Question of gauge for the description of two-photon transitions or light scattering when using incomplete sets

Abstract

In the calculation of two-photon transition probabilities and related radiation-matter interaction problems, the "length" and "velocity" forms of the dipole interaction give results that can differe by many orders of magnitude if a complete set of exact eigenfunctions is not used. These differences have been illustrated in the case of the $1s-2s$ two-photon transition in hydrogen by Bassani, Forney, and Quattropani [Phys. Rev. Lett. 39, 1070 (1977)], and it was concluded by these authors that for this problem the length form of the interaction is far superior to the velocity form when an incomplete set of bound states is used. In order to remove the question of gauge and thereby to improve the accuracy of such calculations when a complete set of exact eigenfunctions is not available, this paper proposes a modified form of the second-order equation. The modified equation, which employs a unique average-frequency approximation, is derived through incorporation of a sum rule that is a direct result of gauge invariance. It contains the length and velocity forms as limiting cases, and, like those equations, is exact for a complete set. Conventional length or velocity equations violate this sum rule for an incomplete set, and therefore are subject to large errors. The use of this modified equation is illustrated for the $1s-2s$ transition of hydrogen. Using only the $2p$ state as the "set" of intermediate states, the modified equation yields results that for all photon energies are within 2% of the exact probability amplitudes. For the $1s-2s$ problem, the accuracy is comparable to that obtained from length- and velocity-form equations employing an ordinary average-frequency approximation and closure. Unlike the modified equation, however, these latter equations require a separate calcualtion of matrix elements of $r^2$ or $p^2$, which may be inconvenient for a many-electron system. The modified equation should therefore be useful when these matrix elements are not easy to calculate, or when the best gauge form for the problem is not known.