Quadratic Hamiltonian in Fourier Space

The kinetic term:

$$\sum_n \frac{p_n^2}{2} = \sum_k \frac12 \vert p_k\vert^2.$$ (17)

The harmonic potential:

$$\sum_n \frac12 (q_{n+1}-q_n)^2 = \sum_k \frac12 \omega_k^2 \vert q_k\vert^2,$$ (18)

with the linear dispersion relation

$$\omega_k = 2 \sin\left( \frac{\pi k}{N} \right).$$ (19)

Hence the quadratic Hamiltonian:

$$H_2 = \sum_k \frac12 \vert p_k\vert^2 + \frac12 \omega_k^2 \vert q_k\vert^2.$$ (20)

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