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[tex]\psi(x,0)= \int_{-\infty}^{\infty}g(k)\exp(ikx)dk[/tex]

and

[tex]g(k)= \int_{-\infty}^{\infty}\psi(0,0)\exp(-ikx)dx[/tex]

one is the fourier transform of the other. some cases to solve this is when we assume a small delta k, so g(k) behaves like a pulse "delta" function.

is there any more general case we can solve this?

i have been thinking hard if we have definite periodic x, say from 0 up to [itex]2\pi L[/itex], is it solvable?

what would be the (periodic) eigen energy function (if it is)?