2. Discrete Fourier Transform

Define the discrete Fourier transform (DFT) of positions and momenta:

	$$\tilde{x}_k$$	$$= \frac{1}{\sqrt{N}} \sum_{j=0}^{N-1} x_j e^{-i k j},$$	$$x_j$$	$$= \frac{1}{\sqrt{N}} \sum_{k=0}^{N-1} \tilde{x}_k e^{i k j},$$	(33)
	$$\tilde{p}_k$$	$$= \frac{1}{\sqrt{N}} \sum_{j=0}^{N-1} p_j e^{-i k j},$$	$$p_j$$	$$= \frac{1}{\sqrt{N}} \sum_{k=0}^{N-1} \tilde{p}_k e^{i k j},$$	(34)

with $k = 2\pi n / N$, $n = 0, \dots, N-1$.