In "Introduction to Quantum Mechanics" by Griffiths, right at the beginning of section 9.1.1 (Time-Dependent Perturbation Theory, The Perturbed System), Griffiths states:
Now suppose we turn on a time-dependent perturbation, $H'(t)$. Since $\psi_a$ and $\psi_b$ constitute a complete set [of the two-level system], the wave function $\Psi (t)$ can still be expressed as a linear combination of them. The only difference is that $c_a$ and $c_b$ are now functions of t:
I don't understand. You modify the Hamiltonian, you modify the solution basis - easy as that. Why on earth does he assume that if you add a time-dependent perturbation to the Hamiltonian the basis (for the two-level system that he considered in the section right before) will remain the same? And if this is indeed a mistake, then how valid is the assumption that the true wave function $\Psi (t)$ is merely a time-dependent linear combination of the two states $\psi_a$ and $\psi_b$?