Equivalence of interaction hamiltonians in the electric dipole approximation

Abstract

The interaction of an atomic system with an externally applied electromagnetic field can be treated in the electric dipole approximation by means of either the minimal coupling (p · A) or direct coupling (d · E) Hamiltonian. It is shown that both methods lead to identical and unambiguous predictions for observable quantities as long as the atomic wavefunctions are transformed when used in the minimal-coupling formulation. The physical meaning of kinetic momentum is used to show that the atomic states must be described by wavefunctions calculated in the absence of an electromagnetic field when using the d · E (but not the p · A) form of the interaction Hamiltonian. When, however, observables are calculated using the common approximations of resonance atomic physics – the two-level approximation and the rotatingwave approximation – the two formulations can lead to measurably different results. This point is illustrated by calculating the induced polarization (and hence the refractive index) of an atomic system for the two exactly soluble cases of the harmonic oscillator and the hydrogen atom.