# Position operator for a system of coupled harmonic oscillators

1. Jul 2, 2012

### mtak0114

Hi

I would like to know if I have a system of coupled harmonic oscillators whether the standard position operator for an uncoupled oscillator is valid, i.e.
$x_i = \frac{1}{\sqrt{2}} (a_i+a_i^\dagger)$?
where i labels the ions.

To give some context I am looking at a problem involving a linear ion trap. The hamiltonian for this system with N ions is given by something like

$H = \frac{1}{2m}\mathbf{p}^T \cdot \mathbf{p} +\omega \mathbf{x}^TB\mathbf{x}$

where B is a coupling matrix and $x_i$ is a component of $\mathbf{x}$

This can be diagonalized in which case the Hamiltonian in the new coordinates (call them $\mathbf{X}$ and $\mathbf{P}$ ) looks like N uncoupled harmonic oscillators (with creation and annihilation operators $A$ and $A^\dagger$) which describe the collective motion of the original system. Now the original position operator $x_i$ is as far as I have seen always written with respect to the the creation and annihilation operators of the new coordinates (uncoupled oscillators $A$ and $A^\dagger$)

My question is would it be valid to write the position operator as I have done so above? and if so why is it more common to use the more commplicated expression in terms of $A$ and $A^\dagger$ as opposed to the one I have written?

cheers

Mark