What is Green function in mathematical physics?

The Green function is the kernel of the integral operator inverse to the differential operator generated by the given differential equation and the homogeneous boundary conditions.

How do you solve Green’s function?

These equations can be written in the more compact forms L[y]=f(x)L[G]=δ(x−ξ). Using these equations, we can determine the solution, y(x), in terms of the Green’s function. Multiplying the first equation by G(x,ξ), the second equation by y(x), and then subtracting, we have GL[y]−yL[G]=f(x)G(x,ξ)−δ(x−ξ)y(x).

What is Green function in scattering theory?

The Green’s function G(→r,→k) is essentially the inverse of the differential operator, (ℏ22m∇2+Ek)G(→r,→k)=δ(→r). This is not a mathematically unique definition: clearly, we can add to G(→r,→k) any solution of the homogeneous equation.

What is Green function in electromagnetic field?

The Green function of a wave equation is the solution of the wave equation for a point source [2]. And when the solution to the wave equation due to a point source is known, the solution due to a general source can be obtained by the principle of linear superposition (see Figure 1).

What are the properties of Green function?

The Green’s function satisfies a homogeneous differential equation for x≠ξ, ∂∂x(p(x)∂G(x,ξ)∂x)+q(x)G(x,ξ)=0,x≠ξ. In the case of the step function, the derivative is zero everywhere except at the jump. At the jump, there is an infinite slope, though technically, we have learned that there is no derivative at this point.

How is Green function derived?

the Green’s function G is the solution of the equation LG = δ, where δ is Dirac’s delta function; the solution of the initial-value problem Ly = f is the convolution (G ⁎ f), where G is the Green’s function.

Is Green function continuous?

The Green function of L is the function G(x,ξ) that satisfies the following conditions: 1) G(x,ξ) is continuous and has continuous derivatives with respect to x up to order n−2 for all values of x and ξ in the interval [a,b].

Is Green function symmetric?

Symmetry of Green’s function certainly can be viewed as a proof of symmetry. However, it would be satisfying if there was a direct argument in the language of the definition of the Green’s function. , and so these two expressions are equal. Summing over all such two-way paths, and then all m gives the result.

Is it Green’s function or Green function?

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.

Who invented Green’s function?

5.1 Overview. Green functions1 are named after the mathematician and physicist George Green born in Nottingham in 1793 who ‘invented’ the Green function in 1828.

What is the scattering theory?

In mathematics and physics, scattering theory is a framework for studying and understanding the scattering of waves and particles. Wave scattering corresponds to the collision and scattering of a wave with some material object, for instance sunlight scattered by rain drops to form a rainbow.

What is Helmholtz wave equation?

Helmholtz’s equation, named after Hermann von Helmholtz, is used in Physics and Mathematics. It is a partial differential equation and its mathematical formula is: ▽ 2 A + k 2 A = 0.

What is Laplace equation in maths?

Laplace’s equation is a special case of Poisson’s equation ∇2R = f, in which the function f is equal to zero. Many physical systems are more conveniently described by the use of spherical or cylindrical coordinate systems.

How do you find non homogeneous differential equations?

  1. a2(x)y″+a1(x)y′+a0(x)y=r(x).
  2. Also, let c1y1(x)+c2y2(x) denote the general solution to the complementary equation. Then, the general solution to the nonhomogeneous equation is given by.
  3. y(x)=c1y1(x)+c2y2(x)+yp(x).

What is Green function in integral equation?

The Green’s function integral equation method (GFIEM) is a method for solving linear differential equations by expressing the solution in terms of an integral equation, where the integral involves an overlap integral between the solution itself and a Green’s function.

What is non homogeneous function?

(Non) Homogeneous systems. Definition 1 A linear system of equations Ax = b is called homogeneous if b = 0, and non-homogeneous if b = 0.

Who is Green’s theorem named after?

The Gauss-Green-Stokes theorem, named after Gauss and two leading English applied mathematicians of the 19th century (George Stokes and George Green), generalizes the fundamental theorem of the calculus to functions of several variables.…

Who discovered Stokes Theorem?

It is named after Sir George Gabriel Stokes (1819–1903), although the first known statement of the theorem is by William Thomson (Lord Kelvin) and appears in a letter of his to Stokes in July 1850. The theorem acquired its name from Stokes’s habit of including it in the Cambridge prize examinations.

What is the formula of scattering?

The “differential cross-section”, dσ/dθ, with respect to the scattering angle is the number of scatterings between θ and θ + dθ per unit flux, per unit range of angle, i.e. dσ dθ = dN(θ) F dθ = π D2 4 cos(θ/2) sin3(θ/2) . dΩ = F dσ dΩ . in analogy to the relation for differential angle dθ.

Why scattering is so important in physics?

Scattering theory is important as it underpins one of the most ubiquitous tools in physics. Almost everything we know about nuclear and atomic physics has been discovered by scattering experiments, e.g. Rutherford’s discovery of the nucleus, the discovery of sub-atomic particles (such as quarks), etc.

What is meant by phase factor?

The phase factor is a unit complex number, i.e. a complex number of absolute value 1. It is commonly used in quantum mechanics. The variable θ appearing in such an expression is generally referred to as the phase.

What is Helmholtz Principle?

In its stronger form, of which we will make great use, the Helmholtz principle states that whenever some large deviation from randomness occurs, a structure is perceived. As a commonsense statement, it states that “we immediately perceive whatever could not happen by chance”.

What is Helmholtz law?

Helmholtz’s first theorem. The strength of a vortex filament is constant along its length. Helmholtz’s second theorem. A vortex filament cannot end in a fluid; it must extend to the boundaries of the fluid or form a closed path.

What are the types of Laplace transform?

Laplace transform is divided into two types, namely one-sided Laplace transformation and two-sided Laplace transformation.

What is the use of Poisson equation?

Poisson’s equation is one of the pivotal parts of Electrostatics, where we would solve the equation to find electric potential from a given charge distribution. In layman’s terms, we can use Poisson’s Equation to describe the static electricity of an object.

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