By James Hamilton
The Aharonov-Bohm influence is linked to cyclic movement. it's considered one of a couple of anholonomic results, and which means the dynamical description depends upon the present place of the procedure and at the direction during which it reached that place. An instance of an anholonomic influence is Foucault's recognized pendulum, which easily demonstrates the Earth's rotation. The Sagnac influence - a gentle beam passing round a circled procedure of mirrors - is one other instance. glossy dynamical advancements corresponding to Hannay's attitude and Berry's section are additional important examples.
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Additional resources for Aharonov-Bohm and other Cyclic Phenomena
We can write A - epAo = K , A being the (ordinary) angular momentum about Oz. 4a) is called the law of conservation of canonical angular momentum. The relation shows very neatly how A is forced to vary by the non-central nature of the Lorentz force. If the path is not a circle p = coast, then, except for the case of the solenoid, pAo will vary, and so must A. 1a), then Ao(p) can be expressed as a gauge invariant quantity [because it equals (27rp)-1 times a flux]. 4a) and therefore K itself, are gauge invariant.
For some account of the theory of these phenomena see Sects. 7 of Ref. 1], and Chap. 11 of Ref. 2]. Consider an area of cross-section of the Fermi surface (with fixed kz and B along Oz). 4b) the energy levels vary. As a result the free energy of the electron gas oscillates; the period of oscillation corresponds to the passage of successive quantized orbits through the Fermi level. 3 Next Approximation and the Radial Oscillations The following is a quick but imprecise method; an accurate treatment is to be found in Sects.
6) and the parts of A relating to single coordinates will be called action variables and labelled J etc. 51). The integration is around a closed path, and in general that will not be a circle. 5) is a special case, in that L 38 4. Classical Physics and the Adiabatic Principle does not explicitly contain 0. 4a) is a consequence. 11)]. Another action variable is J. 7a) Here the integration is over a period of the motion; cf. 8). Clearly J, > O, and Ja = 0 gives a circle p = const. Thus if p(t) = PT + D cos wt , where D is a constant and w is any frequency, Jp = ~2~r(Dw)2 I" sin2wtdt , where r = 21r/w is the period of this vibration.
Aharonov-Bohm and other Cyclic Phenomena by James Hamilton