# Particle in a box

For a particle in a one dimensional infinite squat well of width a, s.t 0<x<a the eignefunctions are given by

$$\psi_{x} (x) = N \sin k_{n} x$$ for 0 < x < a

where $$k_{n} = \frac{n \pi}{a}$$ and n = 1,2,3,...

Consider the Fourier sine series for the function f(x) on teh interval 0<x<a
$$f(x) = \sum_{n=1,2,3,...} c_{n} \psi_{n} (x)$$

Showthat the coefficients of this series are given by
$$c_{n} = \frac{2}{a} \int_{0}^{a} \sin (k_{n} x) f(x) dx$$

do i have to PROVE that the coefficients are given by Cn??

isnt the expression by Cn given by the definition of Cn from teh Foureir series?? Also why is the persiod a? If n was not 1 then the period would not be a, would it/?