Editing Coherent state (section)

{{Short description|Specific quantum state of a quantum harmonic oscillator}}
{{Quantum mechanics}}

In [[physics]], specifically in [[quantum mechanics]], a '''coherent state''' is the specific [[quantum state]] of the [[quantum harmonic oscillator]], often described as a state that has [[dynamical system|dynamics]] most closely resembling the oscillatory behavior of a [[harmonic oscillator|classical harmonic oscillator]]. It was the first example of [[quantum dynamics]] when [[Erwin Schrödinger]] derived it in 1926, while searching for solutions of the [[Schrödinger equation]] that satisfy the [[correspondence principle]].<ref name="schrod">{{cite journal | last=Schrödinger | first=E. | title=Der stetige Übergang von der Mikro- zur Makromechanik | journal=Die Naturwissenschaften | publisher=Springer Science and Business Media LLC | volume=14 | issue=28 | year=1926 | issn=0028-1042 | doi=10.1007/bf01507634 | pages=664–666 | bibcode=1926NW.....14..664S | s2cid=34680073 | language=de}}</ref> The quantum harmonic oscillator (and hence the coherent states) arise in the quantum theory of a wide range of physical systems.<ref name="klau-ska">J.R. Klauder and B. Skagerstam, ''Coherent States'', World Scientific, Singapore, 1985.</ref> For instance, a coherent state describes the oscillating motion of a particle confined in a quadratic [[potential well]] (for an early reference, see e.g. [[Leonard I. Schiff|Schiff's]] textbook<ref>L.I. Schiff, ''Quantum Mechanics'', McGraw Hill, New York, 1955.</ref>). The coherent state describes a state in a system for which the ground-state wavepacket is displaced from the origin of the system. This state can be related to classical solutions by a particle oscillating with an amplitude equivalent to the displacement.

These states, expressed as ''[[eigenvector]]s  of the [[Ladder operator|lowering operator]]'' and forming an ''[[overcompleteness|overcomplete]]'' family, were introduced in the early papers of [[John R. Klauder]], e.g.<ref>{{cite journal | last=Klauder | first=John R | title=The action option and a Feynman quantization of spinor fields in terms of ordinary c-numbers | journal=Annals of Physics | publisher=Elsevier BV | volume=11 | issue=2 | year=1960 | issn=0003-4916 | doi=10.1016/0003-4916(60)90131-7 | pages=123–168| bibcode=1960AnPhy..11..123K }}</ref> 
In the quantum theory of light ([[quantum electrodynamics]]) and other [[boson]]ic [[quantum field theory|quantum field theories]], coherent states were introduced by the work of [[Roy J. Glauber]] in 1963 and are also known as '''Glauber states'''.

The concept of coherent states has been considerably abstracted; it has become a major topic in [[mathematical physics]] and in [[applied mathematics]], with applications ranging from  [[quantization (physics)|quantization]] to [[signal processing]] and [[image processing]] (see [[Coherent states in mathematical physics]]). For this reason, the coherent states associated to the [[quantum harmonic oscillator]] are sometimes referred to as ''canonical coherent states'' (CCS), ''standard coherent states'', ''Gaussian'' states, or oscillator states.