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## Homework Statement

The bottom of an infinite well is changed to have the shape

$$V(x) = \epsilon \sin {\dfrac{\pi x}{b}}, 0 \le x \le b$$

Calculate the energy shifts for all the excited states to first order in ##\epsilon##. Note that the well originally had ##V(x) = 0## for ##0 \le x \le b## and ##V = \infty ## elsewhere.

## Homework Equations

/Attempt at a solution[/B]I know I should use ##<\Psi _n | H_1 | \Psi_n>##, and that ##H_1 = \epsilon \sin {\dfrac{\pi x}{b}}##. If I had ##\Psi _n## all I to do was integrate between ##0## and ##b##. However, I don't have ##\Psi _n##. For the "normal" potential well, I know that ##\Psi _ n = \sqrt{2/b} \sin {\dfrac{n \pi x}{b}}##. However, that is not the case in this exercise.

Any suggestions?