.e., it is not based on any approximation. In particular, this means that the original Coulomb three-body problem can be reduced to the equivalent problem of three interacting, complex (i.e., charged) boson fields. The developed procedure opens new avenues in obtaining analytical and semianalytical (or highly accurate) solutions for various Coulomb three-body problems.
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The exact solution of the proton-neutron isoscalar-isovector (T=0,1) pairing Hamiltonian with nondegenerate single-particle orbits and equal pairing strengths is presented for the first time. The Hamiltonian is a particular case of a family of integrable SO(8) Richardson-Gaudin models. The exact solution of the T=0,1 pairing Hamiltonian is reduced to a problem of 4 sets of coupled nonlinear equations that determine the spectral parameters of the complete set of eigenstates. The microscopic structure of individual eigenstates is analyzed in terms of evolution of the spectral parameters in the complex plane for a system of A=80 nucleons. The spectroscopic trends of the exact solutions are discussed in terms of generalized rotations in isospace.