= Solution
Set $\vartheta_\alpha(t)=\hbar^{-1}\int^t E_\alpha(s)ds$. Differentiating the expansion produces $\dot a_\beta|\varphi_\beta\rangle+a_\beta|\dot\varphi_\beta\rangle-iE_\beta a_\beta|\varphi_\beta\rangle/\hbar$ for each term. The <energy> term cancels $H|\Psi\rangle$ in the <Schrödinger equation>. Projecting the remainder with $\langle\varphi_\alpha|$ and using <orthonormality> yields
$$
\boxed{\dot a_\alpha=-\sum_\beta\langle\varphi_\alpha|\dot\varphi_\beta\rangle a_\beta e^{i(\vartheta_\alpha-\vartheta_\beta)}}.
$$
The diagonal term is purely imaginary and supplies the <geometric phase>. Off-diagonal <instantaneous eigenstate couplings> transfer population; differentiating the <eigenvalue> equation gives $\langle m|\dot n\rangle=\langle m|\dot H|n\rangle/(E_n-E_m)$ when the levels are distinct.
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