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Physical Review A

Sum rules and expansion formula for Stark radiative transitions in the hydrogen atom

David R. Herrick

Published Nov 1, 1975• 24 citations
Publisher

Abstract

An expansion formula for Stark effect transition probabilities is derived in terms of field-free (spherical) transition probabilities. Expansion coefficients are squares of Clebsch-Gordan coefficients from the SO(4) = SU(2) \ifmmode\times\else\texttimes\fi{} SU(2) description of the hydrogen atom. The method provides new sum rules for Stark transitions which explain regularities in previous calculations. Accurate formulas for average Stark transition probabilities obtain from application of the method to an empirical model for field-free transition probabilities. The approximation permits estimation of Stark lifetimes in the large-$n$ regime.

DOI
10.1103/physreva.12.1949

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An expansion formula for Stark effect transition probabilities is derived in terms of field-free (spherical) transition probabilities. Expansion coefficients are squares of Clebsch-Gordan coefficients from the SO(4) = SU(2) \ifmmode\times\else\texttimes\fi{} SU(2) description of the hydrogen atom. The method provides new sum rules for Stark transitions which explain regularities in previous calculations. Accurate formulas for average Stark transition probabilities obtain from application of the method to an empirical model for field-free transition probabilities. The approximation permits estimation of Stark lifetimes in the large-nnn regime.
The radiative transition probabilities in hydrogen have been calculated for all Stark transitions through n=25n=25n=25. In general, the total transition probability and lifetime of a particular state are a function of the parabolic quantum numbers n1{n}_{1}n1​, n2{n}_{2}n2​, and mmm. For m>0m>0m>0 and to within an error of the order of one percent the radiative lifetimes are functions only of nnn, mmm. The radiative lifetimes are presented graphically for all states from n=2n=2n=2 through n=25n=25n=25.
The transformation coefficients connecting the Stark and the angular momentum states belonging to the positive spectrum of the hydrogen atom are calculated by a group- theoretical method which makes use of the O(3,1) symmetry of the states and also by a purely analytical method. The results of the two calculations agree except for an undetermined factor not containing the orbital quantum number l. Complete agreement between the results is achieved by taking the normalization factors for the continuum states to be analytic continuations of those for the bound states. The transformation coefficients turn out to be SU(2) Clebsch-Gordan coefficients (j 1 j 2 l; m 1 m 2 m) with complex j 1 j 2 m 1 m 2 , and physical l,m. From the general theory the well known expansion of the Coulomb scattering function is obtained by giving the magnetic quantum number m and one of the electric quantum numbers n 2 , the special values, 0 and -1, respectively.
A general method is presented for calculating the higher-order terms of series in powers of the DC field for the hydrogen Stark-state wavefunctions, dipole transition matrix elements and corresponding line intensities. Analytical expressions are derived for the second-order Stark effect on the matrix elements and intensities of the Stark lines. Simple formulae for transitions to states with zero parabolic quantum numbers are given. The numerical data for the Lyman and Balmer series and for some transitions between Rydberg levels are analysed in detail.
The Stark effect of the fine structure of hydrogenic atoms is discussed in the light of the quantum mechanics. Explicit formulas are given for the energy levels in either a weak or a strong electric field, including in each case both the relativity corrections and the Stark effect. (The conventional treatment neglects either one or the other.) The results are, except in form, similar to those recently given by Schlapp. In the new mechanics, in contradistinction to the old Bohr theory, there is a linear Stark effect even in weak fields, because of the identity of energy for the ns\mathrm{ns}ns and np1n{p}_{1}np1​ levels in the absence of external fields. This degeneracy is shown to account for the results of the recent experiments of Ornstein, Zernike, and Snoek, who found that the 2s2s2s level in HHH is not metastable.The coefficients are found for the development of the "parabolic" eigenfunctions in terms of the "polar" ones, and a set of numerical values of these coefficients is tabulated.Methods are given for the calculation of the relative intensities in the Stark effect in terms of the already available theoretical fine structure intensities in the absence of fields.
Analyses of the resonant multiphoton ionization of atoms require knowledge of ac Stark energy shifts and of multiphoton, bound-to-bound state, transition amplitudes. In this paper, we consider the three-photon photoionization of hydrogen atoms at frequencies that are at and surrounding the two-photon 1s to 2s resonance. AC energy shift sums of both the 1s and 2s states are calculated as a function of the laser frequency along with two-photon 1s ? 2s resonant transition amplitude sums. These quantities are calculated using an extended version of a method, which has often been employed in a variety of ways, of calculating these sums by expressing them in terms of solutions to a variety of differential equations that are derived from the different sums being evaluated. We demonstrate how exact solutions are obtained to these differential equations, which lead to exact evaluations of the corresponding sums. A variety of different cases are analysed, some involving analytic continuation, some involving real number analysis and some involving complex number analysis. A dc Stark sum calculation of the 2s state is carried out to illustrate the case where analytic continuation, pole isolation and pole subtraction are required and where the calculation can be carried out analytically; the 2s state, ac Stark shift sum calculations involve a case where no analytic continuation is required, but where the solution to the differential equation produces complex numbers owing to the finite photoionization lifetime of the 2s state. Results from these calculations are then used to calculate three-photon ionization probabilities of relevance to an analysis of the multiphoton ionization data published by Kyrala and Nichols (1991 Phys. Rev. A 44, R1450).