Let a(adsbygoogle = window.adsbygoogle || []).push({}); _{+},a_{-}be the ladder operators of the harmonic oscillators. In my book I encountered the hamiltonian:

H = hbarω(a_{+}a_{-}+½) + hbarω_{0}(a_{+}+a_{-})

Now the first term is just the regular harmonic oscillator and the second term can be rewritten with the transformation equations for x and p to the ladder operators as:

hbarω_{0}(a_{+}+a_{-}) = x/(√(2hbar/mω))

My question is: Does this last term just represent a translation in the origin of the harmonic oscillator i.e. the potential is mω^{2}(x-x_{0})^2 where x_{0}is determined by ω_{0}? If so how do I see that algebraically?

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# Harmonic oscillator

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