# Hamiltonian of two level system under radiation

In quantum mechanics, as I understand, the Hamiltonian for a two level system is given as $$\hat{H} = \frac{1}{2}\hbar \omega_0 \hat{\sigma_{z}},$$ does anyone know why the perturbation term of the Hamiltonian, when the two level system is being subjected to an applied electric field $\vec{E} = E_0 \cos(\omega t) \hat{r}$ , is given as $$\vec{d} \cdot E_0 \cos(\omega t) \hat{\sigma_x}?$$ where $\vec{d}$ is the dipole moment $-q \hat{x}$. Also, what is the most general expression for this Hamiltonian of a two level system under the influence of an electric field?

Thanks for any assistance.

• It is not "the" perturbation term, but merely the most important part of the electric field associated with an incoming plane wave. See this webpage on the dipole approximation: farside.ph.utexas.edu/teaching/qmech/Quantum/node118.html – ZeroTheHero Apr 4 '17 at 19:48
• @ZeroTheHero The dipole approximation is usually given as $\vec{d} = e\vec{r}$ for say a Hydrogen atom. What he wrote does seem to be the perturbation term of the Hamiltonian of a dipole perturbed by a sinuoisdal electric field which is usually given by $$H_{I}(t) = e \vec{r} \cdot \vec{E_0}\cos(\omega t) = \vec{d} \cdot \vec{E_0}\cos(\omega t).$$ I'm just not sure where the $\hat{\sigma_x}$ comes from? – Alex Apr 5 '17 at 10:13
• @ZeroTheHero Please have a look at a QM post where I have a proposed answer in the comments. I'm having difficulty making progress on this. Thanks for your time. – user101311 May 4 '17 at 9:13
• @Moses I am travelling and away from the office, with only limited random times to come back to PSE... – ZeroTheHero May 4 '17 at 10:43
• @ZeroTheHero Okay no prob, if you have chance, I would appreciate a look over what I have done but I will put a bounty on the question tomorrow to get more attention. Thanks. – user101311 May 4 '17 at 17:15