| Aharonov-bohm Effect |
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After the 1959 paper was published, Bohm was informed that the effect had been predicted by Rory E. Siday and Werner Ehrenberg a decade earlier; Bohm and Aharonov duly cited this in their second paper (Peat, 1997, p. 192). The most commonly described case, sometimes called the Aharonov-Bohm solenoid effect, is when the wave function of a charged particle passing around a long Solenoid experiences a Phase Shift as a result of the enclosed magnetic field, despite the magnetic field being zero in the region through which the particle passes. This phase shift has been observed experimentally by its effect on interference fringes. (There are also magnetic Aharonov-Bohm effects on bound energies and scattering cross sections, but these cases have not been experimentally tested.) An electric Aharonov-Bohm phenomenon was also predicted, in which a charged particle is affected by regions with different Electrical Potential s but zero electric field, and this has also seen experimental confirmation. A separate "molecular" Aharonov-Bohm effect was proposed for nuclear motion in multiply-connected regions, but this has been argued to be essentially different, depending only on local quantities along the nuclear path (Sjöqvist, 2002). A general review can be found in Peshkin and Tonomura (1989). MAGNETIC AHARONOV-BOHM EFFECT The magnetic Aharonov-Bohm effect can be seen as a result of the requirement that quantum physics be invariant with respect to the Gauge Choice for the vector potential A. This implies that a particle with charge ''q'' travelling along some path P in a region with zero magnetic field () must acquire a phase ; given in SI units by : with a phase difference Δφ between any two paths with the same endpoints therefore determined by the Magnetic Flux Φ through the area between the paths (via Stokes Theorem and ), and given by: : This phase difference can be observed by placing a Solenoid between the slits of a double-slit experiment (or equivalent). A solenoid encloses a magnetic field B, but does not produce any magnetic field outside of its cylinder, and thus the charged particle (e.g. an Electron ) passing outside experiences no classical effect. However, there is a ( Curl -free) vector potential outside the solenoid with an enclosed flux, and so the relative phase of particles passing through one slit or the other is altered by whether the solenoid current is turned on. This corresponds to an observable shift of the interference fringes on the observation plane. The same phase effect is responsible for the ) using a phenomenological model. The magnetic Aharonov-Bohm effect is also closely related to (in Cgs units) for any electric charge ''q'' and magnetic charge ''g''. The magnetic Aharonov-Bohm effect was experimentally confirmed by Osakabe et al. (1986), following earlier work summarized in Olariu and Popèscu (1984). Its scope and application continues to expand. Webb et al. (1985) demonstrated Aharonov-Bohm oscillations in ordinary, non-superconducting metallic rings; for a discussion, see Schwarzschild (1986) and Imry & Webb (1989). Bachtold et al. (1999) detected the effect in carbon nanotubes; for a discussion, see Kong et al. (2004). ELECTRIC AHARONOV-BOHM EFFECT Just as the phase of the wave function depends upon the magnetic vector potential, it also depends upon the scalar electric potential. By constructing a situation in which the electrostatic potential varies for two paths of a particle, through regions of zero electric field, an observable Aharonov-Bohm interference phenomenon from the phase shift has been predicted; again, the absence of an electric field means that, classically, there would be no effect. From the Schrödinger Equation , the phase of an eigenfunction with energy ''E'' goes as . The energy, however, will depend upon the electrostatic potential ''V'' for a particle with charge ''q''. In particular, for a region with constant potential ''V'' (zero field), the electric potential energy ''qV'' is simply added to ''E'', resulting in a phase shift: : where ''t'' is the time spent in the potential. The initial theoretical proposal for this effect suggested an experiment where charges pass through conducting cylinders along two paths, which shield the particles from external electric fields in the regions where they travel, but still allow a varying potential to be applied by charging the cylinders. This proved difficult to realize, however. Instead, a different experiment was proposed involving a ring geometry interrupted by tunnel barriers, with a bias voltage ''V'' relating the potentials of the two halves of the ring. This situation results in an Aharonov-Bohm phase shift as above, and was observed experimentally in 1998. MATHEMATICAL INTERPRETATION In the terms of modern Differential Geometry , the Aharonov-Bohm effect can be understood to be the Holonomy of the complex-valued Line Bundle representing the electromagentic field. The Connection on the line bundle is given by the Electromagnetic Potential ''A'', and thus the electromagnetic Field Strength is the Curvature of the line bundle ''F=dA''. The integral of ''A'' around a closed loop is the holonomy, which, by Stokes Theorem , is the magnetic field threading the loop. Thus the wave function of the electron can be seen to be directly coupled to the complex line bundle representing the electromagnetic field. See also a related effect, the Berry Phase . REFERENCES
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