Position operator for a system of coupled harmonic oscillators

[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.
[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

 

[eljose79]

Hello Mark,

Thank you for your question. The standard position operator for an uncoupled harmonic oscillator, x_i = \frac{1}{\sqrt{2}} (a_i+a_i^\dagger), is indeed valid for a system of coupled harmonic oscillators. This is because the position operator is a linear combination of the creation and annihilation operators, which are valid for both coupled and uncoupled oscillators.

In terms of the Hamiltonian for a system of coupled harmonic oscillators, it is more common to use the more complicated expression in terms of the creation and annihilation operators (A and A^\dagger) because it allows for a more straightforward calculation of the system's energy levels and wavefunctions. The expression x_i = \frac{1}{\sqrt{2}} (a_i+a_i^\dagger) is derived from the more complicated expression and is often used as a shorthand notation.

I hope this answers your question. If you have any further inquiries, please don't hesitate to ask.