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Physical Review

Linearization of the Pairing Hamiltonian

Richard R. Chasman

Published Apr 20, 1967• 6 citations
Publisher

Abstract

The pairing-Hamiltonian eigenstates are used to test linearization procedures for the equations of motion of pair-creation operators. It is found that standard linearization techniques are inadequate and additional terms are suggested by a detailed examination of pairing wave functions. The new linearization procedure is applied with considerable success to the pairing vibration problem. Possible extension of the results to more general two-body interactions is discussed.

DOI
10.1103/physrev.156.1197

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A single-intermediate-state (SIS) approximation is developed for dealing with the pairing Hamiltonian. With the aid of this approximation, expressions are obtained for the ground-state energy of an ($n+2$)-particle system in terms of the ground-state wave function of the $n$-particle system. Results are obtained for excited-state energies in the $n$- and ($n\ifmmode\pm\else\textpm\fi{}1$)-particle systems. Numerical comparisons are made with reliable results obtained with other techniques. It is observed that the results obtained from the SIS approximation are quite accurate. The SIS approximation is also used to point out one source of error inherent in the BCS solution of the pairing problem.

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The pairing-Hamiltonian eigenstates are used to test linearization procedures for the equations of motion of pair-creation operators. It is found that standard linearization techniques are inadequate and additional terms are suggested by a detailed examination of pairing wave functions. The new linearization procedure is applied with considerable success to the pairing vibration problem. Possible extension of the results to more general two-body interactions is discussed.
A method for describing the motion of many-body systems that interact solely through two-body forces is introduced. The general form of the Hamiltonian is found, and applied to the case of particles interacting through Hooke's law forces. The general form of the wave function is exhibited and the means of antisymmetrization discussed. It is shown that the kinetic energy of the center of mass can be separated from the Hamiltonian by transforming to a new set of external (pair c.m.) coordinates. A particular approximation to the system boundary conditions is applied to light nuclei, and some of the implications of the use of current nucleon-nucleon forces discussed.
A systematic procedure is developed for solving the eigenvalue problem for a broad class of Hamiltonian operators containing no terms higher than quadratic in generalized coordinates and th e ir conjugate momenta.The development is oriented toward practical applications in the area of the many-body problem.The procedure accomplishes a canonical reduction of such a Hamiltonian to the form of the Hamiltonian for a collection of non interacting bosons, with the eigenvalues of the Hamiltonian expressed in terms of the solutions to a single secular equation.Application of the results to systems of interacting identical bosons is discussed, including a presentation of a useful calculatlonal technique.The procedures developed are illu s tra te d by detailed treatments of two specific problems of interest in physics.The f i r s t problem considered is an exactly solvable separable potential model of p a rtic le fie ld theory.This problem consists of the description of a collection of lig h t bosons interacting with an in fin ite ly heavy boson via a simple separable po ten tia l.Interest in this problem centers on its use as a test case for approximation techniques to be used on more complicated systems.The second and more r e a lis tic problem investigated is the polaron problem of solid state physics.This problem involves the Iv description of the motion of a single conduction electron within an ionic solid.The polaron consists of the conduction electron together with its self-induced polarization field.The polaron problem is oif interest not only because of its value in solid state physics, but also because it, too, is useful as a testing ground for approximation techniques.Finally, broad applicational aspects of the procedures developed are discussed.
A new criterion is proposed for linearizing many-body equations of motion which is well defined and is related to a stationary principle. The method is applied to the problem of correlation in a narrow sss band and the results differ considerably from Hubbard's treatment of this problem.
Recently I proposed [Jia, Phys. Rev. C 88, 044303 (2013)] a new method for solving the pairing Hamiltonian with the pair-condensate wave-function ansatz based on the Heisenberg equations of motion for the density matrix operators. In this work an improved version is given by deriving the relevant equations more carefully. I evaluate both versions in a large ensemble with random interactions, and the accuracy of the methods is given statistically in terms of root-mean-square derivations from the exact results. The widely used variational calculation is also done and the results and computing-time costs are compared.
The problem of the exact diagonalization of the Hamiltonian of an assembly of N bilinearly interacting bosons is discussed in what concerns the eigenvalues as well as for the expression of the new boson operators in terms of the old ones. The method is equivalent to the standard equation-of-motion approach, nevertheless sensibly more concise. Three sets of operational rules are indicated, and their use is exhibited in some examples. In some particular cases of practical importance (for example, when all the coefficients of the Hamiltonian are real), the research of the eigenvalues has been compacted as much as possible.
A single-intermediate-state (SIS) approximation is developed for dealing with the pairing Hamiltonian. With the aid of this approximation, expressions are obtained for the ground-state energy of an (n+2n+2n+2)-particle system in terms of the ground-state wave function of the nnn-particle system. Results are obtained for excited-state energies in the nnn- and ($n\ifmmode\pm\else\textpm\fi{}1$)-particle systems. Numerical comparisons are made with reliable results obtained with other techniques. It is observed that the results obtained from the SIS approximation are quite accurate. The SIS approximation is also used to point out one source of error inherent in the BCS solution of the pairing problem.