Editing WKB approximation (section)

== Brief history ==
This method is named after physicists [[Gregor Wentzel]], [[Hendrik Anthony Kramers]], and [[Léon Brillouin]], who all developed it in 1926.<ref name=Wentzel-1926/><ref name=Kramers-1926/><ref name=Brillouin-1926/><ref>{{harvnb|Hall|2013}} Section 15.1 </ref> In 1923,<ref name=Jefferys-1924/> mathematician [[Harold Jeffreys]] had developed a general method of approximating solutions to linear, second-order differential equations, a class that includes the [[Schrödinger equation]].  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.<ref>{{cite book |first=Robert Balson |last=Dingle |title=Asymptotic Expansions: Their Derivation and Interpretation |publisher=Academic Press |year=1973 |isbn=0-12-216550-0 }}</ref>

Earlier appearances of essentially equivalent methods are: [[Francesco Carlini]] in 1817,<ref name=Carlini-1817/> [[Joseph Liouville]] in 1837,<ref name=Liouville/> [[George Green (mathematician)|George Green]] in 1837,<ref name=Green-1837/> [[Lord Rayleigh]] in 1912<ref name=Rayleigh-1912/> and [[Richard Gans]] in 1915.<ref name=Gans-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.<ref>{{cite book
 | title = Atmosphere-ocean dynamics
 | author = Adrian E. Gill
 | publisher = Academic Press
 | year = 1982
 | isbn = 978-0-12-283522-3
 | page = [https://archive.org/details/atmosphereoceand0000gill/page/297 297]
 | url = https://archive.org/details/atmosphereoceand0000gill
 | url-access = registration
 | quote = Liouville-Green WKBJ WKB.
 }}</ref><ref>
{{cite book
 | chapter = A Survey on the Liouville–Green (WKB) approximation for linear difference equations of the second order
 |author1=Renato Spigler  |author2=Marco Vianello
  |name-list-style=amp | title = Advances in difference equations: proceedings of the Second International Conference on Difference Equations : Veszprém, Hungary, August 7–11, 1995     
 |editor1=Saber Elaydi |editor2=I. Győri |editor3=G. E. Ladas | publisher = CRC Press
 | year = 1998
 | isbn = 978-90-5699-521-8
 | page = 567
 | chapter-url = https://books.google.com/books?id=a36iXw5_VzcC&dq=Liouville-Green+WKBJ+WKB+LG&pg=PA567
 }}</ref>

The important contribution of Jeffreys, Wentzel, Kramers, and Brillouin to the method was the inclusion of the treatment of [[stationary point|turning points]], connecting the [[evanescent wave|evanescent]] and [[oscillation|oscillatory]] solutions at either side of the turning point. For example, this may occur in the Schrödinger equation, due to a [[potential energy]] hill.