Wave function in terms of Basis Functions

[Domnu]

Problem
We have the function $$g(x)=x(x-a) \cdot e^{ikx}$$. Express $$g(x)$$ in the form

$$\sum_{n=1}^\infty a_n \psi_n (x)$$

where

$$\psi_n = \sqrt{\frac{2}{a}} \sin \(\frac{n\pi x}{a}\)$$

Solution
I have absolutely no clue as to how to start... I know a bit about Fourier series, but here, the function $$g(x)$$ has an infinite period.

 

[Domnu]

Actually wait... we can just let $$g(x)$$ go from 0 to a, since it's supposed to model a particle in a box from potential walls 0 to a.

 

[EngageEngage]

my guess would be that you could do it from 0 to a. So, I believe the problem now is to find the expansion coefficients An. To do this you will have to use Fourier's principle of orthogonality. I'm not 100 percent sure though, but it seems reasonable.

 

[maverick280857]

The set $\{\psi_{n}(x)\}$ is complete, which means you can express any (well behaved, piecewise continuous) function as a linear combination of its elements. The coefficients which act as weights are to be determined by Fourier's theorem.

These are actually eigenfunctions for a particle in a 1D box. Eigenfunctions are complete by definition.