# Possible Error in Assumption - Griffiths Quantum Mechanics

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 $\psi_b$?

• At any fixed time, the state $\psi(t)$ can be written as a linear combination $c_a(t)\psi_a+c_b(t)\psi_b$. Now just think of $c_a$ and $c_b$ as functions of time. – WillO Sep 28 '16 at 3:16

That's why it's called perturbation. You use the Hamiltonian $H_0$ and you get a set of eigenfunctions. Then you add a perturbative Hamiltonian $H'$. Though you tweaked the Hamiltonian, the original eigenfunctions remains the same. You can always calculate the perturbative eigenfunctions using iteration method, but your original eigenfunctions is still related to $H_0$.