Calculation of Higher-Order Foldy-Wouthuysen Transformation Hamiltonian

Abstract

The Foldy-Wouthuysen Hamiltonian of a light atomic system that has an 𝑚𝛼 8 contribution to energy levels is calculated. The case of a Dirac-Coulomb field is discussed. The results can be used for relativistic and radiative corrections to energy levels in the low-energy part. A divergent operator 𝛿 2 (𝑟) emerges. This is probably due to the nature of the point-like charge source. The effective method of radiation calculation may be re-checked.

The high-order nonrelativistic Hamiltonian of electromagnetic system

“…And we find that there may be some imperceptible mistakes in Ref. [16]. and should be rechecked independently.…”

“…The FW Hamiltonian up to mα 8 was obtained by Ref. [16]. However, the equivalence of FW Hamiltonian and NRQED Hamiltonian hasn't been fully tested.…”

High-order Hamiltonian obtained by Foldy–Wouthuysen transformation up to the order of mα 8

“…[22] At the order of mα 8 , this result corrected the coefficient error in the work by Mei. [9] By comparing the two results item by item, one can see that their difference is only in the coefficients of some items. By subtracting one equation from the other, we get the different terms…”

“…[7] For helium, the complete formula for the mα 7 contribution to energy levels was derived recently. [8] At the order of mα 8 , the derivation of the high-order correction operators has already begun by dif-ferent methods, [9][10][11] but the correctness of the results has yet to be verified.…”

“…In the field of calculation of atomic and molecular systems, FW transformation is often used to obtain the correction of the low-energy part in the Lamb shift, [17] which contributes to the hyperfine structure of helium, H + 2 , and HD + . [18][19][20] At present, with the application of the FW transformation, the relativistic corrections of a Dirac particle in an external electromagnetic field are evaluated to the order of mα 8 , [9] but there are some numerical errors in the coefficients, which are corrected in this paper.…”