Particle in a Box: Fourier Sine Series Coefficients

[stunner5000pt]

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 the 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 the Foureir series?? Also why is the persiod a? If n was not 1 then the period would not be a, would it/?

 

[StatusX]

You have a definition of f(x), so just plug that into the integral. All of the terms will drop out except the one with c_n. I don't understand your question about the period.

 

[physics girl phd]

Are you talking about the a in the denominator of the k_n? This puts everything in the right length scale, so the eigenfunction's conditions at the boundary of the box are met. Does that make sense?