  We show that in a quantum adiabatic evolution, even though the adiabatic
approximation is valid, the total phase of the final state indicated by the
adiabatic theorem may evidently differ from the actual total phase. This
invalidates the application of the linearity and the adiabatic approximation
simultaneously. Besides, based on this observation we point out the mistake in
the traditional proof for the adiabatic theorem. This mistake is the root of
the troubles that the adiabatic theorem has met. We also show that a similar
mistake remains in some recent modifications of the traditional adiabatic
theorem attempting to eliminate the troubles.
