Calculation of Hydrogenic Bethe Logarithms for Rydberg States

We describe the calculation of hydrogenic (one-loop) Bethe logarithms for all states with principal quantum numbers n <= 200. While, in principle, the calculation of the Bethe logarithm is a rather easy computational problem involving only the nonrelativistic (Schroedinger) theory of the hydrogen atom, certain calculational difficulties affect highly excited states, and in particular states for which the principal quantum number is much larger than the orbital angular momentum quantum number. Two evaluation methods are contrasted. One of these is based on the calculation of the principal value of a specific integral over a virtual photon energy. The other method relies directly on the spectral representation of the Schroedinger-Coulomb propagator. Selected numerical results are presented. The full set of values is available at quant-ph/0504002.Comment: 10 pages, RevTe

Self-energy values for P states in hydrogen and low-Z hydrogenlike ions

We describe a nonperturbative (in Zalpha) numerical evaluation of the one-photon electron self energy for 3P_{1/2}, 3P_{3/2}, 4P_{1/2} and 4P_{3/2} states in hydrogenlike atomic systems with charge numbers Z=1 to 5. The numerical results are found to be in agreement with known terms in the expansion of the self energy in powers of Zalpha and lead to improved theoretical predictions for the self-energy shift of these states.Comment: 3 pages, RevTe

Perturbation Approach to the Self Energy of non-S Hydrogenic States

We present results on the self-energy correction to the energy levels of hydrogen and hydrogenlike ions. The self energy represents the largest QED correction to the relativistic (Dirac-Coulomb) energy of a bound electron. We focus on the perturbation expansion of the self energy of non-S states, and provide estimates of the so-called A60 perturbative coefficient, which can be considered as a relativistic Bethe logarithm. Precise values of A60 are given for many P, D, F and G states, while estimates are given for other electronic states. These results can be used in high-precision spectroscopy experiments in hydrogen and hydrogenlike ions. They yield the best available estimate of the self-energy correction of many atomic states.Comment: 18 pages (in 2-column format), 21 figures. Version 2 (June 20, 2003) includes minor modification

Two-Loop Bethe Logarithms

We calculate the two-loop Bethe logarithm correction to atomic energy levels in hydrogen-like systems. The two-loop Bethe logarithm is a low-energy quantum electrodynamic (QED) effect involving multiple summations over virtual excited atomic states. Although much smaller in absolute magnitude than the well-known one-loop Bethe logarithm, the two-loop analog is quite significant when compared to the current experimental accuracy of the 1S-2S transition: it contributes -8.19 and -0.84 kHz for the 1S and the 2S state, respectively. The two-loop Bethe logarithm has been the largest unknown correction to the hydrogen Lamb shift to date. Together with the ongoing measurement of the proton charge radius at the Paul Scherrer Institute its calculation will bring theoretical and experimental accuracy for the Lamb shift in atomic hydrogen to the level of 10^(-7).Comment: 4 pages, RevTe

Thermal Correction to the Molar Polarizability of a Boltzmann Gas

Metrology in atomic physics has been crucial for a number of advanced determinations of fundamental constants. In addition to very precise frequency measurements, the molar polarizability of an atomic gas has recently also been measured very accurately. Part of the motivation for the measurements is due to ongoing efforts to redefine the International System of Units (SI), for which an accurate value of the Boltzmann constant is needed. Here we calculate the dominant shift of the molar polarizability in an atomic gas due to thermal effects. It is given by the relativistic correction to the dipole interaction, which emerges when the probing electric field is Lorentz transformed into the rest frame of the atoms that undergo thermal motion. While this effect is small when compared to currently available experimental accuracy, the relativistic correction to the dipole interaction is much larger than the thermal shift of the polarizability induced by blackbody radiation

Electron Self-Energy for the K and L Shells at Low Nuclear Charge

A nonperturbative numerical evaluation of the one-photon electron self-energy for the K- and L-shell states of hydrogenlike ions with nuclear charge numbers Z =1 to 5 is described. Our calculation for the 1S1/2 state has a numerical uncertainty of 0.8 Hz in atomic hydrogen, and for the L-shell states (2S1/2 , 2P1/2 , and 2P3/2) the numerical uncertainty is 1.0 Hz. The method of evaluation for the ground state and for the excited states is described in detail. The numerical results are compared to results based on known terms in the expansion of the self-energy in powers of Zα

Lamb Shift of 3P and 4P States and the Determination of α

The fine-structure interval of P states in hydrogenlike systems can be determined theoretically with high precision, because the energy levels of P states are only slightly influenced by the structure of the nucleus. Therefore a measurement of the fine structure may serve as an excellent test of QED in bound systems, or alternatively as a means of determining the fine-structure constant a with very high precision. In this paper an improved analytic calculation of higher-order binding corrections to the one-loop self-energy of 3P and 4P states in hydrogenlike systems with a low nuclear charge number Z is presented. The method of calculation has been described earlier by Jentschura and Pachucki [Phys. Rev. A 54, 1853 (1996)], and is applied here to the excited P states. Because of the more complicated nature of the wave functions and the bound-state poles corresponding to decay of the excited states, the calculations are more complex. Comparison of the analytic results to the extrapolated numerical data for high-Z ions [Mohr and Kim, Phys. Rev. A 45, 2727 (1992)] serves as an independent test of the analytic evaluation. Theoretical values for the Lamb shift of the P states and for the fine-structure splittings are given