Editing Airy function (section)

==Applications==

=== Quantum mechanics ===
The Airy function is the solution to the [[time-independent Schrödinger equation]] for a particle confined within a triangular [[potential well]] and for a particle in a one-dimensional constant force field. For the same reason, it also serves to provide uniform semiclassical approximations near a turning point in the [[WKB approximation]], when the potential may be locally approximated by a linear function of position. The triangular potential well solution is directly relevant for the understanding of electrons trapped in semiconductor [[heterojunction]]s.

=== Optics ===

A transversally asymmetric optical beam, where the electric field profile is given by the Airy function, has the interesting property that its maximum intensity ''accelerates'' towards one side instead of propagating in a straight line as is the case in symmetric beams. This is at expense of the low-intensity tail being spread in the opposite direction, so the overall momentum of the beam is of course conserved.

=== Caustics ===

The Airy function underlies the form of the intensity near an optical directional [[caustic (optics)|caustic]], such as that of the [[rainbow]] (called supernumerary rainbow). Historically, this was the mathematical problem that led Airy to develop this special function. In 1841, [[William Hallowes Miller]] experimentally measured the analog to supernumerary rainbow by shining light through a thin cylinder of water, then observing through a telescope. He observed up to 30 bands.<ref>[[iarchive:transactionsofca07camb/page/n249/mode/2up|Miller, William Hallowes. "On spurious rainbows." ''Transactions of the Cambridge Philosophical Society'' 7 (1848): 277.]]</ref>

=== Probability ===

In the mid-1980s, the Airy function was found to be intimately connected to [[Chernoff's distribution]].<ref>{{cite journal|title=Chernoff's distribution and differential equations of parabolic and Airy type|last1=Groeneboom|first1=Piet|last2=Lalley|first2=Steven|last3=Temme|first3=Nico|journal=[[Journal of Mathematical Analysis and Applications]]|volume=423|issue=2|pages=1804–1824|year=2015|doi=10.1016/j.jmaa.2014.10.051 |s2cid=119173815 |doi-access=free|arxiv=1305.6053}}</ref>

The Airy function also appears in the definition of [[Tracy–Widom distribution]] which describes the law of largest eigenvalues in [[Random matrix]]. Due to the intimate connection of random matrix theory with the [[Kardar–Parisi–Zhang equation]], there are central processes constructed in KPZ such as the [[Airy process]].<ref>{{cite book|last1=Quastel|first1=Jeremy|last2=Remenik|first2=Daniel|title=Topics in Percolative and Disordered Systems |chapter=Airy Processes and Variational Problems |series=Springer Proceedings in Mathematics & Statistics |year= 2014|volume=69 |pages=121–171 |doi=10.1007/978-1-4939-0339-9_5 |arxiv=1301.0750 |isbn=978-1-4939-0338-2 |s2cid=118241762 |chapter-url=https://link.springer.com/chapter/10.1007/978-1-4939-0339-9_5}}</ref>