# 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

## 2 Answers

A basis is a set of wave functions such that a any wave function can be formed as a linear combination of basis wave functions. Often you choose them to be eigenfunctions of the Hamiltonian. But you don't have to.

If you change the Hamiltonian, you change the egienfunctions, so you change the most common choice for a basis.

• I don't think this answer addresses the underlying point... at least from a wave mechanics point of view, there's nothing guaranteeing the two states are the solution set for the modified hamiltonian. That is, maybe the solution to the perturbed system is not spanned by the unperturbed solution set. – anon01 Sep 28 '16 at 5:56
• True. I addressed the point that I thought was causing confusion. Perhaps the other answer addresses your point better. – mmesser314 Sep 28 '16 at 13:03

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 $H_0$.