Show that if is real, the geometric phase vanishes. (Problems 10.3and 10.4 are examples...

Show that if is real, the geometric phase vanishes. (Problems 10.3and 10.4 are examples of this.) You might try to beat the rap by tracking an unnecessary (but perfectly legal) phase factor onto the eigenfunctions: where is an arbitrary (real) function. Try it. You'll get a nonzero geometric phase, all right, but note what happens when you put it back into Equation 10.23. And for a closed loop it gives zero. Moral: For nonzero Berry's phase, you need (i) more than one time-dependent parameter in the Hamiltonian, and (ii) a Hamiltonian that yields nontrivially complex eigenfunctions.

V(x) = -αδ(x)