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

Collins Discovery Encyclopedia, 1st edition © HarperCollins Publishers 2005

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.

McGraw-Hill Concise Encyclopedia of Physics. © 2002 by The McGraw-Hill Companies, Inc.

perturbation

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.

Collins Dictionary of Astronomy © Market House Books Ltd, 2006

perturbation

McGraw-Hill Dictionary of Scientific & Technical Terms, 6E, Copyright © 2003 by The McGraw-Hill Companies, Inc.