Consider two energy eigenstates of a quantum system with Hamiltonian  (for example, this could be the Hamiltonian of a particle in a  potential, like the Hydrogen atom or the Alkali atoms):
We want to consider the time dependent Hamiltonian
where  is the potential of the electromagnetic field. Treating the potential as a perturbation, we can expect the eigenstates of the perturbed Hamiltonian to be some mixture of the eigenstates of the original Hamiltonian with time dependent coefficients:
Plugging this into the time dependent Schrödinger equation
taking the inner product with each of  and , and using the orthogonality condition of eigenstates , we arrive at two equations in the coefficients  and : 
where . The two terms in parentheses are dipole matrix elements dotted into the polarization vector of the electromagnetic field. In considering the spherically symmetric spatial eigenfunctions  of the Hydrogen atom potential, the diagonal matrix elements go to zero, leaving us with
or
Here , where  is the Rabi Frequency.