Aharonov- Bohm effect

B. Vector-Potential Theory The Molecular Aharonov-Bohm Effect... 

M. Peshkin and A, Tonomura, The Aharonov Bohm Effect, Springer-Verlag, Berlin, 1989. 

The phase-change nale, also known as the Ben phase [101], the geometric phase effect [102,103] or the molecular Aharonov-Bohm effect [104-106], was used by several authors to verify that two near-by surfaces actually cross, and are not repelled apart. This point is of particular relevance for states of the same symmetry. The total electronic wave function and the total nuclear wave function of both the upper and the lower states change their phases upon being bansported in a closed loop around a point of conical intersection. Any one of them may be used in the search for degeneracies. 

The full quantum mechanical study of nuclear dynamics in molecules has received considerable attention in recent years. An important example of such developments is the work carried out on the prototypical systems H3 [1-5] and its isotopic variant HD2 [5-8], Li3 [9-12], Na3 [13,14], and HO2 [15-18], In particular, for the alkali metal trimers, the possibility of a conical intersection between the two lowest doublet potential energy surfaces introduces a complication that makes their theoretical study fairly challenging. Thus, alkali metal trimers have recently emerged as ideal systems to study molecular vibronic dynamics, especially the so-called geometric phase (GP) effect [13,19,20] (often referred to as the molecular Aharonov-Bohm effect [19] or Berry s phase effect [21]) for further discussion on this topic see [22-25], and references cited therein. The same features also turn out to be present in the case of HO2, and their exact treatment assumes even further complexity [18],... 

A brief review is given on electronic properties of carbon nanotubes, in particular those in magnetic fields, mainly from a theoretical point of view. The topics include a giant Aharonov-Bohm effect on the band gap and optical absorption spectra, a magnetic-field induced lattice distortion and a magnetisation and susceptibility of ensembles, calculated based on a k p scheme. 

In Sec. 2 the effective mass equation is introduced and the band structure is discussed with a special emphasis on an Aharonov-Bohm effect. Optical absorption spectra are discussed in Sec. 3. A lattice instability, in particular induced by a magnetic field perpendicular to the tube axis, is discussed in Sec. 4 and magnetic properties of ensembles of CNTs are discussed in Sec. 5. 

In the presence of a magnetic flux, the boundary condition is changed hy the Aharonov-Bohm effect and the band gap exhibits an oscillation between 0 and 2nylL with period 0q is shown in Fig. 3. 

Electronic properties of CNTs, in particular, electronic states, optical spectra, lattice instabilities, and magnetic properties, have been discussed theoretically based on a k p scheme. The motion of electrons in CNTs is described by Weyl s equation for a massless neutrino, which turns into the Dirac equation for a massive electron in the presence of lattice distortions. This leads to interesting properties of CNTs in the presence of a magnetic field including various kinds of Aharonov-Bohm effects and field-induced lattice distortions. 

C2H-molecule (1,2) and (2,3) conical intersections, 111-112 H3 molecule, 104-109 Wigner rotation matrix and, 89-92 Yang-Mills field, 203-205 Aharonov-Anandan phase, properties, 209 Aharonov-Bohm effect. See Geometric phase effect... 

The term a, therefore plays the role of a vector potential in electromagnetic theory, with a particularly close connection with the Aharonov-Bohm effect, associated with adiabatic motion of a charged quantal system around a magnetic... 

The group space of 0(3) is doubly connected (i.e., non-simply connected) and can therefore support an Aharonov-Bohm effect (Section V), which is described by a physical inhomogeneous term produced by a rotation in the internal gauge space of 0(3) [24]. The existence of the Aharonov-Bohm effect is therefore clear evidence for an extended electrodynamics such as 0(3) electrodynamics, as argued already. A great deal more evidence is reviewed in this article in favor of 0(3) over U(l). For example, it is shown that the Sagnac effect [25] can be described accurately with 0(3), while U(l) fails completely to describe it. 

The 0(3) group is homomorphic with the SU(2) group, that of 2 x 2 unitary matrices with unit determinant [6]. It is well known that there is a two to one mapping of the elements of SU(2) onto those of 0(3). However, the group space of SU(2) is simply connected in the vacuum, and so it cannot support an Aharonov-Bohm effect or physical potentials. It has to be modified [26] to SU(2)/Z2 SO(3). 

Therefore, this is a statement of our fundamental hypothesis, specifically, that the topology of the vacuum defines the field equations through group and gauge field theory. Prior to the inference and empirical verification of the Aharonov-Bohm effect, there was no such concept in classical electrodynamics, the ether having been denied by Lorentz, Poincare, Einstein, and others. Our development of 0(3) electrodynamics in this chapter, therefore, has a well-defined basis in fundamental topology and empirical data. In the course of the development of... 

It is useful to go through this derivation in detail because it produces the inhomogeneous term responsible for the Aharonov-Bohm effect in 0(3) electrodynamics. The effect of the rotation may be written as... 

The Aharonov-Bohm effect is self-inconsistent in U(l) electrodynamics because [44] the effect depends on the interaction of a vector potential A with an electron, but the magnetic field defined by = V x A is zero at the point of interaction [44]. This argument can always be used in U(l) electrodynamics to counter the view that the classical potential A is physical, and adherents of the received view can always assert in U(l) electrodynamics that the potential must be unphysical by gauge freedom. If, however, the Aharonov-Bohm effect is seen as an effect of 0(3) electrodynamics, or of SU(2) electrodynamics [44], it is easily demonstrated that the effect is due to the physical inhomogeneous term appearing in Eq. (25). This argument is developed further in Section VI. 

Another example of a physical effect of this type is the Aharonov-Bohm effect, which is supported by a multiply connected vacuum configuration such as that described by the 0(3) gauge group [6]. The Aharonov-Bohm effect is a gauge transform of the true vacuum, where there are no potentials. In our notation, therefore the Aharonov-Bohm effect is due to terms such as (1/ )8 , depending on the geometry chosen for the experiment. It is essential for the Aharonov-Bohm effect to exist such that (1/ )8 be physical, and not random. It follows therefore that the vacuum configuration defined by the... 

The 0(3) Proca equation (856) does not have this artificial constraint on the potentials, which are regarded as physical in this chapter. This overall conclusion is self-consistent with the inference by Barrett [104] that the Aharonov-Bohm effect is self-consistent only in 0(3) electrodynamics, where the potentials are, accordingly, physical. 

The Aharonov-Bohm effect requires topological consideration [1], (i.e., a structured vacuum), and there exist conservation laws of topological origin, the simplest one is given by the sine-Gordon equation, which also appears in the discussion of 0(3) electrodynamics by Evans and Crowell [5]. 

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