PHYSICAL REVIEW A VOLUME 47, NUMBER 1 JANUARY 1993 Second-order corrections to the wave functions of a Coulomb-field electron in a weak uniform harmonic electric field Viorica Florescu and Adam Halasz Faculty of Physics, University of Bucharest, Bucharest Ma-gurele MG 11, -R 76-900 Romania Mircea Marinescu* Institute for Gravity and Space Science, Bucharest Mag-urele MG 6, R-7690-0 Romania (Received 11 October 1991; revised manuscript received 17 July 1992) For an electron in the presence of a Coulomb field and a weak uniform harmonic electric field, and for states which are Coulomb bound states of definite angular momentum in the remote past, we present an- alytic expressions for the tensorial fields, which are the basic quantities that are used to construct the second-order correction to the electron wave function. The calculation is based on solving the inhomo- geneous differential equations connected with this correction in position space. The results are useful for the evaluation of several higher-order radiative processes. PACS number|, 's): 32.80. — t 03.65.Ge I. INTRODUCTION Some recent papers have reconsidered the problem of fi'nding an analytic expression for the first-order correc- tion to the wave function of an electron in the Coulomb field of a fixed nucleus, in interaction with an electromag- netic monochromatic plane wave, a problem initially dis- cussed by Podolsky [1] in 1928. For long wavelengths, the problem reduces to the interaction with the homo- geneous monochromatic electric field 6'= Cocoscot . (1) Under the assumption that the electron — electric-field interaction vanishes adiabatically in the remote past, compact analytic solutions have been derived for the fol- lowing initially stationary Coulomb states: the ground state [2,3], any spherical excited state [3], the Stark (or parabolic) states [4, 5], and the continuum partial waves or scattering states [6,7]. In Refs. [4] and [7] one finds also analytic expressions for the first-order perturbed wave functions, based on the Sturmian expansion of the radial Coulomb Green's function, a procedure which is very successful in the study of high-order radiative pro- cesses. Our analytic equations [3] allowed a systematic study of several two-photon processes in hydrogen, lead- ing to alternative expressions to the exist- ing ones: bound-state — bound-state [8,9], bound-state— continuum-state [10], and continuum-state — continuum- state transitions have been considered [11]. By now the first-order correction has been derived or rederived [12] with several methods. A powerful method seems to be the one which exploits the inhomogeneous equation satisfied by the corrections to the wave func- tions. In fact, this method, which has been used previ- ously in the ground-state case [2, 13], allowed us to study the excited-states case. Nevertheless, our published re- sults [3] on excited states do not present this method. It is the purpose of this paper (Secs. II and III) to show how the method works for the determination of the second- order corrections to the wave functions of initially bound states of definite angular momentum. More explicit re- sults will be given for some particular cases is Sec. IV. Special attention is paid in Sec. V to the ground-state case. Some possible applications, mentioned in the Sec. VI, are three-photon (bound-state — bound-state or bound-state — continuum-state) and four-photon (bound- state — bound-state) transitions. From our analytic results one can also construct the first- and second-order correc- tions in a perturbative expansion of the Floquet solution [14] attached to the bound states of hydrogen. Because some results found in our study are also useful in other related problems, as, for instance, the attempt to treat the corresponding relativistic problem [15], we present our calculation in some detail, especially through Appendices A — D. From a technical point of view, Appendix C is the most important. For the sake of clarity we have tried to maintain a cer- tain correspondence between the content of this paper and that of Ref. [3]. II. SECOND-ORDER CORRECTION TO THE BOUND STATES WAVE FUNCTIONS AND THE KET VECTORS iw;J. „t (Q', 0) ) In this paper we study the ket vector ~w, „t (II', 0)) =Go(Q')P;Go(A)P ~nlm ), (2) where Go is the Coulomb resolvent operator and P the momentum operator. As we shall see in the following, the entity (2) has a direct connection to the second-order correction to the state vector ~%(t) ) of an electron which, in the presence of a fixed nucleus, undergoes the infiuence of the electric field (1) turned on adiabatically, and which in the remote past was a stationary state of definite angular momentum. In the mathematical trans- formations described here, A and 0' will be treated as complex parameters. In the physical applications we meet values of 0 or (and) 0' near the real axis, with a 394 QC1993 The American Physical Society