Generalized Sturm expansions of the Coulomb Green’s function and two-photon Gordon formulas
| Autor: | S. I. Marmo, N. L. Manakov, A. A. Krylovetsky |
|---|---|
| Rok vydání: | 2001 |
| Předmět: | |
| Zdroj: | Journal of Experimental and Theoretical Physics. 92:37-60 |
| ISSN: | 1090-6509 1063-7761 |
| DOI: | 10.1134/1.1348460 |
| Popis: | The radial component of the Coulomb Green’s function (CGF) is written in the form of a double series in Laguerre polynomials (Sturm’s functions in the Coulomb problem), which contains two free parameters α and α′. The obtained result is applicable both in the nonrelativistic case and for the CGF of the squared Dirac equation with a Coulomb potential. The CGF is decomposed into the resonance and potential components (the latter is a smooth function of energy) for α = α′. In the momentum representation, the CGF with the free parameters is written in the form of an expansion in four-dimensional spherical functions. The choice of the parameters α and α ′ in accordance with the specific features of the given problem radically simplifies the calculation of the composite matrix elements for electromagnetic transitions. Closed analytic expressions (in terms of hypergeometric functions) are obtained for the amplitudes of bound-bound and bound-free two-photon transitions in the hydrogen atom from an arbitrary initial state ¦nl〉, which generalize the known (one-photon) Gordon formulas. The dynamic polarizability tensor components αnlm(ω) for an arbitrary n are expressed in terms of the hypergeometric function 2 F 1 depending only on l and $$\tilde \omega $$ and through the polynomial functions $$f_{nl} (\tilde \omega )$$ of frequency $$\tilde \omega = {{\hbar \omega } \mathord{\left/ {\vphantom {{\hbar \omega } {\left| {E_n } \right|}}} \right. \kern-\nulldelimiterspace} {\left| {E_n } \right|}}$$ . The Rydberg (n ≫ 1) and threshold (ℏω ∼ ¦ E n¦) asymptotic forms of polarizabilities are investigated. |
| Databáze: | OpenAIRE |
| Externí odkaz: |
|