Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

Physical Review

Study of the Perturbation Series for the Ground State of a Many-Fermion System. II

George Allen Baker Jr., Bob Hill, R. J. McKee

Published Aug 17, 1964• 7 citations
Publisher

Abstract

We derive through fourth order in the potential strength the many-fermion perturbation series for a velocity-dependent force which exactly simulates, in two-body scattering, a hard core. We use the Pad\'e approximant method to sum the complete series and the ladder and Brueckner approximation series. We compare these results with those of the $K$-matrix procedure and find that the $K$-matrix procedure fails to converge for the Brueckner approximation in the only interesting region. We also investigate near saturation, a system something like ${\mathrm{He}}^{3}$. We find that there is not, as some have speculated, a magic cancellation that would cause the ring diagrams to be of little importance there.

DOI
10.1103/physrev.135.a922

Related papers

Study of the Perturbation Series for the Ground-State Energy of a Many-Fermion System

George Allen Baker Jr., J. L. Gammel, Bob Hill

In order to investigate the validity of the Brueckner and the simple sum of the ladder diagrams approximations to the energy of an infinite system of fermions, we have calculated as a function of density all the terms in the perturbation expansion through the fourth order in the strength of the potential. We have done this for a repulsive, square-well, two-body potential spin-\textonehalf{} fermions. We are able to construct rigorous upper and lower bounds from the coefficients for the value of the ladder approximation and find that the standard solution procedures give reasonably accurate results (within a few percent). We find that the error in the solution as obtained in practice to the equations of the Brueckner approximation is large compared to the size of its departure from the ladder approximation. We further find for low-to-moderate densities and for low-to-moderate potential strengths that the Brueckner approximation both as a sum of a certain class of diagrams and as computed in practice lies above the ladder approximation while the complete perturbation theory lies below it. This result arises from the neglect of the ring diagrams by the Brueckner approximation.

Physical Review•Nov 1, 1963•29 citations

Constructive approach to the ground-state energy of a hard-core square-well fermion system

George Allen Baker Jr.

I am looking at the old problem of the ground-state energy of quantum many-Fermion systems. My tools will be a version of perturbation theory. It is plain that the ordinary, low-density rearrangement can not be used directly to compute the saturation energy and density of a system for which the two-body potential has a strong repulsive core and an attractive portion beyond the core, because the path of continuation from low-density to saturation density passes through the two-phase (vacuum, liquid) region. Likewise it is difficult to supply an expansion directly in lambda for fixed rho (density) because of the difficulty of the strong repulsive core and the tediousness of the calculations. My approach here is to expand the energy in a double series in k/sub F/, the Fermi momentum, and lambda, the strength of the attractive part of the potential. The plan is then to sum up, by the Pade method or a generalization of it, the k/sub F/ expansions of the coefficients of each power of lambda into a function of k/sub F/. The coefficient of lambda/sup 0/ is just the repulsive-core energy itself.

•Sep 1, 1984•0 citations

Energy of the Many-Fermion Normal System

Kleber S. Masterson, K. Sawada

Perturbation theoretic equations for several properties of the zero-temperature many-fermion normal system are rederived from general considerations. Though none of the equations are new, some of the derivations are, and taken together they form a brief and simple summary of many-fermion relations and a basis for further investigation of the basic perturbation description of the many-fermion normal system: the Brueckner-Goldstone expansion. A "change-of-parameter" technique is then developed and employed to investigate the possible use of the true momentum density instead of the unperturbed Fermi distribution in the Brueckner-Goldstone expansion and in the Brueckner $K$-matrix approximation. The result is a simpler, approximate perturbation series for the interaction energy, whose accuracy for nuclear matter is estimated to be approximately \ifmmode\pm\else\textpm\fi{}2 MeV. The new approximation is exact to the fourth order.

Physical Review•Mar 2, 1964•7 citations

Summability of the Many-Fermion Perturbation Series with Attractive Forces

George Allen Baker Jr.

The summability of the many-fermion perturbation series in the presence of attractive forces is considered. We find that, although it is an asymptotic series, the potential-strength perturbation series is summable. We find that some rearrangements to handle the hard-core potentials also lead to the correct sum; however, the ``hole-line'' summation procedure using Brandow's choice for intermediate state energies is either wrong or deceptive in its convergence.

Journal of Mathematical Physics•Aug 1, 1970•7 citations

Singularity Structure of the Perturbation Series for the Ground-State Energy of a Many-Fermion System

George Allen Baker Jr.

Calculational procedures are developed for the ground-state energy of a many-fermion system. The entire theory is developed from the many-body Schr\"odinger equation via perturbation theory. The linked cluster expansion is derived by elementary methods. The analytic singularities of the energy function are analyzed, and procedures designed to avoid or overcome them. Results for the calculation of the energy in the presence of a purely repulsive two-body force are reviewed, and the situation is found to be reasonably satisfactory, though good results are restricted to weak potentials when the system is not dilute. Results for an attractive potential surrounding a strongly repulsive core are reviewed, and procedures discribed for calculations in this case. A sample calculation with an assessment of the probable errors is reviewed.

Reviews of Modern Physics•Oct 1, 1971•117 citations

Λ10 Approximation for Self-Bound Many-Fermion Systems

Robert D. Puff, A.S. Reiner, L. Wilets

We investigate several approximations for self-bound many-fermion systems, each of which employs the Green's-function formalism together with a suitable factorization to close the otherwise infinite hierarchy of coupled integral equations. The particular approximations, which attempt to include the important two-body correlations for short-range forces, differ in the choice of propagator $\ensuremath{\Lambda}$ for two-particle intermediate states in the $T$ matrix used to calculate the two-body correlation function. The advantages and shortcomings of the choice ${\ensuremath{\Lambda}}_{00}\ensuremath{\sim}{{G}_{1}}^{0}{{G}_{1}}^{0}$ have been noted previously and are not treated here. We show that the choice ${\ensuremath{\Lambda}}_{10}\ensuremath{\sim}{{G}_{1}}^{0}{G}_{1}$, which appears in a "natural" factorization, is formulated in a manner which preserves much of the simplicity of the ${\ensuremath{\Lambda}}_{00}$ approximation. (Numerical calculations using the ${\ensuremath{\Lambda}}_{10}$ approximation are presented by Foster in the following paper.) The ground-state energies for three $\ensuremath{\Lambda}$ approximations (${\ensuremath{\Lambda}}_{00}$, ${\ensuremath{\Lambda}}_{10}$, and ${\ensuremath{\Lambda}}_{11}\ensuremath{\sim}{G}_{1}{G}_{1}$) are expanded through third order in the interaction strength to facilitate comparison with each other and with the Brueckner-Goldstone series. Numerical comparison of these third-order contributions for a square-well potential is presented. ${\ensuremath{\Lambda}}_{10}$ is in dramatically closer agreement with the full perturbation theory (for ${{k}_{F}}^{0}c\ensuremath{\lesssim}1$) than are the other $\ensuremath{\Lambda}$ approximations or Brueckner theory.

Physical Review•Sep 23, 1966•23 citations
We derive through fourth order in the potential strength the many-fermion perturbation series for a velocity-dependent force which exactly simulates, in two-body scattering, a hard core. We use the Pad'e approximant method to sum the complete series and the ladder and Brueckner approximation series. We compare these results with those of the KKK-matrix procedure and find that the KKK-matrix procedure fails to converge for the Brueckner approximation in the only interesting region. We also investigate near saturation, a system something like He3{\mathrm{He}}^{3}He3. We find that there is not, as some have speculated, a magic cancellation that would cause the ring diagrams to be of little importance there.
In order to investigate the validity of the Brueckner and the simple sum of the ladder diagrams approximations to the energy of an infinite system of fermions, we have calculated as a function of density all the terms in the perturbation expansion through the fourth order in the strength of the potential. We have done this for a repulsive, square-well, two-body potential spin-\textonehalf{} fermions. We are able to construct rigorous upper and lower bounds from the coefficients for the value of the ladder approximation and find that the standard solution procedures give reasonably accurate results (within a few percent). We find that the error in the solution as obtained in practice to the equations of the Brueckner approximation is large compared to the size of its departure from the ladder approximation. We further find for low-to-moderate densities and for low-to-moderate potential strengths that the Brueckner approximation both as a sum of a certain class of diagrams and as computed in practice lies above the ladder approximation while the complete perturbation theory lies below it. This result arises from the neglect of the ring diagrams by the Brueckner approximation.
I am looking at the old problem of the ground-state energy of quantum many-Fermion systems. My tools will be a version of perturbation theory. It is plain that the ordinary, low-density rearrangement can not be used directly to compute the saturation energy and density of a system for which the two-body potential has a strong repulsive core and an attractive portion beyond the core, because the path of continuation from low-density to saturation density passes through the two-phase (vacuum, liquid) region. Likewise it is difficult to supply an expansion directly in lambda for fixed rho (density) because of the difficulty of the strong repulsive core and the tediousness of the calculations. My approach here is to expand the energy in a double series in k/sub F/, the Fermi momentum, and lambda, the strength of the attractive part of the potential. The plan is then to sum up, by the Pade method or a generalization of it, the k/sub F/ expansions of the coefficients of each power of lambda into a function of k/sub F/. The coefficient of lambda/sup 0/ is just the repulsive-core energy itself.
Perturbation theoretic equations for several properties of the zero-temperature many-fermion normal system are rederived from general considerations. Though none of the equations are new, some of the derivations are, and taken together they form a brief and simple summary of many-fermion relations and a basis for further investigation of the basic perturbation description of the many-fermion normal system: the Brueckner-Goldstone expansion. A "change-of-parameter" technique is then developed and employed to investigate the possible use of the true momentum density instead of the unperturbed Fermi distribution in the Brueckner-Goldstone expansion and in the Brueckner KKK-matrix approximation. The result is a simpler, approximate perturbation series for the interaction energy, whose accuracy for nuclear matter is estimated to be approximately \ifmmode\pm\else\textpm\fi{}2 MeV. The new approximation is exact to the fourth order.
Calculational procedures are developed for the ground-state energy of a many-fermion system. The entire theory is developed from the many-body Schr\"odinger equation via perturbation theory. The linked cluster expansion is derived by elementary methods. The analytic singularities of the energy function are analyzed, and procedures designed to avoid or overcome them. Results for the calculation of the energy in the presence of a purely repulsive two-body force are reviewed, and the situation is found to be reasonably satisfactory, though good results are restricted to weak potentials when the system is not dilute. Results for an attractive potential surrounding a strongly repulsive core are reviewed, and procedures discribed for calculations in this case. A sample calculation with an assessment of the probable errors is reviewed.
We investigate several approximations for self-bound many-fermion systems, each of which employs the Green's-function formalism together with a suitable factorization to close the otherwise infinite hierarchy of coupled integral equations. The particular approximations, which attempt to include the important two-body correlations for short-range forces, differ in the choice of propagator $\ensuremath{\Lambda}$ for two-particle intermediate states in the TTT matrix used to calculate the two-body correlation function. The advantages and shortcomings of the choice ${\ensuremath{\Lambda}}_{00}\ensuremath{\sim}{{G}_{1}}^{0}{{G}_{1}}^{0}$ have been noted previously and are not treated here. We show that the choice ${\ensuremath{\Lambda}}_{10}\ensuremath{\sim}{{G}_{1}}^{0}{G}_{1}$, which appears in a "natural" factorization, is formulated in a manner which preserves much of the simplicity of the ${\ensuremath{\Lambda}}_{00}$ approximation. (Numerical calculations using the ${\ensuremath{\Lambda}}_{10}$ approximation are presented by Foster in the following paper.) The ground-state energies for three $\ensuremath{\Lambda}$ approximations (${\ensuremath{\Lambda}}_{00}$, ${\ensuremath{\Lambda}}_{10}$, and ${\ensuremath{\Lambda}}_{11}\ensuremath{\sim}{G}_{1}{G}_{1}$) are expanded through third order in the interaction strength to facilitate comparison with each other and with the Brueckner-Goldstone series. Numerical comparison of these third-order contributions for a square-well potential is presented. ${\ensuremath{\Lambda}}_{10}$ is in dramatically closer agreement with the full perturbation theory (for ${{k}_{F}}^{0}c\ensuremath{\lesssim}1$) than are the other $\ensuremath{\Lambda}$ approximations or Brueckner theory.