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In classical and quantum mechanics, the geometric phase, Pancharatnam–Berry phase (named after S. Pancharatnam and Sir Michael Berry), Pancharatnam phase or most commonly Berry 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. The phenomenon was first discovered in 1956, and rediscovered in 1984. It can be seen in the Aharonov–Bohm effect and in the conical intersection of potential energy surfaces. 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 both 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. To measure the geometric phase in a wave system, an 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. ==Berry phase in quantum mechanics == In a quantum system at the n-th eigenstate, an adiabatic evolution of the Hamiltonian evolves the system such that it remains 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. From the Schrödinger equation the Berry phase can be calculated to be: : where parametrizes the cyclic adiabatic process. It follows a closed path in the appropriate parameter space. A recent review on the geometric phase effects on electronic properties was given by Xiao, Chang and Niu.〔Di Xiao ''et al.'', Rev. Mod. Phys. 82, 1959 (2010)〕 Geometric phase along the closed path can also be calculated by integrating the Berry curvature over surface enclosed by . 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Geometric phase」の詳細全文を読む スポンサード リンク
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