Abstract
If a time-dependent Hamiltonian is allowed to evolve adiabatically, and if it returns to its original form, then the ground state wavefunction must have picked up the dynamic or(and) the geometric phase factor(s) due to some interaction during the above evolution. Here, we prove that the standard definition for the pure geometric phase is not pure after all and it is composed of both dynamic and geometric phases if the wavefunction is a proper wavefunction. Only with improper wavefunctions, the geometric phase can be shown to be a pure geometric phase. Therefore, any change to this so-called generalized geometric phase implies some interaction-induced changes to both the phase and group momenta of a proper wavefunction. Apart from that, the Pancharatnam phase advance, δ=π−12Ω is also proved to be valid for both integral and half-integral spins to the extent that the original Berry's phase formula is also proved to be a false representation of the geometric phase where Ω is the angle subtended on a sphere.
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