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Equations for one- and two-particle density matrices in Green's function method. Comparison with the random phase approximation

S. P. Kamerdzhiev

Published Oct 1, 1973• 0 citations
OpenAlex

Abstract

Green's function method is applied to the description of nuclear levels-- superposition of the 1p1h and 2p2h configurations. Irreducible amplitudes for the transitions 1p1h yields 1p1h, 1p1h yields 2p2h, 2p2h yields 1p1h, 2p2h yields 2p2h are introduced to obtain the exact equations for eigerfunctions and eigenvalues of the problem-. The renormalization of the equations made it possible to consider all the processes near the Fermi level using the phenomenological effective amplitudes of quasipanticle interaction. The comparison is made with the generalized random-phase method. The approximations are discussed leading in the Green's function method to the equations for one- and two-particle density matrices as the equations obtained in the random--phase approximation. (auth)

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Green's function method is applied to the description of nuclear levels-- superposition of the 1p1h and 2p2h configurations. Irreducible amplitudes for the transitions 1p1h yields 1p1h, 1p1h yields 2p2h, 2p2h yields 1p1h, 2p2h yields 2p2h are introduced to obtain the exact equations for eigerfunctions and eigenvalues of the problem-. The renormalization of the equations made it possible to consider all the processes near the Fermi level using the phenomenological effective amplitudes of quasipanticle interaction. The comparison is made with the generalized random-phase method. The approximations are discussed leading in the Green's function method to the equations for one- and two-particle density matrices as the equations obtained in the random--phase approximation. (auth)
We present an elementary and fairly detailed review of several Green’s function methods for treating nuclear and other many-body systems. We first treat the single-particle Green’s function, by way of which some details concerning linked diagram expansion, rules for evaluating Green’s function diagrams and solution of the Dyson’s integral equation for Green’s function are exhibited. The particle-particle hole-hole (pphh) Green’s function is then considered, and a specific time-blocking technique is discussed. This technique enables us to have a one-frequency Dyson’s equation for the pphh and similarly for other Green’s functions, thus considerably facilitating their calculation. A third type of Green’s function considered is the particle-hole Green’s function. RPA and high order RPA are treated, along with examples for setting up particle-hole RPA equations. A general method for deriving a model-space Dyson’s equation for Green’s functions is discussed. We also discuss a method for determining the normalization of Green’s function transition amplitudes based on its vertex function. Some applications of Green’s function methods to nuclear structure and recent deep inelastic lepton-nucleus scattering are addressed.
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.
The problem of the microscopic description of excited states of the even-even open-shell atomic nuclei is considered. A model is formulated which allows one to go beyond the quasiparticle random phase approximation. The physical content of the model is determined by the quasiparticle time blocking approximation (QTBA) which enables one to include contributions of the two-quasiparticle and the two-phonon configurations, while excluding (blocking) more complicated intermediate states. In addition, the QTBA ensures consistent treatment of ground state correlations in the Fermi systems with pairing. The model is based on the generalized Green function formalism (GGFF) in which the normal and the anomalous Green functions are treated in a unified way in terms of the components of generalized Green functions in a space that is double the size of the usual single-particle space. Modification of the GGFF is considered in the case when the many-body nuclear Hamiltonian contains two-, three-, and other many-particle effective forces.
The problem of correlations is considered in many-fermion systems, although with some modifications the methods are more general. The many-body problem is formulated in terms of reduced one- and two-particle density matrices, and the equations for these are treated by variational and iterative methods. The formalism is applied to a calculation of the energy per particle for a system interacting with Coulomb, screened Coulomb, and singular potentials. Comparisons with the ideas of Brueckner and others are made, and difficulties are discussed.