Inverse of a series (solid state)

Question

I am working with the expression involving the equilibrium displacement ($y_n$) for the $n$th particle in a 1D harmonic lattice in terms of the normal modes coordinates $A_k$. Let me show you the expression:

$$ y_n(t) = \sqrt{\frac{2}{N}} \sum_k A_k(t) \sin(kan) $$

where $k$ is the wave vector and $a$ the lattice spacing.

My question is, how can I invert the series in order to express the $A$'s in terms of $y$'s?. I was thinking about an inverse Fourier transform but since $\imath$ is not involved I'm not sure.

Answer 1

You can use the orthogonality of the sinusoids to do the inversion.

First, multiply both sides by $\sin (l a n)$ where $l$ is an integer: $$y_n(t) \sin(l a n) = \sum_k A_k(t) \sin(k a n) \sin(l a n).$$ Now sum both sides over $n$

$$\sum_n y_n(t) \sin(l a n) = \sum_{k,n} A_k(t) \sin(k a n) \sin(l a n).$$

On the right hand side, the sum over $n$ gives $\delta_{l,k}$ up to pre-factors of $N$ which you get to figure out :) This leaves you with

$$\sum_n y_n(t) \sin(l a n) = A_l(t).$$

Bam.