# Driven quantum mechanical harmonic oscillator

1. Jan 18, 2010

### tommy01

Hi.

I just calculated the quantum mechanical harmonic oscillator with a driving dipole force $$V(x,t) = - x S \sin(\omega t + \phi)$$

I used the Floquet-Formalism. Then I calculated the mean expectation value, in a Floquet-State, of the Hamiltonian over a full Period T indirectly by using the Hellmann-Feynman Theorem.

$$\overline{H_\alpha} = \hbar \omega_0 \left( \alpha + \frac{1}{2} \right) - \frac{S^2}{4m} \frac{(\omega_0^2 + \omega^2)}{(\omega_0^2-\omega^2)^2}$$
So the first Term is the Energyeigenvalue of the "stationary" harmonic oscillator and the other term accounts for the perturbation mentioned above, which is always positive and therefore lowers the enery with a singularity at $$\omega_0 = \omega$$.
Now to my question. Is it physically possible to get a negative energy expectation value around $$\omega_0$$?