Berry Phase (Homework 9, Physics 325)

1. Berry phase for real wave functions

Assuming that $<ψ_n|∇_R|ψ_n>$ is pure imaginary, show that when the wavefunctions $ψ_n(t)$ are real, the net geometric phase in a cyclic process is zero.

For a closed path one has

$0=∮ dR=∮ <ψ_n |ψ_n>dR$

$∇0=0=∇(∮ <ψ_n |ψ_n>dR)=∮ <∇ψ_n |ψ_n>dR+∮ <ψ_n |∇ψ_n>dR=$

$=∮ <ψ_n |∇ψ_n >* dR+∮<ψ_n |∇ψ_n>dR$

But since ψ_n  are real  it means $<ψ_n |∇ψ_n>$ is real  and  $<ψ_n |∇ψ_n >*=<ψ_n |∇ψ_n>$  so that

$0=2∮ <ψ_n |∇ψ_n>dR=2γ/i$   and thus $γ=0$