- 1,170

- 2

_{+},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?