Writing perturbed wave function

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TheBlueDot
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Hello,

I'm trying to write a wave function for a perturbed system. The original (unperturbed) wave function has two solutions for even and odd n-values, for example ##sin(n \pi x)## for even-n and ##cos(n\pi x)## for odd-n. Then, the perturbed wave function also has an even and odd solutions. My question is does ##k^{(0)}## in the attachment also need to be even or odd according to even and odd n-values?Thanks!
 

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