I am trying to derive the energy spectrum of a 1D chain of identical quantum oscillators from its Hamiltonian by Fourier transforming the position and momentum operator.(adsbygoogle = window.adsbygoogle || []).push({});

I came across this: https://en.wikipedia.org/wiki/Phonon#Quantum_treatment

However, I am unsure of the mathematics. Specifically, ## \sum x_l x_{l+m} ## onwards.

I am unsure of how ## \sum x_l x_{l+m} ## and ## \sum p_l^2 ## is derived from the two Fourier transformed coordinates and how the potential energy term is expressed.

Can anybody explain or provide another source that is more clear in the math?

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# I Energy spectrum of a chain of quantum oscillators

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