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##\hat{H} \psi (x) = E \psi (x)##

where E is the eigenvalue of the *disturbed* eigenfunction ##\psi (x)##

and ##E_n## are the eigenvalues of the *undisturbed* Hamiltonian ##\hat{H_0}##

and the *disturbed* Hamiltonian is of the form:

##\hat{H} = \hat{H_0} +{\epsilon} \hat{V} ##

where ## \hat{V} ## is the potential and

then we can expand, using a Taylor expansion, ##\psi (x) ## and ##\epsilon## is a very small (positive?) number

##\psi (x) = {\psi}_0 (x) + {\epsilon}{\psi}_1 (x) + {\epsilon}^2 {\psi}_2 (x) + ... ## (1)

where ##{\psi}_k (x)## is the eigenfunction of ##\hat{H}## corresponding to the eigenvalue ##E_n## (eigenvalues are assumed to be non-degenerate)

We also get, using a Taylor expansion:

##E = E_0 +{\epsilon}{E_1} + {\epsilon}^2 {E_2} + ...## (2)

MY QUESTIONS:

I tried deriving formulae (1) and (2) - i.e. the expansions - on my own but I was unable to do it. Could anyone please show me *in detail* how to obtain those expansions?

Another question is:

How do we know that, when using the Taylor expansion, the coefficient ## {\epsilon} ## applies to ##{\psi}_1 (x)## and not to some other eigenfuncion, the coefficient ## {\epsilon}^2 ## applies to ##{\psi}_2 (x)## and not to some other eigenfuncion, and so on? The same question for eigenvalues.

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# Perturbation Theory - First Order Approximation

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