Lamb shift theory

The Lamb shift theory of P-states is in a better shape than the theory of P-states. Corrections of order a(Za) are now known with uncertainty about 1 Hz for 2P states [10, 11, 12, 13]. 

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In the language of control theory, Tr[p(0)P] is a kinematic critical point [87] if Eq. (4.159) holds, since Tr[e p(0)e- P] = Tr[p(0)P] + Tr(7/[p(0),P]) + O(H ) for a small arbitrary system Hamiltonian H. Since we consider p in the interaction picture, Eq. (4.159) means that the score is insensitive (in first order) to a bath-induced unitary evolution (i.e., a generalized Lamb shift) [88]. The purpose of this assumption is only to simplify the expressions, but it is not essential. Physically, one may think of a fast auxiliary unitary transformation that is applied initially in order to diagonalize the initial state in the eigenbasis of P. 

Calculation of the leading logarithmic corrections of order a Za) Ep to HFS parallels the calculation of the leading logarithmic corrections of order a(Za) to the Lamb shift, described above in Subsect. 3.5.1. Again all leading logarithmic contributions may be calculated with the help of second order perturbation theory (see (3.71)). 

The leading nuclear size correction of order Za) m r )EF may easily be calculated in the framework of nonrelativistic perturbation theory if one takes as one of the perturbation potentials the potential corresponding to the main proton size contribution to the Lamb shift in (6.3). The other perturbation potential is the potential in (9.28) responsible for the main Fermi contribution... 

Prom the theoretical point of view the accuracy of calculations is limited by the magnitude of the yet uncalculated contributions to the Lamb shift. Corrections to the P levels are known now with a higher accuracy than the corrections to the S levels, and do not limit the results of the comparison between theory and experiment. 

Many experiments on the precise measurement of the classic Lamb shift were performed since its experimental discovery in 1947. We have collected modern post 1979 experimental results in Table 12.2. Two entries in this Table are changed compared to the original published experimental results [16, 15]. These alterations reflect recent improvements of the theory used for extraction of the Lamb shift value from the raw experimental data. 

The theory of high order corrections to the Lamb shift described above for H and D may also be applied to other light hydrogenlike ions. The simplest such ion is He+. Originally the classic Lamb shift in 17e+ was measured in [50] by the quenching-anisotropy method with the result L 2Si — 2Pi,17e+) = 14 042.52 (16) MHz. Later the authors of [50] discovered a previously unsuspected source of systematic error in their experiment. Their new measurement of the classic Lamb shift in 17e+ by the anisotropy method resulted in the value L 2Si — 2Pi,He ) = 14 041.13 (17) MHz [51]. Besides the experimental data this result depends also on the theoretical value of the hne structure interval. In [51] the value AE 2Pz —2Pi) = 175 593.50 (2) MHz was used. We recalculated this interval using tire latest theoretical results discussed above and obtained AE 2P3 — 2Pi) = 175 593.33 (1) MHz. Then the value of the... 

This experimental development was matched by rapid theoretical progress, and the comparison and interplay between theory and experiment has been important in the field of metrology, leading to higher precision in the determination of the fundamental constants. We feel that now is a good time to review modern bound state theory. The theory of hydrogenic bound states is widely described in the literature. The basics of nonrelativistic theory are contained in any textbook on quantum mechanics, and the relativistic Dirac equation and the Lamb shift are discussed in any textbook on quantum electrodynamics and quantum field theory. An excellent source for the early results is the classic book by Bethe and Salpeter [6]. A number of excellent reviews contain more recent theoretical results, and a representative, but far from exhaustive, list of these reviews includes [7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17]. 

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