Undergrad Aharonov-Bohm Effect & Adiabatic Approximation

[Silviu]

Hello! I am reading Griffiths book on QM and in the chapter about the adiabatic approximation he introduces the Aharonov-Bohm effect. I am not sure why is this effect an example of adiabatic approximation. The hamiltonian doesn't depend on time as the vector potential is not a function of time so I don't really see how does this fit in the chapter. Griffiths mentions, indeed, that the same effect holds even when the vector potential depends on time, but again, there is nothing mentioned about slow change (which is required for the adiabatic approximation to hold). Can someone explain to me what is Griffiths actually doing there? Thank you!

 

[strangerep]

Can someone explain to me what is Griffiths actually doing there?

Since no one else has answered, here's my $0.02...

Chapter 10 of Griffiths progresses through the following sequence:

-- The adiabatic theorem,

-- Nonholonomic systems (in which interesting measurable phenomena can happen even adiabatically, by transport around a closed loop),

-- Geometric Berry phase (which is an example of such "interesting, measurable" phenomena). He derives the geometric net phase change formula [10.49] using the adiabatic approximation back in [10.40]. See his remarks under point 3 on p338.

-- Then he gives an example of an electron in a magnetic field of constant magnitude, but changing direction, (still using the adiabatic approximation), and derives the net phase change of the electron wave function.

-- Then he moves on the A-B effect (presumably because it involves a net phase change around a closed loop, similar to geometric phase). BUT, as he notes on p348 after eq[10.95], in this case the process of going around the solenoid doesn't have to be adiabatic.

I'm not sure why Griffiths does it this way. Maybe it's because the general idea of net phase change around a closed loop is easier to present at introductory level if one uses the adiabatic approximation (which simplifies some of the math). But the A-B effect, which doesn't need the adiabatic approximation, shows that net phase change around a closed loop is of far more general importance in QM, not merely in the restricted circumstances of the adiabatic approximation.

HTH.