Nonrelativistic quantum electrodynamic Hamiltonian and photon-exchange interaction up to mα8 order

Abstract

We derive the effective Hamiltonian of the Nonrelativistic Quantum Electrodynamic up to mα 8 by using scattering matching approach. At mα 6 order, these results are coincide with Pachucki's, which is obtained by applying Foldy-Wouthuysen transformation. And by using the NRQED Hamiltonian, we derive the photon-exchange interaction in non-retarded approximation and the retardation correction up to mα 8 . The energy shift of the photon-exchange interaction is obtained by studying the pole of the total Green function.a Haoxue Qiao; electronic mail: qhx@whu.edu.cn where the subscript (n) means the mα n . The leading terms E (2) , which is calculated by solving the Schrödinger equation, is at mα 2 order. The high-order energy shift, relativistic, radiation and recoil corrections, are the means value of the high-order Hamiltonian. The precision of the energy is restricted by the accuracy of E (2) , the non-relativistic wave-functions and the order of relativistic, radiation and recoil corrections obtained. As the non-relativistic problem has been solved by using Rayleigh-Ritz variational method in Hylleraas coordinates [4][5][6][7]. The core issue of the high-precision calculating the energy-level is deriving these high-order corrections.The most effective method to do this work is the non-relativistic quantum electrodynamic(NRQED) [8][9][10], which is an effective field theory describing the non-relativistic electromagnetic systems. The key point is the contributions of the virtual photon are separated into the high-energy region ω > mα and low-energy region ω ∽ mα 2 . In the low-energy region, the energy shift can be calculated by using the low-energy effective Hamiltonian. In the High-energy region, the contributions can be treated in QED approach with scattering approximation.The NRQED has achieved a great success in the light Hydrogenlike atoms. Their energy-level has been calculated up to mα 7 [11] In the three-body systems, Helium and Hydrogen molecule ions, the effective Hamiltonian of the energy up to mα 6 order has been complete obtained [12] And at mα 7 order, only the fine-structure corrections are derived [13]. The studying about the mα 8 order corrections are seldom reported.The fine-structure splitting in the 2 3 P J level of Helium is recognized as the best atomic system for the determination of the fine-structure constant α. Comparing with calculations of the mα 7 order energy, the measured result can be used to determine the fine-structure constant with a precision of about 2 × 10 −9 [2] It is necessary to derive the mα 8 order Hamiltonian to reduce the uncertain 2.3 × 10 −10 in CODATA [14] At present, the low-energy effective Hamiltonian of NRQED obtained by using Foldy-Wouthuysen transformation [15] is at mα 6 order. The mα 8 order result hasn't been obtained. In this work, we

The high-order nonrelativistic Hamiltonian of electromagnetic system

“…Using this approach, the mα 8 orders effective Hamiltonians of spin-0,1/2,1 are obtained. The result of spin-1/2 is coincide with the previous obtained by using scattering matching approach [17].…”

“…After correcting the error, This result is coincide with the result obtained by scattering matching approach in Ref. [17].…”

“…However, the equivalence of FW Hamiltonian and NRQED Hamiltonian hasn't been fully tested. In our earlier work [17], the mα 8 order Hamiltonian was obtained by using scattering matching approach at tree diagram level. It is coincide with the FW Hamiltonian at mα 6 order.…”

“…The coefficient in front of [∂ t E, E] in Ref. [17] is double counting error in the two-photon-fermion scattering matching.…”

Theory and computational methods for high precision spectra of few-electron atomic systems

“…A μ~m e cα 2 =α a.u., 可以认为该有效哈密顿量按α的幂 展开. Pachucki得到了α 4 a.u.阶的结果, 最近Zhou等 人 [39] 得到FW哈密顿量的α 6 a.u.阶的结果, 该结果与之 前散射匹配的方法得到的结果相同 [40] . [41,42] .…”