In a review question, we are asked to consider a particle of mass m and charge q in a 1-D harmonic oscillator potential V(x). Light is shined on the harm. osc. with E-field [tex]E=E_{o}cos((\omega)t-kx)[/tex], where [tex]k=\omega/c[/tex].(adsbygoogle = window.adsbygoogle || []).push({});

(Fine, so far. It seems like a Rabi frequency problem, similar to an Ammonia maser, which we studied. I'm assuming we can consider this charge in a harm. osc. as being analagous to an electron in a H-atom?)

I'm not exactly sure how to cross over the math for it though... Also, it's clear that [tex]V(x)=\frac{1}{2}m(\omega_{ho})^2x^2[/tex] is the potential of the charge, I think. Back to the question...

The light frequency [tex]\omega[/tex] is chosen to be resonant with the transition from the ground state to the first excited state of the harm. osc. Find an expression for the Rabi frequency for light induced transitions between the ground state and first excited state assuming the dipole approximation is valid.

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Okay, when I started on this question, I reviewed the Ammonia maser.

For that, the Rabi frequency [tex]\Omega[/tex] is given by [tex]\Omega=\eta/\hbar[/tex], where [tex]\eta=Ed_{o}[/tex].

When the frequency is "resonant", it is

[tex]\omega=\omega_{o}=\frac{2A}{\hbar}[/tex],

where A is the Bohr radius.

I'm not really sure if I'm thinking about this correctly. Can I cross these results over from the (very different) H-atom? Or in the context of the ground-to-first state transition, is it allowed?

Also, what is the "dipole approximation"? I know that it has to do with assuming that the wavelength of the radiation causing the transition is large compared to the size of the system making the transition, but how does that play into this problem quantitatively?

Is there an expression/equation I can use to ammend the assumptions above? There's nothing in my text on it, and I couldn't find anything more than qualitative on the web.

Thanks much for any pointers!!

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# Induced transitions, Dipole Approx.

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