# I Energy spectrum of a chain of quantum oscillators

1. Sep 30, 2016

### 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?

2. Sep 30, 2016

### drvrm

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

3. Sep 30, 2016

### SataSata

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?

4. Oct 1, 2016

### drvrm

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 ....