# Why do "good" quantum states remain stationary under perturbation?

I've been reading the degenerate perturbation theory section of Griffiths QM. He introduces the idea that, if we can find an operator $$\hat A$$ which commutes with $$\hat H^0$$ and $$\hat H'$$, then simultaneous eigenfunctions of $$\hat A$$ and $$\hat H^0$$ (say, $$\psi_a$$ and $$\psi_b$$) will be "good" eigenstates to use in the degenerate theory, satisfying $$W_{ab} \equiv \langle \psi_a^0 | \hat H' \psi_b^0\rangle = 0$$ (which resolves computational issues involving division by zero).

In other words, if the operator $$\hat A$$ commutes with the new Hamiltonian $$\hat H = \hat H^0 + \hat H'$$ and is therefore conserved under the perturbation in the sense that $$\frac{\mathrm d\langle A\rangle}{\mathrm dt} = \frac i \hbar \langle [\hat H, \hat A] \rangle + \left\langle \frac{\partial \hat A}{\partial t}\right\rangle = 0,$$ then the simultaneous eigenstates of $$\hat A$$ and $$\hat H^0$$ are "good".

Intuitively, I understand that the underlying reason why certain states are "good" is that they remain stationary after the perturbation is turned on, whereas other states in the degenerate subspace are no longer stationary.

My question is, how can I concretely prove that $$\langle A\rangle$$ being conserved after the perturbation is turned on leads to its stationary eigenstates remaining stationary under the perturbation?

As for the mathematical proof, you already got it: if $$\langle A \rangle$$ is conserved, $$A$$ commutes with the full Hamiltonian, which implies that they can be simultaneously diagonalized. The basis that simultaneously diagonalizes these two operators is, by definition, what you called a "good" basis.
• Thanks. On further inspection of the concept, I realize that by using the time-independent Schrödinger equation in the first place to develop the degenerate perturbation theory, we are assuming that the states are stationary. It is from this that everything else (constructing and diagonalizing $\mathbf W$) is derived. Jun 10, 2022 at 17:37