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.

Sep 1, 20260 citations

Thermal Green functions

C. Di Castro, Roberto Raimondi

In Chapter 9 we applied the second quantization to the Hartree–Fock method. In this chapter we provide the basic aspects of the thermal Green function method, which is convenient for systematic perturbative expansion beyond the mean-field approximation. Several problems in quantum statistical mechanics can be elegantly and compactly described in terms of the so-called Matsubara Green functions (Matsubara (1955); Abrikosov et al. (1963)). Hence, if on one hand, the density matrix description, often used in the previous chapters, has a more direct physical interpretation, on the other hand the Green function method is far more powerful and takes advantage of a handy diagrammatic perturbation expansion. In the following chapters, we rely on the Green function for a more complete treatment of some many-body problems. To be self contained we introduce here this technical chapter with a short presentation of the by now standard Green function method. For a comprehensive treatment we refer the reader to the many books available and in particular to Abrikosov et al. (1963). The Matsubara Green function The main aim of quantum statistical mechanics is the evaluation of the partition function (see Eqs. (6.31) and (6.32)), e.g. in the grand canonical ensemble In the case of quantum gases one can evaluate Ƶ exactly as we did in Chapter 7. In general, the evaluation of the partition function is a difficult task and it is useful to use perturbation theory to obtain approximate solutions. To set up a perturbation theory, one begins by splitting the Hamiltonian as H = H 0 + H i, where H 0 represents the “easy” part, which can be solved exactly. In the second step, one could naturally think of expanding the exponential of the interacting part H i in order to write the partition function as a perturbative series. However, such a procedure requires extreme care due to the non-commutative nature of operators in quantum mechanics. The way of solving this problem is via the “ time ” ordering of operators in each term of the perturbative expansion. It can be shown, as we shall see in the present chapter, that the quantum statistical average of any time-ordered sequence of operators with respect to the unperturbed Hamiltonian H 0 can be expressed in terms of elementary quantum statistical averages of two field operators. Such elementary averages are called Green functions and in this section we introduce the basic definitions concerning them.

Cambridge University Press eBooksAug 27, 20150 citations