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

Zeitschrift für Naturforschung A

Erweiterung der Neuen Tamm-Dancoff-Methode auf Phasen- übergänge in Vielteilchensystemen / Extension of the New Tamm-Dancoff Method to Phase Transitions in Many-Body Systems

A. Friederich, W. Gerling, K. Bleuler

Published Feb 1, 1975• 3 citations
Publisher

Abstract

Abstract We extend the New Tamm-Dancoff method by introducing intermediate states. In this way we are able to treat with the Green's function method the effect of nearby levels in many-body systems. We formulate the η-and the ζ-function method with intermediate states. Aldready in first order, the ζ-function method yields a whole series of new approximations in addition to known theories such as the Hartree-Fock theory and the Hartree-Bogoliubov theory. As an example we study intensively "the Hartree-Fock theory with intermediate states". The ζ-function method which is based on the η-function method yields in first order in addition to RPA, quasi-particle RPA and other approximations "RPA with intermediate states". We apply the Hartree-Fock theory with intermediate states and RPA with intermediate states to the exactly solvable Lipkin model.

DOI
10.1515/zna-1975-0204

Related papers

Wiek-Regeln mit Zwischenzuständen und die Neue Tamm-Dancoff-Methode mit Zwischenzuständen / On Wick Rules with Intermediate States and the New Tamm-Dancoff Method with Intermediate States

A. Friederich, W. Gerling

Abstract Instead of emphasizing the ground state as is done in Green's function method, we take a finite-dimensional subspace of the Hilbert space: the space of the "intermediate states". A systematic introduction of intermediate states is effected by an extension of the method of generating functionals: we combine the generating functionals of the n-point Green's functions to a "matrix functional" T, and form new matrix functionals, which are matrix functions of T. The aim of this paper is to develop the functional calculus in such a way that the transition from scalar functionals to matrix functionals is straightforward, and the method of obtaining further results becomes clear. Following the lines of Dürr and Wagner we get u η-and ζ-rules with intermediate states". Using them we define a truncation procedure for the equations of motion of the n-point Green's functions, the "New Tamm-Dancoff method with intermediate states". This extension makes it possible to treat the effect of nearby levels in many body systems with Green's functions. In ad-dition to well-known approximations, such as the Hartree-Fock and the Hartree-Bogoliubov theory, the RPA and the quasiparticle RPA, we obtain a series of new approximations. Among these are the "Hartree-Fock theory with intermediate states" and the "random-phase approximation with intermediate states", which we already applied with great success to some exactly soluble models.

Zeitschrift für Naturforschung A•Aug 1, 1976•1 citation

Comparison of Heisenberg’s New Tamm-Dancoff Method with Approximation Schemes used in Many-Body Theory

K. Bleuler, A. Friederich, Herbert Rainer Petry +1 more

In the theoretical investigation of nonrelativistic feimion systems, several characteristic approximation schemes play a decisive role. Examples are the BCS theory of superconductivity, the Hartree-Fock method in atomic and molecular physics and its generalization by means of the Bogoliubov theory (containing also BCS as a special case) used in nuclear physics. Furthermore, the Random-Phase-Approximation (RPA) is of fundamental importance in plasma physics as well as in the explanation of the well known vibrational states of nuclei. In this connection, it should be pointed out that the socälled nuclear models, which to a certain extent brought nuclear physics into miscredit, represent in fact nothing else but a more or less intuitive substitute for special well defined approximation schemes: the introduction of an average potential in heavy nuclei stands for Hartree-Fock, whereas the so-called surface vibrations visualize RPA. In addition, the possibility to describe elementary excitations by the naive concept of independent particle motion is justified theoretically by the Bogoliubov transformation which leads to the so-called quasi-particles.

Vieweg+Teubner eBooks•Jan 1, 1971•3 citations

Many-body methods within the nuclear Lipkin model

Radek Folprecht

Title: Many-body methods within the nuclear Lipkin model Author: Radek Folprecht Institute: Institute of Particle and Nuclear Physics Supervisor: Mgr. František Knapp Ph.D., Institute of Particle and Nuclear Physics Abstract: The subject of the present thesis is the Lipkin model, a two- level degenerated quantum system enhanced by two-body interactions. The exact solutions and properties of the model are reviewed and summarized. Several many-body methods common in nuclear physics are used to solve the eigenvalue problem of the time-independent Schrödinger equation. The many-body methods applied to the Lipkin model in the present thesis are the Hartree-Fock mean-field method, the random phase approximation, the Tamm-Dancoff approximation, and the Bardeen-Cooper-Schrieffer method. We propose an instructive iterative extension to the random phase approxi- mation method. For the listed methods, we provide a comprehensive com- parison of the approximation quality for either the ground state energy or excited state energies. Keywords: Lipkin model, Hartree-Fock mean-field method, random phase approximation, Bardeen-Cooper-Schrieffer method 1

•Jan 1, 2023•0 citations

A New Tamm-Dancoff Method Based on the SO (2N+1) Regular Representation of Fermion Many-Body Systems

Hideo Fukutome

A new Tamm-Dancoff method is developed on the basis of the SO(2N + 1) regular representation formalism of Fermion many-body systems. The Schrodinger equation in the SO(2N + 1) generator coordinate representation is decomposed into symmetry subspaces by means of a Peierls-Yoccoz, projection procedure. The decomposed Schrödinger equation is transformed to a quasi-particle frame by a transformation of the generator coordinate and expanded by the symmetry projected Tamm-Dancoff bases. The Tamm-Danco[f equation and wave function thus obtained satisfy all symmetry requirements at every order of the expansion. The first order Tamm-Dancoff wave function is neither an independent particle nor quasi-particle approximation but represents Bose condensed Fermion pairs (and an unpaired particle) performing a coherent motion. The higher order terms represent fluctuations of Fermion pairs (and the unpaired particle) from the Bose condensate.

Progress of Theoretical Physics•Dec 1, 1978•18 citations

Equations-of-Motion Method and the Extended Shell Model

DAVID J. ROWE

This paper presents the equations-of-motion method as a useful and flexible tool in the study of nuclear spectroscopy. It is partly a review, but also it introduces a new and much more powerful equations-of-motion technique which supercedes the older linearization methods. The older methods worked with operator equations. To obtain closed expressions they had to be linearized in a rather arbitrary manner. The present approach works with the ground-state expectation of operator equations and thereby avoids all problems of linearization. Thus, like the Green's function method, the equations-of-motion method becomes potentially exact. It has many advantages over Green's function methods, however, among which are its greater compactness, simplicity, and the physical insight it yields.The method is first applied to rederive the random phase approximation (RPA) and the quasi-particle RPA (QRPA) and to show precisely what terms they neglect. It is demonstrated that some of these terms have coherent phases. A higher RPA and QRPA are then derived to include these terms. The corrections have some interesting effects: notably, there is a reduction of the effective interaction strength and a stabilization of the nucleus against sudden phase transitions. The equations-of-motion method is also used to generalize, in a very simple and natural way, the Hartree-Fock (HF) and Hartree-Bogolyubov (HB) concepts of independent particles and quasi-particles to nonsimple ground states.The equations-of-motion method is presented as a simple extension of the shell model to the treatment of excitationg of a correlated ground state. By concentrating on the quantities of direct physical interest, the complexity of workins with correlated wavefunctions is avoided.

Reviews of Modern Physics•Jan 1, 1968•1,127 citations

Análise da aproximação Tamm-Dancoff + projeção em um modelo solúvel

Akiyoshi Mizukami, Sergio Augusto Carias

O método de três aproximações sucessivas. Hartree-Fock + Tamm-Dancoff + Projeção é aplicado a um modelo solúvel devido a Lipkin e colaboradores constituído de N fermions distribuídos em dois níveis de energia N-degenerados e com interação via monopolo-monopolo. Testa-se o método comparando seus resultados com as soluções exatas. The three sucessives aproximation Hartree-Fock + Tamm-Dancoff + Projection for many bodies tratament are applied to a model with exact solution developed Lipkin et al. Luck model is constitued of N fermions in two N-fold degenerated levels and interacting via monopole-monopole. The metod is tested by comparing its results with the exact solutions.

Semina Ciências Biológicas e da Saúde•Feb 1, 1980•0 citations
Abstract We extend the New Tamm-Dancoff method by introducing intermediate states. In this way we are able to treat with the Green's function method the effect of nearby levels in many-body systems. We formulate the η-and the ζ-function method with intermediate states. Aldready in first order, the ζ-function method yields a whole series of new approximations in addition to known theories such as the Hartree-Fock theory and the Hartree-Bogoliubov theory. As an example we study intensively "the Hartree-Fock theory with intermediate states". The ζ-function method which is based on the η-function method yields in first order in addition to RPA, quasi-particle RPA and other approximations "RPA with intermediate states". We apply the Hartree-Fock theory with intermediate states and RPA with intermediate states to the exactly solvable Lipkin model.
Abstract Instead of emphasizing the ground state as is done in Green's function method, we take a finite-dimensional subspace of the Hilbert space: the space of the "intermediate states". A systematic introduction of intermediate states is effected by an extension of the method of generating functionals: we combine the generating functionals of the n-point Green's functions to a "matrix functional" T, and form new matrix functionals, which are matrix functions of T. The aim of this paper is to develop the functional calculus in such a way that the transition from scalar functionals to matrix functionals is straightforward, and the method of obtaining further results becomes clear. Following the lines of Dürr and Wagner we get u η-and ζ-rules with intermediate states". Using them we define a truncation procedure for the equations of motion of the n-point Green's functions, the "New Tamm-Dancoff method with intermediate states". This extension makes it possible to treat the effect of nearby levels in many body systems with Green's functions. In ad-dition to well-known approximations, such as the Hartree-Fock and the Hartree-Bogoliubov theory, the RPA and the quasiparticle RPA, we obtain a series of new approximations. Among these are the "Hartree-Fock theory with intermediate states" and the "random-phase approximation with intermediate states", which we already applied with great success to some exactly soluble models.
In the theoretical investigation of nonrelativistic feimion systems, several characteristic approximation schemes play a decisive role. Examples are the BCS theory of superconductivity, the Hartree-Fock method in atomic and molecular physics and its generalization by means of the Bogoliubov theory (containing also BCS as a special case) used in nuclear physics. Furthermore, the Random-Phase-Approximation (RPA) is of fundamental importance in plasma physics as well as in the explanation of the well known vibrational states of nuclei. In this connection, it should be pointed out that the socälled nuclear models, which to a certain extent brought nuclear physics into miscredit, represent in fact nothing else but a more or less intuitive substitute for special well defined approximation schemes: the introduction of an average potential in heavy nuclei stands for Hartree-Fock, whereas the so-called surface vibrations visualize RPA. In addition, the possibility to describe elementary excitations by the naive concept of independent particle motion is justified theoretically by the Bogoliubov transformation which leads to the so-called quasi-particles.
Title: Many-body methods within the nuclear Lipkin model Author: Radek Folprecht Institute: Institute of Particle and Nuclear Physics Supervisor: Mgr. František Knapp Ph.D., Institute of Particle and Nuclear Physics Abstract: The subject of the present thesis is the Lipkin model, a two- level degenerated quantum system enhanced by two-body interactions. The exact solutions and properties of the model are reviewed and summarized. Several many-body methods common in nuclear physics are used to solve the eigenvalue problem of the time-independent Schrödinger equation. The many-body methods applied to the Lipkin model in the present thesis are the Hartree-Fock mean-field method, the random phase approximation, the Tamm-Dancoff approximation, and the Bardeen-Cooper-Schrieffer method. We propose an instructive iterative extension to the random phase approxi- mation method. For the listed methods, we provide a comprehensive com- parison of the approximation quality for either the ground state energy or excited state energies. Keywords: Lipkin model, Hartree-Fock mean-field method, random phase approximation, Bardeen-Cooper-Schrieffer method 1
A new Tamm-Dancoff method is developed on the basis of the SO(2N + 1) regular representation formalism of Fermion many-body systems. The Schrodinger equation in the SO(2N + 1) generator coordinate representation is decomposed into symmetry subspaces by means of a Peierls-Yoccoz, projection procedure. The decomposed Schrödinger equation is transformed to a quasi-particle frame by a transformation of the generator coordinate and expanded by the symmetry projected Tamm-Dancoff bases. The Tamm-Danco[f equation and wave function thus obtained satisfy all symmetry requirements at every order of the expansion. The first order Tamm-Dancoff wave function is neither an independent particle nor quasi-particle approximation but represents Bose condensed Fermion pairs (and an unpaired particle) performing a coherent motion. The higher order terms represent fluctuations of Fermion pairs (and the unpaired particle) from the Bose condensate.
This paper presents the equations-of-motion method as a useful and flexible tool in the study of nuclear spectroscopy. It is partly a review, but also it introduces a new and much more powerful equations-of-motion technique which supercedes the older linearization methods. The older methods worked with operator equations. To obtain closed expressions they had to be linearized in a rather arbitrary manner. The present approach works with the ground-state expectation of operator equations and thereby avoids all problems of linearization. Thus, like the Green's function method, the equations-of-motion method becomes potentially exact. It has many advantages over Green's function methods, however, among which are its greater compactness, simplicity, and the physical insight it yields.The method is first applied to rederive the random phase approximation (RPA) and the quasi-particle RPA (QRPA) and to show precisely what terms they neglect. It is demonstrated that some of these terms have coherent phases. A higher RPA and QRPA are then derived to include these terms. The corrections have some interesting effects: notably, there is a reduction of the effective interaction strength and a stabilization of the nucleus against sudden phase transitions. The equations-of-motion method is also used to generalize, in a very simple and natural way, the Hartree-Fock (HF) and Hartree-Bogolyubov (HB) concepts of independent particles and quasi-particles to nonsimple ground states.The equations-of-motion method is presented as a simple extension of the shell model to the treatment of excitationg of a correlated ground state. By concentrating on the quantities of direct physical interest, the complexity of workins with correlated wavefunctions is avoided.
O método de três aproximações sucessivas. Hartree-Fock + Tamm-Dancoff + Projeção é aplicado a um modelo solúvel devido a Lipkin e colaboradores constituído de N fermions distribuídos em dois níveis de energia N-degenerados e com interação via monopolo-monopolo. Testa-se o método comparando seus resultados com as soluções exatas. The three sucessives aproximation Hartree-Fock + Tamm-Dancoff + Projection for many bodies tratament are applied to a model with exact solution developed Lipkin et al. Luck model is constitued of N fermions in two N-fold degenerated levels and interacting via monopole-monopole. The metod is tested by comparing its results with the exact solutions.