# Nature of Perturbed state in Perturbation Theory?

I'm interested in the Nature of Perturbed state in Perturbation Theory.

The first order perturbed state is given by

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

Where

$$a_{m}=\int\psi_{1}^{(m)*}\psi_{n}^{(0)}d\tau.$$

Now as the Hamiltonian changes while applying perturbation. It's basis should change,My question is, why it is not changing when we expand perturbed state. More precisely what is the logical step behind expanding perturbed state as sum of unperturbed states?

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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.