perturbation(redirected from perturbational)
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perturbation(pŭr'tərbā`shən), in astronomy and physics, small forceforce,
commonly, a "push" or "pull," more properly defined in physics as a quantity that changes the motion, size, or shape of a body. Force is a vector quantity, having both magnitude and direction.
..... Click the link for more information. 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 energyenergy,
in physics, the ability or capacity to do work or to produce change. Forms of energy include heat, light, sound, electricity, and chemical energy. Energy and work are measured in the same units—foot-pounds, joules, ergs, or some other, depending on the system of
..... Click the link for more information. or path of motion. One important effect of perturbations is the advance, or precession, of the perihelionperihelion
, point nearest the sun in the orbit of a body about the sun. See apsis.
..... Click the link for more information. 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 relativityrelativity,
physical theory, introduced by Albert Einstein, that discards the concept of absolute motion and instead treats only relative motion between two systems or frames of reference.
..... Click the link for more information. .
In the solar systemsolar system,
the sun and the surrounding planets, natural satellites, dwarf planets, asteroids, meteoroids, and comets that are bound by its gravity. The sun is by far the most massive part of the solar system, containing almost 99.9% of the system's total mass.
..... Click the link for more information. 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 lawsKepler's laws,
three mathematical statements formulated by the German astronomer Johannes Kepler that accurately describe the revolutions of the planets around the sun. Kepler's laws opened the way for the development of celestial mechanics, i.e.
..... Click the link for more information. 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 systemsplanetary system,
a star and all the celestial bodies bound to it by gravity, especially planets and their natural satellites. Until the last decade of the 20th cent., the only planetary system known was the solar system, which comprises the sun and the surrounding planets,
..... Click the link for more information. .
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.
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 Ψn whose time dependence is given by exp (-iEnt/ℏ), where i = and En 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.
perturbation(per-ter-bay -shŏn) A small disturbance that causes a system to deviate from a reference or equilibrium state. Periodic perturbations cancel out over the period involved and do not affect the system's stability. Secular perturbations have a progressive effect on the system that can cause it eventually to become unstable.
A single planet orbiting the Sun would follow an elliptical orbit according to Kepler's laws; in reality a planet is perturbed from its elliptical orbit by the gravitational effects of the other planets. Likewise the revolution of the Moon around the Earth is perturbed mainly by the Sun and to a much lesser extent by other bodies. The assumptions that the Sun is the only perturbing body and that the Earth orbits the Sun in a fixed elliptical orbit lead to a simplified theory of lunar motion. The orbits of comets and asteroids are strongly perturbed when the body passes close to a major planet, such as Jupiter. The influence of a perturbing body can be calculated from the orbital elements of an osculating orbit, to which corrections are made.