Editing Geometric phase (section)

{{Short description|Phase of a cycle}}
In [[Classical mechanics|classical]] and [[quantum mechanics]], '''geometric phase''' is a [[Phase (waves)|phase]] difference acquired over the course of a [[Period (physics)|cycle]], when a system is subjected to cyclic [[adiabatic process (quantum mechanics)|adiabatic process]]es, which results from the geometrical properties of the [[parameter space]] of the [[Hamiltonian (quantum mechanics)|Hamiltonian]].<ref name=Solem1993>{{cite journal|last1=Solem|first1=J. C.|last2=Biedenharn|first2=L. C.|year=1993|title=Understanding geometrical phases in quantum mechanics: An elementary example|journal=Foundations of Physics|volume=23|issue=2|pages=185–195|bibcode = 1993FoPh...23..185S |doi = 10.1007/BF01883623 |s2cid=121930907}}</ref> The phenomenon was independently discovered by [[S.&nbsp;Pancharatnam]] (1956),<ref>{{cite journal|author=S. Pancharatnam|title=Generalized Theory of Interference, and Its Applications. Part I. Coherent Pencils|journal=Proc. Indian Acad. Sci. A|volume=44|issue=5|pages=247–262|year=1956|doi=10.1007/BF03046050|s2cid=118184376}}</ref> in classical optics and by [[Christopher Longuet-Higgins|H.&nbsp;C. Longuet-Higgins]] (1958)<ref name=Longuet-Higgins1958>{{cite journal|author1=H. C. Longuet Higgins|author2=U. Öpik|author3=M. H. L. Pryce|author4=R. A. Sack|title=Studies of the Jahn-Teller effect .II. The dynamical problem|journal=Proc. R. Soc. A|volume=244|issue=1236|pages=1–16|year=1958|doi=10.1098/rspa.1958.0022 |bibcode=1958RSPSA.244....1L|s2cid=97141844}}See page 12</ref>  in molecular physics; it was generalized by [[Michael Berry (physicist)|Michael Berry]] in (1984).<ref>{{cite journal|author=M. V. Berry|journal=Proceedings of the Royal Society A|title=Quantal Phase Factors Accompanying Adiabatic Changes|volume=392|issue=1802|pages=45–57|year=1984|doi=10.1098/rspa.1984.0023|bibcode = 1984RSPSA.392...45B |s2cid=46623507}}</ref> 
It is also known as the '''Pancharatnam–Berry phase''', '''Pancharatnam phase''', or '''Berry phase'''.
It can be seen in the [[conical intersection]] of [[potential energy surface]]s<ref name=Longuet-Higgins1958/><ref>{{cite journal|author1=G. Herzberg|author2=H. C. Longuet-Higgins|title=Intersection of potential energy surfaces in polyatomic molecules|journal=Discuss. Faraday Soc.|volume=35|pages=77–82|year=1963|doi=10.1039/DF9633500077}}</ref> and in the [[Aharonov–Bohm effect]]. Geometric phase around the conical intersection involving the ground electronic state of the C<sub>6</sub>H<sub>3</sub>F<sub>3</sub><sup>+</sup> molecular ion is discussed on pages 385–386 of the textbook by Bunker and Jensen.<ref>''Molecular Symmetry and Spectroscopy'', 
2nd ed. Philip R. Bunker and Per Jensen, NRC Research Press, Ottawa (1998) [https://volumesdirect.com/products/molecular-symmetry-and-spectroscopy?_pos=1&_sid=90a6edc37&_ss=r]
{{ISBN|9780660196282}}</ref> In the case of the Aharonov–Bohm effect, the adiabatic parameter is the [[magnetic field]] enclosed by two interference paths, and it is cyclic in the sense that these two paths form a loop. In the case of the conical intersection, the [[adiabatic]] parameters are the [[molecular geometry|molecular coordinates]]. Apart from quantum mechanics, it arises in a variety of other [[wave]] systems, such as classical [[optics]]. As a rule of thumb, it can occur whenever there are at least two parameters characterizing a wave in the vicinity of some sort of singularity or hole in the topology; two parameters are required because either the set of nonsingular states will not be [[simply connected]], or there will be nonzero [[holonomy]].

Waves are characterized by [[amplitude]] and [[Phase (waves)|phase]], and may vary as a function of those parameters. The geometric phase occurs when both parameters are changed simultaneously but very slowly (adiabatically), and eventually brought back to the initial configuration. In quantum mechanics, this could involve rotations but also translations of particles, which are apparently undone at the end. One might expect that the waves in the system return to the initial state, as characterized by the amplitudes and phases (and accounting for the passage of time). However, if the parameter excursions correspond to a loop instead of a self-retracing back-and-forth variation, then it is possible that the initial and final states differ in their phases. This phase difference is the geometric phase, and its occurrence typically indicates that the system's parameter dependence is [[Mathematical singularity|singular]] (its state is undefined) for some combination of parameters.

To [[Measurement|measure]] the geometric phase in a wave system, an [[interference (wave propagation)|interference]] [[experiment]] is required. The [[Foucault pendulum]] is an example from [[classical mechanics]] that is sometimes used to illustrate the geometric phase. This mechanics analogue of the geometric phase is known as the [[Hannay angle]].