Consider a bead on a circular wire loop of radius R. The loop rotates about a vertical diameter with angular velocity ω.
(a) Find the effective potential experienced by the bead, including the centrifugal term. Show that above a certain critical angular velocity ωc, the bottom of the loop ceases to be a stable equilibrium, and a new equilibrium appears at some angle θ0=0.
(b) The eigenvalues of the instantaneous Hamiltonian are En±=ℏωc(n+1/2)±ℏωn ... [rest of problem involves Berry phase computation]. Suppose the angular velocity increases very slowly from 0 to some value ω>ωc. What is the geometric phase acquired by the ground state during this process?