= Instantaneous eigenstate coupling
{title2=$A_{mn}=\langle m(t)|\partial_tn(t)\rangle$}
Expanding a <quantum state> in instantaneous <energy eigenstates> separates <dynamical phases> from the coupling $A_{mn}$. <orthonormality> makes $A$ anti-Hermitian. For distinct <eigenvalues>, differentiating the <eigenvalue> equation gives $A_{mn}=\langle m|\dot H|n\rangle/(E_n-E_m)$. Its off-diagonal entries drive nonadiabatic transitions; diagonal entries describe phase changes of the basis and generate the <Berry phase>.
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