The Hamiltonian

Consider a quantum system with a Hamiltonian $$H_0$$. Suppose this system is subject to an external driving force $$F_e (t)$$ such that the full Hamiltonian takes the form

$H = H_0 - BF_e(t) = H_0 + H'$

where $$B$$ is an operator through which this coupling occurs. This is the situation, for example, when the infrared spectrum is measured experimentally - the external force $$F_e (t)$$ is identified with an electric field $$E (t)$$ and $$B$$ is identified with the electric dipole moment operator. If the field $$F_e (t)$$ is inhomogeneous, then $$H$$ takes the more general form

$H = H_0 - \int d^3x\; B({\textbf x}) F_e({\textbf x},t) = H_0 - \sum_{\textbf k} B_{\textbf k}F_{e,{\textbf k}}(t)$

where the sum is taken over Fourier modes. Often, $$B$$ is an operator such that, if $$F_e (t) = 0$$, then

$\langle B \rangle = {{\rm Tr}\left(Be^{-\beta H}\right) \over {\rm Tr}\left(e^{-\beta H}\right)}$

Suppose we take $$F_e (t)$$ to be a monochromatic field of the form

$F_e(t) = F_{\omega}e^{i\omega t}$

Generally, the external field can induce transitions between eigenstates of $$H_0$$ in the system. Consider such a transition between an initial state $$\vert i \rangle$$ and a final state $$\vert f \rangle$$, with energies $$E_i$$ and $$E_f$$, respectively:

 $$H_0\vert i\rangle$$ $$E_i\vert i\rangle$$ $$H_0\vert f\rangle$$ $$E_f\vert f\rangle$$

(see figure below).

This transition can only occur if

$E_f = E_i + \hbar\omega$