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

The first order perturbed state is given by




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.

Addendum. One thing I forgot to mention is why one would want to write the perturbed state in the unperturbed basis. The simple, essential answer is that perturbation theory is so useful precisely because the unperturbed Hamiltonian can be easily diagonalized; namely it's eigenvectors and eigenvalues are known exactly, whereas those of the unperturbed Hamiltonian are not.

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A wavefunction is a complete description for a system where the basis states are all orthogonal. So if you think of a vector as a linear combination of the three basis vectors, any vector can be described by this method (including a different set of basis vectors). Similarly after a perturbation we can describe our new basis vectors as linear combinations of our old ones.

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