Lamb shift, 0 , relativistic theory

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]. 

The aim of this section is to extract from the measurements the values of the Rydberg constant and Lamb shifts. This analysis is detailed in the references [50,61], More details on the theory of atomic hydrogen can be found in several review articles [62,63,34], It is convenient to express the energy levels in hydrogen as the sum of three terms the first is the well known hyperfine interaction. The second, given by the Dirac equation for a particle with the reduced mass and by the first relativistic correction due to the recoil of the proton, is known exactly, apart from the uncertainties in the physical constants involved (mainly the Rydberg constant R0c). The third term is the Lamb shift, which contains all the other corrections, i.e. the QED corrections, the other relativistic corrections due to the proton recoil and the effect of the proton charge distribution. Consequently, to extract i oo from the accurate measurements one needs to know the Lamb shifts. For this analysis, the theoretical values of the Lamb shifts are sufficiently precise, except for those of the 15 and 2S levels. 

The measured hyperfine splittings of 2 3Pi level were in reasonable agreement with the relativistic calculations of [96], and also with non-relativistic calculations corrected for relativistic and QED effects [114,115], The results for the hyperfine corrected 21S o — 23Pl interval in 14N5+ are compared with theory in table 3. QED corrections make up 3.5% of the measured interval. The experiment is hence sensitive to these corrections at the level of 20 ppm, the highest precision for a Lamb shift in any multiply-charged ion. 

This second point of view can be illustrated by an example from the late 1940 s that will play an important role in this chapter. At that time the Schrodinger equation was well established, and its relativistic generalization, the Dirac equation, appeared to describe the spectrum of hydrogen perfectly, though the question of how to apply the Dirac equation to many-electron systems was still open. However, when more precise experiments were carried out, most notably by Lamb and Retherford [1], a small disagreement with theory was found. The attempt to understand this new physics stimulated theoretical efforts that led to the modern form of the first quantum field theory. Quantum Electrodynamics (QED). This small shift, which removes the Dirac degeneracy between the 2si/2 and states, known as the Lamb shift, is an example of a radiative correction. 

The retarded electron-electron interaction presented above arises from the first of the one-photon Feynman diagrams in figure 5.1. In terms of an expansion of the relativistic interactions in powers of 1 /c, this interaction contains the lowest-order terms. As pointed out above, the Breit interaction contains all terms of order c . After the Breit interaction, the lowest-order interactions come from the other two one-photon diagrams, the vacuum polarization and self-energy terms, which are 0 c ). The energy contribution from these two terms is called the Lamb shift, after its discoverer W. E. Lamb Jr. (1952), and its calculation has been an important testing ground for QED theories. 

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