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Physical Review A

Bound-bound two-photon transition matrix elements for the hydrogen atom in the dipole approximation

Tudor A. Marian

Published Apr 1, 1989• 11 citations
Publisher

Abstract

By use of the linear-response-vectors method, the Kramers-Heisenberg matrix elements for arbitrary bound-bound transitions of a nonrelativistic hydrogenlike atom have been evaluated. To illustrate the discussion, Rayleigh and Raman scattering have been selected as relevant examples of second-order processes. We have derived, in a unitary manner, A- and \ensuremath{\varphi}-gauge compact analytic formulas, both for angular momentum and Stark states. Our results are expressed in terms of two new generalized hypergeometric functions with four variables, $_{1}\mathrm{F}_{\mathrm{E}}$ and $_{2}\mathrm{F}_{\mathrm{E}}$, which are finite sums of Gauss hypergeometric functions. Special attention is given to the two-photon transitions involving the atomic ground state. For angular momentum states, we mention the connection that we have established between our general results and earlier ones.

DOI
10.1103/physreva.39.3816

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Physica Scripta•Jan 1, 2003•2 citations
By use of the linear-response-vectors method, the Kramers-Heisenberg matrix elements for arbitrary bound-bound transitions of a nonrelativistic hydrogenlike atom have been evaluated. To illustrate the discussion, Rayleigh and Raman scattering have been selected as relevant examples of second-order processes. We have derived, in a unitary manner, A- and \ensuremath{\varphi}-gauge compact analytic formulas, both for angular momentum and Stark states. Our results are expressed in terms of two new generalized hypergeometric functions with four variables, 1FE_{1}\mathrm{F}_{\mathrm{E}}1​FE​ and 2FE_{2}\mathrm{F}_{\mathrm{E}}2​FE​, which are finite sums of Gauss hypergeometric functions. Special attention is given to the two-photon transitions involving the atomic ground state. For angular momentum states, we mention the connection that we have established between our general results and earlier ones.
In this paper, we present an efficient alternative method for the evaluation of the three-photon transition matrix element in the dipole approximation from the ground state to bound states in atomic hydrogen. This method is a variation of the Dalgarno–Lewis method for the treatment of the second-order Stark effect in the hydrogen atom. In this approach, the infinite double sum over the complete set of states including the continuum states present in the third-order perturbation theory result is treated exactly. The closed analytical expression obtained for the matrix element, as a function of incident photon energy, clearly displays all singularities present in the original third-order perturbation theory result.
The two-photon decay mode of hydrogenic atoms from an arbitrary state (n1{n}_{1}n1​, l1{l}_{1}l1​, m1{m}_{1}m1​) to an arbitrary state (n2{n}_{2}n2​, l2{l}_{2}l2​, m2{m}_{2}m2​) is studied within the framework of nonrelativistic quantum mechanics. In the dipole approximation, these decay rates, which involve infinite summation over intermediate states, are derived exactly via a general second-order matrix element obtained by Kelsey and Macek and an implicit technique introduced by Dalgarno and Lewis. The results are expressed in terms of hypergeometric functions. For transitions ${n}_{1}s\ensuremath{\rightarrow}{n}_{2}s$, our results reduce to those of Klarsfeld whose starting point is the Coulomb Green's function. For transitions to the ground state, an alternative expression involving a simple one-dimensional integral is presented. The decay rate of the 2s2s2s metastable state of atomic hydrogen is calculated as an illustration of the method. The result, $\frac{1}{\ensuremath{\tau}}=8.2284$ ${\mathrm{sec}}^{\ensuremath{-}1}$, agrees with Klarsfeld. For transitions of ${n}_{1}s\ensuremath{\rightarrow}1s$ and ${n}_{1}d\ensuremath{\rightarrow}1s ({n}_{1}\ensuremath{\ge}3)$, the transition rates exhibit interesting and unexpected structures. In particular, "zeros" are found in the two-photon emission spectrum indicating that two-photon emission is not possible at certain frequencies. Physically, these "zeros" are the result of destructive interference between the radiating dipole terms associated with the sum over intermediate states. In addition to the emission spectrum the expected coincidence signal between two detectors monitoring the two photons simultaneously emitted during a two-photon transition is calculated as a function of the angle between the detectors. The angular distribution for the ${n}_{1}d\ensuremath{\rightarrow}1s$ transitions is shown to be significantly different from the ${n}_{1}s\ensuremath{\rightarrow}1s$ transitions. Finally, a possible experiment is suggested to test the results presented in this paper.
The interaction of a low-intensity laser field with a nonrelativistic one-electron atom is considered to the first order of perturbation theory and in the electric dipole approximation. The radiation field is turned on adiabatically modifying the initial unperturbed atomic state, which is either an angular-momentum eigenstate ∣nlm〉|\mathrm{nlm}〉∣nlm〉 or a Stark state ∣nnem〉|{\mathrm{nn}}_{e}m〉∣nne​m〉. The first-order correction to the wave function is expressed both in the length and velocity gauges in terms of a vector function called the linear-response vector and depending on the field-free energy eigenstate. We derive in a unifying manner the linear-response vectors in the position representation, as closed-form contour integrals, starting from a unique generating function built up with the Coulomb Green function. The linear-response vectors are then evaluated in momentum space via a Fourier transformation: they are obtained as integral representations and also in explicit form, as generalized hypergeometric functions. With reference to the static limit, we complement earlier results and find the reduced linear-response vectors in momentum space, as Fourier transforms of their coordinate-representation counterparts. The low-frequency behavior of the length-gauge first-order correction to the ground-state wave function is established in coordinate as well as in momentum space. We finally point out the high-frequency limit of the linear response in the velocity gauge.
Within the framework of second order time dependent perturbation theory and non relativistic dipole approximation, the exact expressions of the radial transition amplitude from 3d, 3p and 3s excited states, were obtained from the integral representation of the Green Coulomb function. Below threshold and above one photon ionization threshold, it can be expressed as an analytic combination of three Appell hypergeometric functions of first order, one of which is degenerated; while at threshold, it can be obtained in terms of an expansion of Gauss confluent hypergeometric functions. The angular coefficients are numerically evaluated and analysed for some available laser wavelengths.