REMARKS ON THE FORMULATIONS OF THE ADIABATIC THEOREM

By comparing the adiabatic limit of the exact solutions of the time-dependent Schrodinger equation for spin in rotating magnetic field and for harmonic oscilator with time-dependent frequency with the solutions obtained using the quantum adiabatic theorem we have demonstrated the complete agreement of the two sets of solutions and the importance of phase fixing condition for this agreement. We argue that the notions like "familiar dynamical phase" of the "usual quantum adiabatic theorem" and "an additional phase" of "geometrical origin" have been based on the unjustified neglection of the mentioned condition by applying the quantum adiabatic theorem. There is nothing to add to the quantum adiabatic theorem in which time-dependent eigenbasis satisfies phase fixing condition.