# Nature of perturbed state in perturbation theory?

I'm interested in the “nature” of the perturbed state in perturbation theory in quantum physics.

The first order perturbed state is given by

$$\psi^{(1)}_{n}=\Sigma_{m}a_{m}\psi^{(0)}_{m} ~,$$

where the perturbed state $\psi^{(1)}_{n}$ is expanded as a sum over unperturbed states $\psi^{(0)}_{m}$ (i.e., eigenstates of the unperturbed Hamiltonian).

Now as the Hamiltonian changes upon applying the perturbation, its basis should also change accordingly. My question is, why the basis does not change when we add a perturbation to the system? More precisely, what is the justification for expanding the perturbed state as a sum of unperturbed states?

• It seems there is misuse or misunderstanding of the word “basis”. As the basis of the Hilbert space, one can use any complete set of states. The thing that changes upon adding a perturbation, is the set of the eigen-states of the Hamiltonian (not necessarily the basis of the Hilbert space). We use the same basis before and after adding the perturbation: that basis is the eigen-states of the unperturbed Hamiltonian --- as described in the answer below. Commented Dec 2, 2015 at 18:39

Suppose that the Hamiltonian is written as $$H = H^{(0)} + V$$ It is assumed that both $H^{(0)}$, the unperturbed hamiltonian, and $H$, the perturbed hamiltonian, are self-adjoint. In particular, there is an orthonormal basis $\psi^{(0)}_m$ for the Hilbert space consisting of eigenvectors of $H^{(0)}$, and a an orthonormal basis $\psi_m$ consisting of eigenvectors of $H$.
One can choose to write any state $\psi$ in either basis. In particular, one can choose to write the first order perturbed state in the basis $\psi^{(0)}_m$ of $H^{(0)}$ and then use this to derive lots of useful things.