  A general quantum adiabatic theorem with and without the time-dependent
orthogonalization is proven, which can be applied to understand the origin of
activation energies in chemical reactions. Further proofs are also developed
for the oscillating Schwinger Hamiltonian to establish the relationship between
the internal (due to time-dependent eigenfunctions) and external (due to
time-dependent Hamiltonian) time scales. We prove that this relationship needs
to be taken as an independent quantum adiabatic approximation criterion. We
give four examples, including logical expositions based on the spin-1/2
two-level system to address the gapped and gapless (due to energy level
crossings) systems, as well as to understand how does this theorem allows one
to study dynamical systems such as chemical reactions.
