By James Hamilton
The Aharonov-Bohm impact is linked to cyclic movement. it's considered one of a couple of anholonomic results, and which means the dynamical description is determined by the present place of the approach and at the course wherein it reached that place. An instance of an anholonomic impact 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. smooth dynamical advancements corresponding to Hannay's attitude and Berry's section are additional necessary examples.
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Extra info for Aharonov-Bohm and other Cyclic Phenomena
3) below]. 8) with the equality holding for a Ganssian minimum uncertainty profile. We arbitrarily choose the condition: 1 6 A~opz < -~( Pz)2~ , with d~ (Spz)2~ = e ~ z This should ensure the validity of Pryce's argument. 7a) is to be valid. This result can be understood in another fashion. Scanning along a line y = const, in Fig. 5, fringes appear at separation (co/bo)[zf I, and that is equivalent to separation Izf[ on x = - f . The coherent wave front should cover at least one such unit, in order that the fringes remain stable with respect to each other.
7) will exist for closed orbits. The adiabatic theorem (cf. Sect. 4 below, and in more detail in Sect. 3) then states that Jo does not change when the flux ~ is varied sufficiently slowly. Thus for any closed orbit around the solenoid, Eq. 8cl where J0 is constant. This result is also a more general form of the result in Sect. 7. 8b). 8b), K is an adiabatic constant. Thus K does not vary when the flux 9 is changed sufficiently slowly. 3 Angle-Action Equations of Motion A conservative periodic dynamical system such as we are considering, can be described in terms of canonical conjugate pairs of variables: (Ja, Ca) , (Jp, Cp) r Cp are angle variables, but in general they are not geometric angles.
The change in r is given by an alteration in the relation between CA(Z) and r that was shown in Sect. 10; changing can thus be seen as a change in the boundary condition that the wave function must obey on going once around the surface of the solenoid. The situation is somewhat similar to the polarization of the vacuum around an atomic nucleus of charge +Ze. Changing Z will change the polarization whether or not any electron is present. The difference between the two cases is that it may be easier to envisage the electric field arising from the vacuum polarization, than it is to remember the difference between wave 48 5.
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