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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.

[itex]x_i = \frac{1}{\sqrt{2}} (a_i+a_i^\dagger)[/itex]?

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

[itex]H = \frac{1}{2m}\mathbf{p}^T \cdot \mathbf{p} +\omega \mathbf{x}^TB\mathbf{x} [/itex]

where B is a coupling matrix and [itex]x_i[/itex] is a component of [itex]\mathbf{x}[/itex]

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

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 [itex]A[/itex] and [itex]A^\dagger[/itex] as opposed to the one I have written?

cheers

Mark

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# Position operator for a system of coupled harmonic oscillators

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