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Green's functions can be expanded in terms of the basis elements (harmonic functions) which are determined using the separable coordinate systems for the linear partial differential equation. There are many expansions in terms of special functions for the Green's function. In the case of a boundary put at infinity with the boundary condition ...
Green's functions are also useful tools in solving wave equations and diffusion equations. In quantum mechanics, Green's function of the Hamiltonian is a key concept with important links to the concept of density of states. The Green's function as used in physics is usually defined with the opposite sign, instead.
In many-body theory, the term Green's function (or Green function) is sometimes used interchangeably with correlation function, but refers specifically to correlators of field operators or creation and annihilation operators. The name comes from the Green's functions used to solve inhomogeneous differential equations, to which they are loosely ...
See Green's functions for the Laplacian or [2] for a detailed argument, with an alternative. It can be further verified that the above identity also applies when ψ is a solution to the Helmholtz equation or wave equation and G is the appropriate Green's function.
The GW approximation (GWA) is an approximation made in order to calculate the self-energy of a many-body system of electrons. [1] [2] [3] The approximation is that the expansion of the self-energy Σ in terms of the single particle Green's function G and the screened Coulomb interaction W (in units of =)
The main mathematical object in the Keldysh formalism is the non-equilibrium Green's function (NEGF), which is a two-point function of particle fields. In this way, it resembles the Matsubara formalism , which is based on equilibrium Green functions in imaginary-time and treats only equilibrium systems.
In quantum field theory, correlation functions, often referred to as correlators or Green's functions, are vacuum expectation values of time-ordered products of field operators. They are a key object of study in quantum field theory where they can be used to calculate various observables such as S-matrix elements.
For the imaginary time Schrödinger equation, instead, we propagate forward in time using a convolution integral with a special function called a Green's function. So we get (, +) = (, ′,) (′,) ′. Similarly to classical mechanics, we can only propagate for small slices of time; otherwise the Green's function is inaccurate.