Abstract
The Green's function method has applications in several fields in Physics, from classical differential equations to quantum many-body problems. In the quantum context, Green's functions are correlation functions, from which it is possible to extract information from the system under study, such as the density of states, relaxation times and response functions. Despite its power and versatility, it is known as a laborious and sometimes cumbersome method. Here we introduce the equilibrium Green's functions and the equation-of-motion technique, exemplifying the method in discrete lattices of non-interacting electrons. We start with simple models, such as the two-site molecule, the infinite and semi-infinite one-dimensional chains, and the two-dimensional ladder. Numerical implementations are developed via the recursive Green's function, implemented in Julia, an open-source, efficient and easy-to-learn scientific language. We also present a new variation of the surface recursive Green's function method, which can be of interest when simulating simultaneously the properties of surface and bulk.
Highlights
The Green’s functions method is a powerful mathematical tool to solve linear differential equations
Before we examine the development of the Green’s functions in quantum mechanics, we shall review some of the general properties of a Green’s function
Which verifies to be a correct result in electrostatics [11]. This is a quite simple example, the Green’s function technique as presented can be applied to other physical problems described by linear differential equations
Summary
The Green’s functions method is a powerful mathematical tool to solve linear differential equations. These functions were named after the English miller,. Physicist and mathematician George Green (17931841) [1,2,3] His seminal work “An essay on the application of mathematical analysis to the theories of electricity and magnetism” (1828) [4] developed a theory of partial differential equations with gen-. The Green’s functions were born as auxiliary functions for solving boundary-value problems. The latter are differential equations with constraining boundary conditions, which specify values that the solution or its normal derivative take on the boundary of the domain. Before we examine the development of the Green’s functions in quantum mechanics, we shall review some of the general properties of a Green’s function
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