perturbation

perturbation (pŭr'tərbā`shən), in astronomy and physics, small force or other influence that modifies the otherwise simple motion of some object. The term is also used for the effect produced by the perturbation, e.g., a change in the object's energy or path of motion. One important effect of perturbations is the advance, or precession, of the perihelion of a planet, which can be described as a slow rotation of the entire planetary orbit. A residual advance in the perihelion of Mercury provided a valuable test of Einstein's general theory of relativity.

In the solar system the dominant force is the gravitational force exerted by the sun on each planet; assuming that this is the only force, the simple elliptical orbits described by Kepler's laws are derived. However, the perturbations caused by the gravitational interaction of the planets among themselves change and complicate the curve of these orbits. The study of perturbations has led to important discoveries in astronomy. Within the solar system, the existence and position of Neptune was predicted because of the deviations of Uranus from its computed path. Likewise, Pluto was discovered by its effect on Neptune. Beyond the confines of the solar system, perturbations in the orbits of stars caused by the gravitational forces of orbiting bodies have led to the discovery of a number of extrasolar planetary systems.

In the atom the dominant force is the electrical force between the nucleus and the electrons; this force determines the characteristic structure, or energy levels, of the atom. The forces exerted by the electrons among themselves are perturbations that slightly modify this structure.

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perturbation

A disturbance or irregularity. For example, a "perturbation in an input signal" that is not properly dealt with may cause erroneous output or a system failure.

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perturbation

1. Physics a secondary influence on a system that modifies simple behaviour, such as the effect of the other electrons on one electron in an atom

2. Astronomy a small continuous deviation in the inclination and eccentricity of the orbit of a planet or comet, due to the attraction of neighbouring planets

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Perturbation (quantum mechanics)

An expansion technique useful for solving complicated quantum-mechanical problems in terms of solutions for simple problems. Perturbation theory in quantum mechanics provides an approximation scheme whereby the physical properties of a system, modeled mathematically by a quantum-mechanical description, can be estimated to a required degree of accuracy. Such a scheme is useful because very few problems occurring in quantum mechanics can be solved analytically. Consequently an approximation technique must be employed in order to give an approximate analytic solution or to provide suitable algorithms for a numerical solution. Even for problems which admit an exact analytic solution, the exact solution may be of such mathematical complexity that its physical interpretation is not apparent. For these situations, perturbation techniques are also desirable.

Here the discussion of the application of perturbation techniques to quantum mechanics is limited to the domain of nonrelativistic quantum theory. Applications of a similar but mathematically more intricate nature have also been made in quantum electrodynamics and quantum field theory. See Quantum electrodynamics, Quantum field theory, Quantum mechanics

Perturbation theory is applied to the Schrdinger equation, HΨ = (H₀ + λV)Ψ = iℏ(∂/∂t)Ψ [where ℏ is Planck's constant h divided by 2π, and (∂/∂t) represents partial differentiation with respect to the time variable t], for which the exact hamiltonian H is split into two parts: the approximate (unperturbed) time-independent hamiltonian H₀ whose solutions of the corresponding Schrdinger equation are known analytically, and the perturbing potential λV. The basic idea is to expand the exact solution Ψ in terms of the solution set of the unperturbed hamiltonian H₀ by means of a power series in the coupling constant λ. Such a procedure is expected to be successful if the system characterized by the unperturbed hamiltonian closely resembles that characterized by the exact hamiltonian. Supposedly the differences are not singular in character, but change as a continuous function of the parameter λ.

Perturbation theory is used in two contexts to provide information about the state of the system, which in quantum mechanics is determined by the wave function Ψ. If λV is time-independent, an objective may be to find the stationary states of the system Ψ_n whose time dependence is given by exp (-iE_nt/ℏ), where i = and E_n represents the energy of the stationary state labeled by n. If λV is either time-independent or time-dependent, an objective may be to find the time evolution of a state which at some specified time was a stationary state of the unperturbed hamiltonian. The perturbing potential is then considered as causing transitions from the original state to other states of the unperturbed hamiltonian, and application of time-dependent perturbation theory provides the probability of such transitions.