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Physical Review

Approach to the Nuclear Many-Body Problem by a Variationally Determined Transformation

A. Mann, H. Nissimov, I. Unna

Published Oct 20, 1969• 5 citations
Publisher

Abstract

Ground-state correlations are introduced via a unitary operator ${e}^{\mathrm{iF}}$, which is determined by the variational principle. It is surmised that the diagonal part of the transformed Hamiltonian ${H}^{\ensuremath{'}}={e}^{\ensuremath{-}iF}H{e}^{\mathrm{iF}}$ provides a better starting point for any approximate calculation than the corresponding part of $H$. The method yields good results for a solvable model.

DOI
10.1103/physrev.186.1104

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Ground-state correlations are introduced via a unitary operator eiF{e}^{\mathrm{iF}}eiF, which is determined by the variational principle. It is surmised that the diagonal part of the transformed Hamiltonian H\ensuremath′=e\ensuremath−iFHeiF{H}^{\ensuremath{'}}={e}^{\ensuremath{-}iF}H{e}^{\mathrm{iF}}H\ensuremath′=e\ensuremath−iFHeiF provides a better starting point for any approximate calculation than the corresponding part of HHH. The method yields good results for a solvable model.
The short range repulsion between nucleons is treated by a unitary correlation operator which shifts the nucleons away from each other whenever their uncorrelated positions are within the replusive core. By formulating the correlation as a transformation of the relative distance between particle pairs, general analytic expressions for the correlated wave functions and correlated operators are given. The decomposition of correlated operators into irreducible n-body operators is discussed. The one- and two-body-irreducible parts are worked out explicitly and the contribution of three-body correlations is estimated to check convergence. Ground state energies of nuclei up to mass number A=48 are calculated with a spin-isospin-dependent potential and single Slater determinants as uncorrelated states. They show that the deduced energy- and mass-number-independent correlated two-body Hamiltonian reproduces all "exact" many-body calculations surprisingly well.
We present a generalization of the Hartree-Fock (HF) method incorporating two-body correlations. The correlations are introduced by means of a unitary operator eiF{e}^{\mathrm{iF}}eiF, where FFF is a Hermitian one- and two-body operator to be determined by the variational principle. Assuming that the matrix elements of FFF are small so that powers of FFF higher than the second may be neglected, a set of linear equations determining FFF is obtained. The relation between our method and methods used to treat singular interactions is discussed. The method is applied to two recently suggested models and is compared with the HF method, second-order perturbation theory, and with the random-phase approximation. It is found that our method yields very good results within the range of validity of our approximations, which seems to call forth applications to more realistic problems. Possible applications to the study of collective phenomena in nuclei are indicated.
"A theory is developed to allow the systematic calculation of the properties of quantum fluids given the microscopic two-body potential. Physical arguments are used to construct a correlation operator that approximately transforms the states of a non-interacting system into the corresponding states of the interacting-system. The theory can be regarded as a microscopic realization of Landau's assumption of a one-to-one correspondence between the non-interacting and interacting states. It is variational since the correlation operator is determined by minimizing E(p) the expectation value of the ground-state energy. We first give a survey of various other many-body theories including the simpler Jastrow variational method. Differential equations are derived to calculate the correlation operator and the related Jastrow correlation"" function. A comparison of results in several Bose systems with various methods is given. The Fermi hypernetted chain integral equations are generalized to allow the calculation of E(p) in Fermi systems, using the correlation operator wave function. The method is applied to liquid 3He and the v2-model of nuclear matter. In 3He the correlation operator gives results in much closer agreement with experiment than that with the Jastrow wave function. A new set of hypernetted chain equations are then derived to allow the calculation of the excited st;ate energies in Bose systems. The theory is then applied to the calculation of the single quasi-particle excitation spectrum of liquid 4He. The spectrum is found to be a stringent test on the form of the correlation operator, deficiencies in the correlation operator giving corresponding errors in the spectrum. Good results are obtained for the density dependence of the roton energy. A variational principle for the free energy is used to generalize the correlation operator technique to finite temperatures. Results are given for the finite temperature properties of the v8-model of nuclear matter."