What's the evolution of a state under an adiabatic evolution? [duplicate]

For an initial state $|\Psi\rangle_0$ as the ground state of a Hamiltonian $H(0)$, if it undergoes an adiabatic evolution $H(t)$ to reach the ground state $|\Psi\rangle_1$ of $H(1)$.

Then what's the evolution from the state $|\Psi\rangle_0$ and $|\Psi\rangle_1$?

Can the evolution be written in a form given by $H(t)$ something as $exp(-i\int H(t))dt$? If this is the case, then what's the difference between an adiabatic evolution and a non-adiabatic evolution?

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• @ Adam Thanks. Yes, the time-ordered integration seems to be the solution. But then there is no difference between the adiabatic and the non-adiabatic evolution since this applies to a normal non-adiabatic evolution too. Does the adiabatic evolution is just to control the Hamiltonian $H(t)$ so that the excited state will not be generated during the evolution (or just the evolution is SLOW enough)? – XXDD Aug 23 '17 at 9:59