Time Dependent Perturbation Theory of Radiation-Matter Interactions

The Hamiltonian is conventionally written in terms of the position coordinates Qi (x, if Cartesian) and their conjugate momenta p. From Eq. 1.58, the latter are 

4 TIME-DEPENDENT PERTURBATION THEORY OF RADIATION-MATTER INTERACTIONS 

The quantum mechanical Hamiltonian operator corresponding to the classical Hamiltonian (1.74) is 

The parentheses surrounding the quantity V A in Eq. 1.76 indicate that V operates only on the vector potential A immediately following it. The choice of Coulomb gauge (V A = 0) for electromagnetic waves propagating in free space ( = 0) reduces the Hamiltonian to 

As a concrete example, the vector potential for a linearly polarized monochromatic electromagnetic plane wave with wave vector k may be written 

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