The Adiabatic Approximation, Aharonov–Bohm, and Berry Phases

Eliahu Cohen, Hugo Larocque, Frédéric Bouchard +3 more

Yves Brihaye, Piotr Kosiński

Lei Yian, ZENG JIN-YAN, 中国科学院理论物理研究所;北京大学物理系

Berry pointed out that the Aharonov-Bohm phase could be considered as a Berry phase.In his argument,he gave up the adiabatic approximation,and the nonstationarity of quantum states was not concerned.In this paper we re-investigate Berry’s proof,and find that his final statement is correct,and in his argument,the adiabatic approximation should be used.In the AB effect,a moving electron is described as a moving wave packet (nonstationary),and the appearance of AB phase originates from the requirement that the evolution of the wave packet must satisfy the Schrdinger equation.

Ali Mostafazadeh

Cássio Cecato Favarato

In this work, we make a review of the literature that the so-called Berry’s geometric phase. A recent issue, which has been employed in various areas of physics. For this, we use, in a number of references, in which many were critical to the theoretical foundation of this work. At first, we focus on the study of the time evolution of quantum systems governed by Hamiltonians operators whose time dependence is manifested by a set of parameters called classical fields. The adiabatic approximation, is justified mathematically when demonstrated the adiabatic theorem. Under these conditions, we obtain geometric phases - Berry’s phase - associated with these systems in different representations, each displaying its particularity. Finally, as a particular case of geometric phase, we studi from the viewpoint of duality transformations of Maxwell’s equations, the dual Aharonov-Bohm effect, extracting the geometric Berry’s phase associated with the dynamics of the magnetic monopole in the presence of the electric vector potential

Kazuo Fujikawa

By analyzing an exactly solvable model in the second quantized formulation which allows a unified treatment of adiabatic and non-adiabatic geometric phases, it is shown that the topology of the adiabatic Berry's phase, which is characterized by the singularity associated with possible level crossing, is trivial in a precise sense. This topology of the geometric phase is quite different from the topology of the Aharonov-Bohm effect, where the topology is specified by the external local gauge field and it is exact for the slow as well as for the fast motion of the electron.