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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 Green's function to be used in the above integral is one which vanishes on the boundary: (,) = for and . Such a Green's function is usually a sum of the free-field Green's function and a harmonic solution to the differential equation.
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.
Reciprocity is also a basic lemma that is used to prove other theorems about electromagnetic systems, such as the symmetry of the impedance matrix and scattering matrix, symmetries of Green's functions for use in boundary-element and transfer-matrix computational methods, as well as orthogonality properties of harmonic modes in waveguide ...
When the transfer function and the Laplace transform of the input are known, this convolution may be more complicated than the alternative of multiplying two functions in the frequency domain. The impulse response, considered as a Green's function, can be thought of as an "influence function": how a point of input influences output.
In mathematics, influence function is used to mean either: a synonym for a Green's function ; Influence function (statistics) , the effect on an estimator of changing one point of the sample