13.1.1: 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

\[H = H_0 - BF_e(t) = H_0 + H' \nonumber \]

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

\[ \begin{align*} H &= H_0 - \int d^3x\; B({\textbf x}) F_e({\textbf x},t) \\[4pt] &= H_0 - \sum_{\textbf k} B_{\textbf k}F_{e,{\textbf k}}(t) \end{align*}\]

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

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

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

\[ F_e(t) = F_{\omega}e^{i\omega t} \nonumber \]

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:

\[ \begin{align*} H_0\vert i\rangle &= E_i\vert i\rangle \\[4pt] H_0\vert f\rangle &= E_f\vert f\rangle \end{align*}\]  

(see figure below).

This transition can only occur if

\[ E_f = E_i + \hbar\omega \nonumber \]