Publication

Effective-State Approach to 2nd-Order Perturbation-Theory

Alfredo Pasquarello
1992
Journal Articles

Abstract

We show that the complete set of intermediate states in second-order perturbation expressions can be replaced by a finite number of states, provided the subspace they span contains one effective state, solution of an inhomogeneous equation. It is therefore not necessary that the subspace be complete. The existence of this effective state guides the choice of the basis in approximate calculations. As an example, we apply these considerations to two-photon transition rates in hydrogen.

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Alfredo Pasquarello
1992
Journal Articles

Abstract

Official source

Related concepts (21)

Perturbation theory

In mathematics and applied mathematics, perturbation theory comprises methods for finding an approximate solution to a problem, by starting from the exact solution of a related, simpler problem. A critical feature of the technique is a middle step that breaks the problem into "solvable" and "perturbative" parts. In perturbation theory, the solution is expressed as a power series in a small parameter . The first term is the known solution to the solvable problem. Successive terms in the series at higher powers of usually become smaller.

Wave equation

The (two-way) wave equation is a second-order linear partial differential equation for the description of waves or standing wave fields - as they occur in classical physics - such as mechanical waves (e.g. water waves, sound waves and seismic waves) or electromagnetic waves (including light waves). It arises in fields like acoustics, electromagnetism, and fluid dynamics. Single mechanical or electromagnetic waves propagating in a pre-defined direction can also be described with the first-order one-way wave equation, which is much easier to solve and also valid for inhomogeneous media.

Helmholtz equation

In mathematics, the Helmholtz equation is the eigenvalue problem for the Laplace operator. It corresponds to the linear partial differential equation where ∇2 is the Laplace operator, k2 is the eigenvalue, and f is the (eigen)function. When the equation is applied to waves, k is known as the wave number. The Helmholtz equation has a variety of applications in physics, including the wave equation and the diffusion equation, and it has uses in other sciences.

Related publications (1)

Effective-State Approach to 2nd-Order Perturbation-Theory

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MATHICSE Technical Report : A stochastic collocation method for the second order wave equation with a discontiuous random speed