Green’s Function Method

Giampaolo Co’

T.T.S. Kuo, Yiharn Tzeng

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.

Т. Б. Қоштыбаев, Sh. Aliev, А.Т. Жавлиева +1 more

This article is devoted to the theoretical investigation of processes in macroscopic systems based on wave particle duality and quantum statistical distributions using the Green’s function method. In particular, it pro vides a detailed description of the properties and characteristics of the Green’s function for an ideal Fermi liquid, single- and two-particle Green’s functions, as well as Green’s functions at limiting temperatures and for the Dirac and Schrödinger equations. The study analyzes the application of the Green’s function method to processes in many-particle quantum systems and to solving equations that describe the constituent particles of such systems. For a given operator and set of boundary conditions, a Green’s function can be defined, al lowing the solution of the corresponding equation to be expressed in integral form. A distinctive feature of this method is the application of mathematical-physics tools, such as Fourier methods, integral transforms, special functions, and mathematical analysis, to problems in quantum physics governed by the principles of quantum and statistical mechanics. The relevance of this study stems from the application of the Green’s function method to the analysis of quantum phenomena. This approach is considered one of the effective methods for solving partial differential equations describing various physical processes. It involves transform ing the original differential equation into an equivalent integral equation involving Green’s functions, which can then be solved. The transformation involves passing to the momentum representation and applying Fouri er transforms using methods of complex analysis.

Ingvar Lindgren

Gin-yih Tsaur, Jyhpyng Wang

The initial value problem in quantum mechanics is most conveniently solved by the Green function method. Instead of the conventional methods of eigenfunction expansion and path integration, we present a new method for constructing the Green functions systematically. By using suitable elementary transformations, one of the conjugate variables in the Hamiltonian can be eliminated and the Green function for the simplified Hamiltonian can be easily derived. We then obtain the Green function for the original Hamiltonian by the reverse sequence of the elementary transformations. The method is illustrated for the linear potential, the harmonic oscillator, the centrifugal potential, and the centripetal barrier oscillator.

Wolfgang Nolting