In
mathematical physics
Mathematical physics refers to the development of mathematical methods for application to problems in physics. The '' Journal of Mathematical Physics'' defines the field as "the application of mathematics to problems in physics and the developm ...
, the WKB approximation or WKB method is a method for finding approximate solutions to linear differential equations with spatially varying coefficients. It is typically used for a semiclassical calculation in
quantum mechanics
Quantum mechanics is a fundamental theory in physics that provides a description of the physical properties of nature at the scale of atoms and subatomic particles. It is the foundation of all quantum physics including quantum chemistry, ...
in which the wavefunction is recast as an exponential function, semiclassically expanded, and then either the amplitude or the phase is taken to be changing slowly.
The name is an initialism for Wentzel–Kramers–Brillouin. It is also known as the LG or Liouville–Green method. Other often-used letter combinations include JWKB and WKBJ, where the "J" stands for Jeffreys.
Brief history
This method is named after physicists
Gregor Wentzel,
Hendrik Anthony Kramers, and
Léon Brillouin, who all developed it in 1926. In 1923, mathematician
Harold Jeffreys
Sir Harold Jeffreys, FRS (22 April 1891 – 18 March 1989) was a British mathematician, statistician, geophysicist, and astronomer. His book, ''Theory of Probability'', which was first published in 1939, played an important role in the revival ...
had developed a general method of approximating solutions to linear, second-order differential equations, a class that includes the
Schrödinger equation
The Schrödinger equation is a linear partial differential equation that governs the wave function of a quantum-mechanical system. It is a key result in quantum mechanics, and its discovery was a significant landmark in the development of th ...
. The Schrödinger equation itself was not developed until two years later, and Wentzel, Kramers, and Brillouin were apparently unaware of this earlier work, so Jeffreys is often neglected credit. Early texts in quantum mechanics contain any number of combinations of their initials, including WBK, BWK, WKBJ, JWKB and BWKJ. An authoritative discussion and critical survey has been given by Robert B. Dingle.
Earlier appearances of essentially equivalent methods are:
Francesco Carlini
Francesco Carlini (January 7, 1783 – August 29, 1862) was an Italian astronomer. Born in Milan, he became director of the Brera Astronomical Observatory there in 1832. He published ''Nuove tavole de moti apparenti del sole'' in 1832. In 1810, ...
in 1817,
Joseph Liouville
Joseph Liouville (; ; 24 March 1809 – 8 September 1882) was a French mathematician and engineer.
Life and work
He was born in Saint-Omer in France on 24 March 1809. His parents were Claude-Joseph Liouville (an army officer) and Thérèse ...
in 1837,
George Green in 1837,
Lord Rayleigh
John William Strutt, 3rd Baron Rayleigh, (; 12 November 1842 – 30 June 1919) was an English mathematician and physicist who made extensive contributions to science. He spent all of his academic career at the University of Cambridge. A ...
in 1912 and
Richard Gans
Richard is a male given name. It originates, via Old French, from Old Frankish and is a compound of the words descending from Proto-Germanic ''*rīk-'' 'ruler, leader, king' and ''*hardu-'' 'strong, brave, hardy', and it therefore means 'stro ...
in 1915. Liouville and Green may be said to have founded the method in 1837, and it is also commonly referred to as the Liouville–Green or LG method.
The important contribution of Jeffreys, Wentzel, Kramers, and Brillouin to the method was the inclusion of the treatment of
turning points, connecting the
evanescent
Evanescent may refer to:
* Evanescent (dermatology)
Evanescent skin lesions, like wheals, are those that last for less than 24 hours before resolving.James, William; Berger, Timothy; Elston, Dirk (2005). ''Andrews' Diseases of the Skin: Clinical ...
and
oscillatory
Oscillation is the repetitive or periodic variation, typically in time, of some measure about a central value (often a point of equilibrium) or between two or more different states. Familiar examples of oscillation include a swinging pendulum ...
solutions at either side of the turning point. For example, this may occur in the Schrödinger equation, due to a
potential energy
In physics, potential energy is the energy held by an object because of its position relative to other objects, stresses within itself, its electric charge, or other factors.
Common types of potential energy include the gravitational potenti ...
hill.
Formulation
Generally, WKB theory is a method for approximating the solution of a differential equation whose ''highest derivative is multiplied by a small parameter'' . The method of approximation is as follows.
For a differential equation
:
assume a solution of the form of an
asymptotic series In mathematics, an asymptotic expansion, asymptotic series or Poincaré expansion (after Henri Poincaré) is a formal series of functions which has the property that truncating the series after a finite number of terms provides an approximation to ...
expansion
: