Relativistic two-photon decay rates with the Lagrange-mesh method

Abstract

Relativistic two-photon decay rates of the 2s 1/2 and 2p 1/2 states towards the 1s 1/2 ground state 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. Highly accurate values are obtained by a simple calculation involving different meshes for the initial, final, and intermediate wave functions and for the calculation of matrix elements. The accuracy of the results with a Coulomb potential is improved by several orders of magnitude in comparison with benchmark values from the literature. The general requirement of gauge invariance is also successfully tested, down to rounding errors. The method provides high accuracies for two-photon decay rates of a particle in other potentials and is applied to a hydrogen atom embedded in a Debye plasma simulated by a Yukawa potential.

Hyperspherical Harmonics Expansion on Lagrange Meshes for Bosonic Systems in One Dimension

“…However, the simplicity of (30) is then lost in the calculation of some matrix elements connecting different K values since the associated meshes and Gauss quadratures then differ. These matrix elements can nevertheless be accurately calculated in many cases with a third Gauss quadrature [16]. This leads to a hybrid calculation where some matrix elements are calculated like in the previous subsection and a different method of evaluation is used for the other ones.…”

Vacuum polarization and finite nuclear size effects in the two-photon decay of hydrogen-like ions

“…In the past, a large number of methods has been proposed to perform this summation. For example the Coulomb Green's function approach [16,20,21] and various finite basis set methods [4][5][6] were both successfully applied to perform secondorder calculations. In this work, we will also use finite basis sets constructed from B-spline functions.…”

“…Theoretical investigations of two-photon transitions in hydrogen-like ions, therefore, have become a testbed for the development of second-order computational approaches. Several of such methods have been developed during the last decades leading to more precise predictions of the decay rates [4][5][6]. The increased accuracy of the calculations has allowed to investigate how the total and differential rates are influenced by nuclear and even QED effects.…”

POLALMM: A program to compute polarizabilities for nominal one-electron systems using the Lagrange-mesh method

“…A detailed description of the LMM can be found in Ref. [1] with its application to solve the Dirac equation and estimate energies [6], polarizabilities [7] and multiphoton transition amplitudes [8]. The theory of semiempirical-core-potential approaches is presented in detail in a paper by Filippin et al [3] and shall therefore only be summarized in Section 2.…”