Long-wavelength approximation in on- and off-resonance transitions

In view of the possibilities of laser spectroscopy with very short pulses and relatively high frequencies, we examine aspects of the theory of atom-field interactions that are related to violation of the condition of the long-wavelength approximation (LWA) according to which kr<<1, where r is 'of the order of atomic dimensions' and k is the magnitude of the wave vector. On- and off-resonance transitions are considered, with kr being larger than unity due to the large extent of the two wave functions involved in the coupling matrix element. The implementation of the analysis uses bound-free transition matrix elements with the n=50 and 80 hydrogenic functions as initial states and values of k up to 27 eV, which is sufficient to produce kr>1, thereby rendering the LWA inoperative. In spite of this, it is shown that, for on-resonance transitions, the results from the use of the well-known multipole operators resulting from the LWA [e.g., the electric dipole approximation (EDA)] agree with those from the exact expressions derived here from the application of the multipolar Hamiltonian. This numerical agreement is proven analytically. As a test of the kind of convergence of the multipole series expansion for small values of k and large r, it is shown that the lowest-order ratio of electric dipole to quadrupole matrix elements decreases rapidly within 1.0 atomic unit above threshold. Finally, it is shown that off-resonance couplings lead to differences between the full-interaction operator and the EDA which cannot be neglected. In the extreme case of intrashell couplings for the n=50 shell, calculation shows that the 50p state is coupled to angular momentum states up to l=21, compared to the electric dipole coupling of Δl=±1