Energy spectrum of a chain of quantum oscillators

[SataSata]

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.

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?

 

[drvrm]

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

well i saw a very lucid treatment of your problem in the following-
<
https://ocw.mit.edu/courses/...quantum.../MIT22_51F12_Ch9.pdf>
by P Cappellaro - ‎2011
i think it may help,thanks

 

[SataSata]

well i saw a very lucid treatment of your problem in the following-
<
https://ocw.mit.edu/courses/...quantum.../MIT22_51F12_Ch9.pdf>
by P Cappellaro - ‎2011
i think it may help,thanks

Thank you for the help. However, in the notes you provided, the final Hamiltonian does not have a $\hbar$, but in the wiki, there is a $\hbar$. Is that a mistake or am I missing something here?

 

[drvrm]

the final Hamiltonian does not have a ℏℏ\hbar, but in the wiki, there is a ℏℏ\hbar. Is that a mistake or am I missing something here?

many a time physicists use such units as h bar=c=1 esp. in high energy physics but you may check whether the dimensional requirements need a hbar and if it is necessary you can always put in Planck's constant ...