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

Krylovetsky, A. A.; Manakov, N. L.; Marmo, S. I.

doi:10.1088/0953-4075/38/3/n01

Cited by 6 publications

(6 citation statements)

References 12 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…However, the LMM is sometimes also able to exactly calculate more complicated physical quantities. An interesting example is given by static polarizabilities for which a general analytical expression is available but its existence is not well known [118,119]. The exact static polarizabilities of hydrogen were also published for a limited number of states in Refs.…”

Section: Static Polarizabilitiesmentioning

confidence: 99%

See 1 more Smart Citation

The Lagrange-mesh method

2015

View full text Add to dashboard Cite

The Lagrange-mesh method is an approximate variational method taking the form of equations on a grid thanks to the use of a Gauss-quadrature approximation. The variational basis related to this Gauss quadrature is composed of Lagrange functions which are infinitely differentiable functions vanishing at all mesh points but one. This method is quite simple to use and, more importantly, can be very accurate with small number of mesh points for a number of problems. The accuracy may however be destroyed by singularities of the potential term. This difficulty can often be overcome by a regularization of the Lagrange functions which does not affect the simplicity and accuracy of the method.The principles of the Lagrange-mesh method are described, as well as various generalizations of the Lagrange functions and their regularization. The main existing meshes are reviewed and extensive formulas are provided which make the numerical calculations simple. They are in general based on classical orthogonal polynomials. The extensions to non-classical orthogonal polynomials and periodic functions are also presented.Applications start with the calculations of energies, wave functions and some observables for bound states in simple solvable models which can rather easily be used as exercises by the reader. The Dirac equation is also considered. Various problems in the continuum can also simply and accurately be solved with the Lagrange-mesh technique including multichannel scattering or scattering by non-local potentials. The method can be applied to three-body systems in appropriate systems of coordinates. Simple atomic, molecular and nuclear systems are taken as examples. The applications to the timedependent Schrödinger equation, to the Gross-Pitaevskii equation and to Hartree-Fock calculations are also discussed as well as translations and rotations on a Lagrange mesh.

show abstract

Section: Static Polarizabilitiesmentioning

confidence: 99%

“…This formula has first been proven analytically in Refs. [118,119] and rediscovered on the basis of LMM results in Ref. [123].…”

Section: Static Polarizabilitiesmentioning

confidence: 99%

The Lagrange-mesh method

2015

View full text Add to dashboard Cite

show abstract

“…In the nonrelativistic case, the values are computed using Eqs. (5) and (6), and are compared with results reported in Refs. [29,30].…”

Section: B Yukawa Potentialmentioning

confidence: 94%

“…Let us start with the nonrelativistic case and show that the Lagrange-mesh approximations of Eqs. (4) and (6) are identical. This property is valid because of a consistent use of Lagrange functions and Gauss quadratures.…”

Section: Appendix A: Equivalence Between Mesh Expressions For Polarizmentioning

confidence: 99%

See 1 more Smart Citation

Relativistic polarizabilities with the Lagrange-mesh method

2014

View full text Add to dashboard Cite

Relativistic dipolar to hexadecapolar polarizabilities of the ground state and some excited states of hydrogenic atoms are calculated by using numerically exact energies and wave functions obtained from the Dirac equation with the Lagrange-mesh method. This approach is an approximate variational method taking the form of equations on a grid because of the use of a Gauss quadrature approximation. The partial polarizabilities conserving the absolute value of the quantum number κ are also numerically exact with small numbers of mesh points. The ones where |κ| changes are very accurate when using three different meshes for the initial and final wave functions and for the calculation of matrix elements. The polarizabilities of the n = 2 excited states of hydrogenic atoms are also studied with a separate treatment of the final states that are degenerate at the nonrelativistic approximation. The method provides high accuracies for polarizabilities of a particle in a Yukawa potential and is applied to a hydrogen atom embedded in a Debye plasma.

show abstract