In classical and quantum mechanics, geometric phase is a phase difference acquired over the course of a cycle, when a system is subjected to cyclic adiabatic processes, which results from the geometrical properties of the parameter space of the Hamiltonian.[1] The phenomenon was independently discovered by S. Pancharatnam (1956),[2] in classical optics and by H. C. Longuet-Higgins (1958)[3] in molecular physics; it was generalized by Michael Berry in (1984).[4]
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 surfaces[3][5] and in the Aharonov–Bohm effect. Geometric phase around the conical intersection involving the ground electronic state of the C6H3F3+ molecular ion is discussed on pages 385–386 of the textbook by Bunker and Jensen.[6] 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 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, 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 singular (its state is undefined) for some combination of parameters.
In a quantum system at the n-th eigenstate, an adiabatic evolution of the Hamiltonian sees the system remain in the n-th eigenstate of the Hamiltonian, while also obtaining a phase factor. The phase obtained has a contribution from the state's time evolution and another from the variation of the eigenstate with the changing Hamiltonian. The second term corresponds to the Berry phase, and for non-cyclical variations of the Hamiltonian it can be made to vanish by a different choice of the phase associated with the eigenstates of the Hamiltonian at each point in the evolution.
However, if the variation is cyclical, the Berry phase cannot be cancelled; it is invariant and becomes an observable property of the system. By reviewing the proof of the adiabatic theorem given by Max Born and Vladimir Fock, in Zeitschrift für Physik51, 165 (1928), we could characterize the whole change of the adiabatic process into a phase term. Under the adiabatic approximation, the coefficient of the n-th eigenstate under adiabatic process is given by
where is the Berry's phase with respect to parameter t. Changing the variable t into generalized parameters, we could rewrite the Berry's phase into
where parametrizes the cyclic adiabatic process. Note that the normalization of implies that the integrand is imaginary, so that is real. It follows a closed path in the appropriate parameter space. Geometric phase along the closed path can also be calculated by integrating the Berry curvature over surface enclosed by .
Examples of geometric phases
Foucault pendulum
One of the easiest examples is the Foucault pendulum. An easy explanation in terms of geometric phases is given by Wilczek and Shapere:[7]
How does the pendulum precess when it is taken around a general path C? For transport along the equator, the pendulum will not precess. [...] Now if C is made up of geodesic segments, the precession will all come from the angles where the segments of the geodesics meet; the total precession is equal to the net deficit angle which in turn equals the solid angle enclosed by C modulo 2π. Finally, we can approximate any loop by a sequence of geodesic segments, so the most general result (on or off the surface of the sphere) is that the net precession is equal to the enclosed solid angle.
To put it in different words, there are no inertial forces that could make the pendulum precess, so the precession (relative to the direction of motion of the path along which the pendulum is carried) is entirely due to the turning of this path. Thus the orientation of the pendulum undergoes parallel transport. For the original Foucault pendulum, the path is a circle of latitude, and by the Gauss–Bonnet theorem, the phase shift is given by the enclosed solid angle.[8]
Derivation
This section may need to be cleaned up. It has been merged from Foucault pendulum.
Parallel transport of a vector around a closed loop on the sphere: The angle by which it twists, α, is proportional to the area inside the loop.
In a near-inertial frame moving in tandem with the Earth, but not sharing the rotation of the Earth about its own axis, the suspension point of the pendulum traces out a circular path during one sidereal day.
At the latitude of Paris, 48 degrees 51 minutes north, a full precession cycle takes just under 32 hours, so after one sidereal day, when the Earth is back in the same orientation as one sidereal day before, the oscillation plane has turned by just over 270 degrees. If the plane of swing was north–south at the outset, it is east–west one sidereal day later.
This also implies that there has been exchange of momentum; the Earth and the pendulum bob have exchanged momentum. The Earth is so much more massive than the pendulum bob that the Earth's change of momentum is unnoticeable. Nonetheless, since the pendulum bob's plane of swing has shifted, the conservation laws imply that an exchange must have occurred.
Rather than tracking the change of momentum, the precession of the oscillation plane can efficiently be described as a case of parallel transport. For that, it can be demonstrated, by composing the infinitesimal rotations, that the precession rate is proportional to the projection of the angular velocity of Earth onto the normal direction to Earth, which implies that the trace of the plane of oscillation will undergo parallel transport. After 24 hours, the difference between initial and final orientations of the trace in the Earth frame is α = −2π sin φ, which corresponds to the value given by the Gauss–Bonnet theorem. α is also called the holonomy or geometric phase of the pendulum. When analyzing earthbound motions, the Earth frame is not an inertial frame, but rotates about the local vertical at an effective rate of 2π sin φ radians per day. A simple method employing parallel transport within cones tangent to the Earth's surface can be used to describe the rotation angle of the swing plane of Foucault's pendulum.[9][10]
From the perspective of an Earth-bound coordinate system (the measuring circle and spectator are Earth-bounded, also if terrain reaction to Coriolis force is not perceived by spectator when he moves), using a rectangular coordinate system with its x-axis pointing east and its y-axis pointing north, the precession of the pendulum is due to the Coriolis force (other fictitious forces as gravity and centrifugal force have not direct precession component, Euler's force is low because Earth's rotation speed is nearly constant). Consider a planar pendulum with constant natural frequency ω in the small angle approximation. There are two forces acting on the pendulum bob: the restoring force provided by gravity and the wire, and the Coriolis force (the centrifugal force, opposed to the gravitational restoring force, can be neglected). The Coriolis force at latitude φ is horizontal in the small angle approximation and is given by
where Ω is the rotational frequency of Earth, Fc,x is the component of the Coriolis force in the x-direction and Fc,y is the component of the Coriolis force in the y-direction.
Graphs of precession period and precession per sidereal day vs latitude. The sign changes as a Foucault pendulum rotates anticlockwise in the Southern Hemisphere and clockwise in the Northern Hemisphere. The example shows that one in Paris precesses 271° each sidereal day, taking 31.8 hours per rotation.
A second example is linearly polarized light entering a single-mode optical fiber. Suppose the fiber traces out some path in space, and the light exits the fiber in the same direction as it entered. Then compare the initial and final polarizations. In semiclassical approximation the fiber functions as a waveguide, and the momentum of the light is at all times tangent to the fiber. The polarization can be thought of as an orientation perpendicular to the momentum. As the fiber traces out its path, the momentum vector of the light traces out a path on the sphere in momentum space. The path is closed, since initial and final directions of the light coincide, and the polarization is a vector tangent to the sphere. Going to momentum space is equivalent to taking the Gauss map. There are no forces that could make the polarization turn, just the constraint to remain tangent to the sphere. Thus the polarization undergoes parallel transport, and the phase shift is given by the enclosed solid angle (times the spin, which in case of light is 1).
Stochastic pump effect
A stochastic pump is a classical stochastic system that responds with nonzero, on average, currents to periodic changes of parameters.
The stochastic pump effect can be interpreted in terms of a geometric phase in evolution of the moment generating function of stochastic currents.[11]
Spin 1⁄2
The geometric phase can be evaluated exactly for a spin-1⁄2 particle in a magnetic field.[1]
Geometric phase defined on attractors
While Berry's formulation was originally defined for linear Hamiltonian systems, it was soon realized by Ning and Haken[12] that similar geometric phase can be defined for entirely different systems such as nonlinear dissipative systems that possess certain cyclic attractors. They showed that such cyclic attractors exist in a class of nonlinear dissipative systems with certain symmetries.[13] There are several important aspects of this generalization of Berry's phase: 1) Instead of the parameter space for the original Berry phase, this Ning-Haken generalization is defined in phase space; 2) Instead of the adiabatic evolution in quantum mechanical system, the evolution of the system in phase space needs not to be adiabatic. There is no restriction on the time scale of the temporal evolution; 3) Instead of a Hermitian system or non-hermitian system with linear damping, systems can be generally nonlinear and non-hermitian.
Exposure in molecular adiabatic potential surface intersections
There are several ways to compute the geometric phase in molecules within the Born–Oppenheimer framework. One way is through the "non-adiabatic coupling matrix" defined by
where is the adiabatic electronic wave function, depending on the nuclear parameters . The nonadiabatic coupling can be used to define a loop integral, analogous to a Wilson loop (1974) in field theory, developed independently for molecular framework by M. Baer (1975, 1980, 2000). Given a closed loop , parameterized by where is a parameter, and . The D-matrix is given by
(here is a path-ordering symbol). It can be shown that once is large enough (i.e. a sufficient number of electronic states is considered), this matrix is diagonal, with the diagonal elements equal to where are the geometric phases associated with the loop for the -th adiabatic electronic state.
For time-reversal symmetrical electronic Hamiltonians the geometric phase reflects the number of conical intersections encircled by the loop. More accurately,
where is the number of conical intersections involving the adiabatic state encircled by the loop
An alternative to the D-matrix approach would be a direct calculation of the Pancharatnam phase. This is especially useful if one is interested only in the geometric phases of a single adiabatic state. In this approach, one takes a number of points along the loop with and then using only the j-th adiabatic states computes the Pancharatnam product of overlaps:
In the limit one has (see Ryb & Baer 2004 for explanation and some applications)
Geometric phase and quantization of cyclotron motion
An electron subjected to magnetic field moves on a circular (cyclotron) orbit.[2] Classically, any cyclotron radius is acceptable. Quantum-mechanically, only discrete energy levels (Landau levels) are allowed, and since is related to electron's energy, this corresponds to quantized values of . The energy quantization condition obtained by solving Schrödinger's equation reads, for example, for free electrons (in vacuum) or for electrons in graphene, where .[3] Although the derivation of these results is not difficult, there is an alternative way of deriving them, which offers in some respect better physical insight into the Landau level quantization. This alternative way is based on the semiclassical Bohr–Sommerfeld quantization condition
which includes the geometric phase picked up by the electron while it executes its (real-space) motion along the closed loop of the cyclotron orbit.[14] For free electrons, while for electrons in graphene. It turns out that the geometric phase is directly linked to of free electrons and of electrons in graphene.
^S. Pancharatnam (1956). "Generalized Theory of Interference, and Its Applications. Part I. Coherent Pencils". Proc. Indian Acad. Sci. A. 44 (5): 247–262. doi:10.1007/BF03046050. S2CID118184376.
^G. Herzberg; H. C. Longuet-Higgins (1963). "Intersection of potential energy surfaces in polyatomic molecules". Discuss. Faraday Soc. 35: 77–82. doi:10.1039/DF9633500077.
^Molecular Symmetry and Spectroscopy,
2nd ed. Philip R. Bunker and Per Jensen, NRC Research Press, Ottawa (1998) [1]ISBN9780660196282
^Somerville, W. B. (1972). "The Description of Foucault's Pendulum". Quarterly Journal of the Royal Astronomical Society. 13: 40. Bibcode:1972QJRAS..13...40S.
^Hart, John B.; Miller, Raymond E.; Mills, Robert L. (1987). "A simple geometric model for visualizing the motion of a Foucault pendulum". American Journal of Physics. 55 (1): 67–70. Bibcode:1987AmJPh..55...67H. doi:10.1119/1.14972.
Baer, M. (1975). "Adiabatic and diabatic representations for atom-molecule collisions: Treatment of the collinear arrangement". Chemical Physics Letters. 35 (1): 112–118. Bibcode:1975CPL....35..112B. doi:10.1016/0009-2614(75)85599-0.
TX 31 redirects here. The term may also refer to Texas's 31st congressional district. State highway in Texas State Highway 31Route informationMaintained by TxDOTLength149.813 mi[1] (241.101 km)Existed1919–presentMajor junctionsWest end US 84 near WacoMajor intersections I-45 / US 287 in Corsicana US 175 in Athens US 69 in Tyler US 271 in Tyler US 259 in Kilgore I-20 near KilgoreEast end US 80 / Spur 63 at Longvi…
Amber Liu劉逸雲Amber Liu pada acara Style Icon Asia 2016Nama Tionghoa劉逸雲Nama Tionghoa劉逸雲 (Tradisional)Nama Tionghoa刘逸云 (Sederhana)PinyinLiú Yìyún (Mandarin)JyutpingLau4 Jat6 Wan4 (Kanton)Nama LahirAmber Josephine LiuLahir18 September 1992 (umur 31)Los Angeles, California, Amerika SerikatNama Lain엠버 (Amber)Pekerjaan Penyanyi-penulis lagu rapper penari Genre K-pop InstrumenVokalrappinggitardrumLabel S.M. Entertainment Situs webamber.smtown.com Amber Liu Hanzi tradi…
Жирные кислоты — алифатические одноосновные карбоновые кислоты с открытой цепью, содержащиеся в этерифицированной форме в жирах, маслах и восках растительного и животного происхождения. Жирные кислоты, как правило, содержат неразветвлённую цепь из чётного числа атом…
البحرية الملكية الدولة مملكة إنجلترا(القرن الخامس عشر - 1707)مملكة بريطانيا العظمى(1707 - 1801)المملكة المتحدة(1801 - الوقت الحاضر) الإنشاء أواخر القرن الخامس عشر/ أوائل القرن السادس عشر - حتى الآن النوع قوات بحرية الحجم 98 سفينة170 طائرة جزء من القوات المسلحة البريطانية المقر الرئيسي م
Mario + The Lapins CrétinsKingdom BattleLogo du jeu.Développeur Ubisoft MilanUbisoft ParisÉditeur UbisoftDistributeur NintendoRéalisateur Davide Soliani (créatif)Scénariste NintendoCompositeur Grant Kirkhope[1]Producteur Xavier ManzanaresDébut du projet 2014[2]Date de sortie INT : 29 août 2017 Genre Tactique au tour par tourMode de jeu 1 à 2 joueursPlate-forme Nintendo SwitchLangue MultilingueMoteur SnowdropVersion 1.7.561410Ventes 10 millions (août 2022)[3]Évaluation ACB : …
Canada based mining company First Quantum MineralsFormerlyFirst Quantum VenturesZeal CapitalXenium ResourcesTypePublicTraded asTSX: FMS&P/TSX 60 componentIndustryMiningFounded1983HeadquartersToronto, Ontario, Canada[1]Key peopleRobert Harding (Chairman)[2] Tristan Pascall (CEO)ProductsCopperNumber of employees20,000 (2021)Websitewww.first-quantum.com First Quantum Minerals is a Canadian-based mining and metals company[3] whose principal activities include mineral…
Inilah daftar orang-orang penting yang terbunuh yang tak lengkap; yang merupakan orang penting yang dibunuh, biasanya karena alasan politis maupun ideologis. Pembunuhan di Afrika Afrika Selatan Shaka, (1828), Raja Zulu, dekat Stanger (kini KwaDukuza) oleh Dingane dan Mhlangana Umthalangana (1828), pangeran Zulu, saudara Shaka. Hendrik Verwoerd, (1966), Perdana Menteri Afrika Selatan, ditikam di parlemen oleh Dimitri Tsafendas Onkgopotse Tiro, (1974), pemimpin mahasiswa Afrika Selatan Vernon Nkad…
Fernsehsendung Titel Eurovision Young Musicians Produktionsland verschiedene Genre Musik Erscheinungsjahre seit 1982 Ausstrahlungs-turnus alle 2 Jahre Produktions-unternehmen verschiedene Produktion Europäische Rundfunkunion Premiere 11. Mai 1982 Moderation verschiedene Moderatorin Sabine Heinrich und deutsche Teilnehmerin Judith Stapf vor dem Wettbewerb 2014 Eurovision Young Musicians (EYM) ist ein Musikwettbewerb im Bereich der Klassischen Musik für europäische Jugendliche im Alter zwi…
مناطق أوكرانيا الجغرافيةRaions of Ukraine (after Oct. 2020).svgالمناطق الإدارية بعد الإصلاح الإداري عام 2020نوع التقسيمالمستوى الثاني من التقسيمالدولة Ukraineالإنشاء1922العدد136 (اعتباراً من 2020)السكان~150,000المناطق1,200 كـم2 (460 ميل2)الحكومةمجلس المنطقةالتقسيم الأقل1470 مجتمع إقليمي جزء من سلسل…
This article needs additional citations for verification. Please help improve this article by adding citations to reliable sources. Unsourced material may be challenged and removed.Find sources: Solitary practitioner – news · newspapers · books · scholar · JSTOR (November 2013) (Learn how and when to remove this template message) Part of a series onWitchcraft Neopagan Wicca Feminist Dianic By region Africa Asia Europe Goetia Roma Latin America Middle East…
Peña Falconera Huevo de MorranoPeña Falconera and the Sierra de SevilHighest pointElevation766 m (2,513 ft)Coordinates42°12′10″N 0°06′14″W / 42.20278°N 0.10389°W / 42.20278; -0.10389GeographyPeña Falconera Huevo de MorranoSpain LocationHuesca Province, AragonParent rangePre-PyreneesGeologyMountain typeConglomerateClimbingFirst ascentMarch of 1960Easiest routeFrom Morrano The Peña Falconera or Huevo de Morrano (Morrano egg) is a rock formatio…
Passagiersboot: de Koegelwieck vaart tussen Harlingen, Terschelling en Vlieland Een passagiersschip is een schip met als hoofdfunctie het vervoeren van passagiers. Volgens de Internationale Maritieme Organisatie, Europese en nationale wetten is een passagiersschip een schip dat meer dan 12 passagiers vervoert.[1] Dit zijn dus mensen die niet werkzaam zijn op een schip. Een passagiersschip dient te voldoen aan bijzondere eisen. Zo dient op een zeegaand passagiersschip een arts aan boord t…
السفارة السعودية في كوبا السعودية كوبا الإحداثيات 23°06′50″N 82°25′57″W / 23.114°N 82.4324°W / 23.114; -82.4324 البلد كوبا المكان هافانا السفير فيصل بن فلاح البيضاني الحربي الموقع الالكتروني سفارة المملكة العربية السعودية السعودية في كوبا تعديل مصدري - تعديل سفارة المم
Oliver Bendel (2015) Oliver Bendel (* 1968 in Ulm) ist ein deutscher Wirtschaftsinformatiker, Szientist und Buchautor. Leben Bendel studierte an der Universität Konstanz Philosophie, Germanistik und Informationswissenschaft und promovierte an der Universität St. Gallen in Wirtschaftsinformatik. Seit 2009 ist er Dozent unter anderem für Wirtschaftsinformatik und Betriebsökonomie an der Hochschule für Wirtschaft der Fachhochschule Nordwestschweiz.[1] Er veröffentlichte Romane im Leip…
هذه المقالة يتيمة إذ تصل إليها مقالات أخرى قليلة جدًا. فضلًا، ساعد بإضافة وصلة إليها في مقالات متعلقة بها. (يوليو 2019) 31°59′53.05″N 35°52′47.3″E / 31.9980694°N 35.879806°E / 31.9980694; 35.879806 جريدة الديار الأردنيةمعلومات عامةالنوع جريدة يومية[1]التأسيس 2003الثمن 250 فلس أردنيشخصيات ها
Participation of British Empire and Commonwealth in World War II Propaganda poster promoting the joint war-effort of the British Empire and Commonwealth, 1939See also: Military history of the United Kingdom during World War II and Diplomatic history of World War IIvteBritish Commonwealth & Empire Campaigns of World War IICampaigns fought exclusively or largely by the armed forces of the UK, the Dominions, India and/or crown colonies Europe Arctic Battle of the Barents Sea Evacuation of Norwa…
State park in Utah, USA Wasatch Mountain State ParkLocation of Wasatch Mountain State Park within the State of UtahShow map of UtahWasatch Mountain State Park (the United States)Show map of the United StatesLocationWasatch, Utah, United StatesCoordinates40°30′10″N 111°32′14″W / 40.50278°N 111.53722°W / 40.50278; -111.53722Area21,592 acres (87.38 km2)Elevation5,900 ft (1,800 m)Established1961Named forthe Wasatch RangeVisitors360383 (in F…
2011 video game 2011 video gameEverybody's Golf 6Japanese cover artDeveloper(s)Clap HanzPublisher(s)Sony Computer EntertainmentSeriesEverybody's GolfPlatform(s)PlayStation Vita, PlayStation 3ReleasePlayStation VitaJP: 17 December 2011NA: 15 February 2012EU: 22 February 2012AU: 23 February 2012PlayStation 3JP: 22 November 2012NA: 23 July 2013PAL: 24 July 2013Genre(s)SportsMode(s)Single-player, multiplayer Everybody's Golf 6 (みんなのGOLF 6, Minna no Gorufu 6), known in Europe as Everybody's G…
Late Neolithic archaeological culture Hamangia cultureHorizonOld EuropePeriodNeolithic, ChalcolithicDatescirca 5250 BC — circa 4,500 BCType siteDurankulakPreceded byKaranovo culture, Starcevo cultureFollowed byVarna culture, Boian culture, Gumelnița culture See also: Old Europe (archaeology) The Hamangia culture is a Late Neolithic archaeological culture of Dobruja (Romania and Bulgaria) between the Danube and the Black Sea and Muntenia in the south. It is named after the site of Baia-Hamangi…