Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

Physical Review A

Higher-Order Corrections to the Momentum-Translation Approximation

Bhaskar J. Choudhury

Published Oct 1, 1973• 2 citations
Publisher

Abstract

A general analytical result has been obtained for the calculation of higher-order corrections to the momentum-translation approximation. Explicit analytical results are given for the calculation of higher-order corrections for bound-bound and bound-free transitions for hydrogenic atoms. Numerical results are given for $1S$ to $2S$ transitions in hydrogen.

DOI
10.1103/physreva.8.1759

Related papers

Comment on analytical calculations of two-photon transition amplitudes for the hydrogen atom

Крыловецкий Александр Абрамович, Nikolai L. Manakov, S. I. Marmo

It is shown that analytical results for amplitudes of two-photon transitions from an arbitrary bound state of the hydrogen atom published in two recent papers in this journal reduce to an essentially simpler form that considerably simplifies both analysis and numerical evaluations.

Journal of Physics B Atomic Molecular and Optical Physics•Jan 25, 2005•6 citations

Further properties of hydrogenic wave functions

J D Hey

The generalized and canonical momentum operators in spherical polar coordinates are derived and discussed, and the radial operators applied to the evaluation of the Heisenberg uncertainty relation for bound states of hydrogenic atoms. The two Pasternack relations, which connect the expectation values 〈rk〉 for values of k differing by an integer, are examined, as well as a generalization of the first relation, and the advantages which they hold for computational purposes discussed. An earlier calculation by the present author, of the electron impact ionization cross section for hydrogen atoms in the 2S metastable state, is compared with other experimental data from the literature. The calculation of the wavelengths of transitions between hydrogen-like levels in highly excited ions possessing a few additional, tightly bound, electrons is described and compared with experimental data.

American Journal of Physics•Aug 1, 1993•28 citations

Higher-order effects in multiphoton transitions

Bhaskar J. Choudhury

I have obtained a general analytic result for calculating any higher-order correction to the transition amplitude for $N$-photon absorption by writing the time-dependent perturbation theory in the $S$-matrix form. In obtaining this result, I have made use of the concept of average energy as has been defined by Bebb and Gold. A formal expression has been obtained for the upper bound of the higher-order correction and the results are discussed for the case of the hydrogen atom.

Physical Review A•Dec 1, 1974•0 citations

Second-order radiative bound-free transitions in hydrogen atoms

S. Klasfeld

Lettere al nuovo cimento della societa italiana di fisica/Lettere al nuovo cimento•Oct 1, 1969•37 citations

Momentum-Translation Approximation outside Its Domain of Applicability

John E. Rogerson, Robert N. De Witt, Howard R. Reiss

The momentum-translation approximation is valid for electromagnetic transitions in bound systems caused by external radiation of photon energy much less than the transition energy. Its utility is explored here for two types of problems that violate that restriction. One application is to resonant bound-bound multiphoton transitions of low multiphoton order, and results are compared to recent perturbative calculations. The other application is to both the transition rate and the spectrum of two-photon spontaneous emission from the metastable $2S$ state in hydrogen. Predicted transition rates are too low, but the spectrum agrees well with the known results. All momentum-translation results follow from simple closed-form analytical expressions.

Physical Review A•Jan 1, 1973•12 citations

On the inadequacy of the momentum-translation approximation: multiphoton ionization of atomic hydrogen

A. Rachman, G Laplanche

The cross section for multiphoton ionization of hydrogen is computed using the momentum translation method (MTA) in the cases of linearly and circularly polarized light, alternatively. Extended numerical comparison is made with well-established results obtained from the Nth-order time-dependent perturbation theory in the weak intensity limit. The MTA does not reproduce the values given by perturbation theory and considerable differences are observed. Higher order iterations than the lower one are introduced and an analysis of the leading terms of the development is made: at weak intensities the zeroth-order term that defines the MTA cancels with higher order terms and the final result obtained after making (N-1 iterations is none other than the well-known transition amplitude deduced from standard perturbation theory, defined by the remaining terms of the last iteration after mutual cancellation with lower-order ones. According to the results obtained in this work, it is not possible to affirm that the MTA is adequate to calculate atomic multiphoton ionization cross sections.

Journal of Physics B Atomic and Molecular Physics•Apr 21, 1975•2 citations
A general analytical result has been obtained for the calculation of higher-order corrections to the momentum-translation approximation. Explicit analytical results are given for the calculation of higher-order corrections for bound-bound and bound-free transitions for hydrogenic atoms. Numerical results are given for 1S1S1S to 2S2S2S transitions in hydrogen.
The generalized and canonical momentum operators in spherical polar coordinates are derived and discussed, and the radial operators applied to the evaluation of the Heisenberg uncertainty relation for bound states of hydrogenic atoms. The two Pasternack relations, which connect the expectation values 〈rk〉 for values of k differing by an integer, are examined, as well as a generalization of the first relation, and the advantages which they hold for computational purposes discussed. An earlier calculation by the present author, of the electron impact ionization cross section for hydrogen atoms in the 2S metastable state, is compared with other experimental data from the literature. The calculation of the wavelengths of transitions between hydrogen-like levels in highly excited ions possessing a few additional, tightly bound, electrons is described and compared with experimental data.
I have obtained a general analytic result for calculating any higher-order correction to the transition amplitude for NNN-photon absorption by writing the time-dependent perturbation theory in the SSS-matrix form. In obtaining this result, I have made use of the concept of average energy as has been defined by Bebb and Gold. A formal expression has been obtained for the upper bound of the higher-order correction and the results are discussed for the case of the hydrogen atom.
The momentum-translation approximation is valid for electromagnetic transitions in bound systems caused by external radiation of photon energy much less than the transition energy. Its utility is explored here for two types of problems that violate that restriction. One application is to resonant bound-bound multiphoton transitions of low multiphoton order, and results are compared to recent perturbative calculations. The other application is to both the transition rate and the spectrum of two-photon spontaneous emission from the metastable 2S2S2S state in hydrogen. Predicted transition rates are too low, but the spectrum agrees well with the known results. All momentum-translation results follow from simple closed-form analytical expressions.
The cross section for multiphoton ionization of hydrogen is computed using the momentum translation method (MTA) in the cases of linearly and circularly polarized light, alternatively. Extended numerical comparison is made with well-established results obtained from the Nth-order time-dependent perturbation theory in the weak intensity limit. The MTA does not reproduce the values given by perturbation theory and considerable differences are observed. Higher order iterations than the lower one are introduced and an analysis of the leading terms of the development is made: at weak intensities the zeroth-order term that defines the MTA cancels with higher order terms and the final result obtained after making (N-1 iterations is none other than the well-known transition amplitude deduced from standard perturbation theory, defined by the remaining terms of the last iteration after mutual cancellation with lower-order ones. According to the results obtained in this work, it is not possible to affirm that the MTA is adequate to calculate atomic multiphoton ionization cross sections.