Aharonov-Bohm and other Cyclic Phenomena by James Hamilton

By James Hamilton

The Aharonov-Bohm impact is linked to cyclic movement. it really is considered one of a few anholonomic results, and which means the dynamical description will depend on the present place of the approach and at the direction wherein it reached that place. An instance of an anholonomic impression is Foucault's recognized pendulum, which easily demonstrates the Earth's rotation. The Sagnac impact - a mild beam passing round a turned around method of mirrors - is one other instance. smooth dynamical advancements resembling Hannay's attitude and Berry's section are additional valuable examples.

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11]) the quantum number of the angular momentum about the axis of symmetry will be unaltered provided the change 6~ in the (inaccessible) flux is made sufficiently slowly. 20a) on taking the expectation value with respect to the circulating wave packet. The AB effect is an assertion about the phase of the (circulating) wave function in the case of an inaccessible flux that is unaltered in time, relative to the phase when there is zero flux. Thus the AB effect is not an absolute, but a relative effect.

9], there are surveys of these arguments. 22] (see also Ref. 23]). Consider an infinite cylindrical solenoid as in Fig. 23]. In order to see what is wrong with this argument, it is useful to remember the basic idea of the gauge transformation in the case of wave mechanics. 10a) Thus the electron probability destribution I~b(x)l2 is unaltered by the gauge transformation. 13a). 11a) to the eigenvalue of Lz. 13), as the total component of the electron's angular momentum about the symmetry axis Oz.

The total energy is E = (M + n + 1)/iwB . 10) The quantum numbers M and n relate to the azimuthal and the radial motion respectively. 9c). The corresponding wave functions are obvious. 6). 4a) defines the flux through the mean radius P0 of the wave packet in terms of M. (iii) The Gauge. The role of the gauge is important. 1), but in Appendix H, for the same physical problem, the gauge A = (-By, O,O) is used. Different gauges may give different wave functions r for the same problem, but they should yield the same gaugeinvariant operators, or expectation values, such as the current density j(x).

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