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I must misunderstood somewhere, couldn't figure out the following, any helps will be greatly appreciated.

The first order correction of the pertubated energy is:

[itex]\leftψ

_{n}

^{0}\langle[/itex] H

^{'}[itex]\rightψ

_{n}

^{0}\rangle[/itex]

Where:

ψ

_{n}

^{0}

Is the solution of the unpertubated Hamiltonian.

My question is can ψ

_{n}

^{0}be the general solution to the Hamiltonian or has to be a specified state vector?

i.e.,

ψ

_{n}

^{0}= aψ

_{1}

^{0}+bψ

_{2}

^{0}

Or has to be:

ψ

_{n}

^{0}=ψ

_{1}

^{0}

If it can be the superposition of the eigenstates, then how to construct the first order wave function?

Thanks in advance:)