D ec 1 99 8 Semiclassical time – dependent propagation in three dimensions : How accurate is it for a Coulomb potential ?

A unified semiclassical time propagator is used to calculate the semiclassical time-correlation function in three cartesian dimensions for a particle moving in an attractive Coulomb potential. It is demonstrated that under these conditions the singularity of the potential does not cause any difficulties and the Coulomb interaction can be treated as any other non-singular potential. Moreover, by virtue of our three-dimensional calculation, we can explain the discrepancies between previous semiclassical and quantum results obtained for the one-dimensional radial Coulomb problem. 3.65.Sq,3.65.G,31.50 Typeset using REVTEX 1 Semiclassical propagation in time has been studied intensively in two dimensions [1–4]. There are by far not as many applications to higher dimensional problems, in particular not in connection with the singular Coulomb potential. Our motivation for this study is threefold: Firstly to see, if the advanced semiclassical propagation techniques in time, namely the Herman-Kluk propagator [5–7], can be implemented for realistic problems of scattering theory involving long range forces. Secondly to see, if we can avoid to regularize the Coulomb singularity in the classical equations of motion if we work in three (cartesian) dimensions, and thirdly, to clarify the reason for the small, but pertinent discrepancies with the quantum result in two previous, one–dimensional semiclassical calculations of the hydrogen spectrum from the time domain [8,9]. As it will turn out, the Coulomb problem with the Hamiltonian (we work in atomic units unless stated otherwise) H = p 2 + Z |r| (1) can be propagated in time semiclassically without taking any special care of the singularity in the potential which poses a lot of difficulties for the one-dimensional radial problem if Z < 0, i.e., if the potential is attractive as in the case of hydrogen (Z = −1) which we take as an example in the following. The relevant information in the time domain is the autocorrelation function c(t) = 〈ψ|K|ψ〉 (2) where K(r, r′, t) = 〈r|e|r〉 (3) is the propagator in the coordinate representation. By diagonalizing K in Eq. (2) one can express the autocorrelation function with the time evolution operator U(t), c(t) = 〈ψ(0)|U(t)|ψ(0)〉 ≡ 〈ψ(0)|ψ(t)〉. (4) This form has the obvious interpretation of correlating the time evolving wavefunction ψ(t) at each time with its value at t = 0. The extraction of the energy spectrum from Eq. (4) is routinely performed by Fourier transform, σ(ω) = ∫ c(t) e dt. (5) Expanding formally ψ(t) in terms of eigenfunctions ψ(t) = ∑ nlm anlmΦnlm e iEnt/h̄ (6) and inserting Eq. (6) into Eq. (5) one sees that σ(ω) = ∑ n δ(ω − En/h̄) bn. (7) Hence, the power spectrum σ(ω) exhibits peaks at the eigenenergies of the system with weights given by 2