Lamb Shifts of the Energy Levels

Theoretical results described above find applications in numerous high precision experiments with hydrogen, deuterium, helium, muonium, muonic hydrogen, etc. Detailed discussion of all experimental results in comparison with theory would require as much space as the purely theoretical discussion above. We will consider below only some applications of the theory, intended to serve as illustrations, their choice being necessarily somewhat subjective and incomplete (see also detailed discussion of phenomenology in the recent reviews [1, 2]). 

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Lamb Shifts of the Energy Levels 239 Table 12.2. Classic 251/2 — 2Pi/2 Lamb Shift... 

In 1947 Lamb and Retherford observed the 22-p3/2 — 225 1/2 transition using microwave techniques and found it to have a wavenumber 0.0354 cm 1 less than predicted by Dirac. The corresponding shift of the energy level, known as the Lamb shift, is shown in Figure 7.8 the 22.P,, 2 level is not shifted. Later, Lamb and Retherford observed the 22S1/2 — 22P1/2 transition directly with a wavenumber of 0.0354 cm-1. Quantum electrodynamics is the name given to the modified Dirac theory which accounts for the Lamb shift. 

By comparison of one quarter of the IS1 — 25 transition frequency with the 25 — 45 and 25 — 4D transition frequency, the main energy contributions described by the simple Rydberg formula are eliminated. The remaining difference frequency (about 5 GHz) is determined by well known relativistic contributions, the hyperfine interaction, and a combination of Lamb shifts. Since quantum electrodynamic contributions scale roughly as 1/n3 with the principal quantum number, the Lamb shift of the 15 level is the largest. 

Self energy and vacuum polarization of order a and the nuclear size account for the measured Lamb shift in hydrogenlike heavy ions at the current level of accuracy. Radiative corrections of the order contribute to the Lamb shift of the lsi/2 state and amount to about 1 eV for uranium. Facing higher precision in experiments, these corrections have to be evaluated to yield a reliable Lamb shift calculation. 

Note the emergence of the last term in (3.4) which lifts the characteristic degeneracy in the Dirac spectrum between levels with the same j and / = j 1/2. This means that the expression for the energy levels in (3.4) already predicts a nonvanishing contribution to the classical Lamb shift E 2Si) — E 2Pi). Due to the smallness of the electron-proton mass ratio this extra term is extremely small in hydrogen. The leading contribution to the Lamb shift, induced by the QED radiative correction, is much larger. 

Let us start systematic discussion of such corrections with the recoil corrections to the leading contribution to the Lamb shift. The most important observation here is that the mass dependence of all corrections of order a." Za.Y obtained above is exact, as was proved in [1, 2], and there is no additional mass dependence beyond the one already present in (3.7)-(3.24). This conclusion resembles the similar conclusion about the exact mass dependence of the contributions to the energy levels of order (Za) m discussed above, and it is valid essentially for the same reason. The high frequency part of these corrections is generated only by the one photon exchanges, for which we know the exact mass dependence, and the only mass scale in the low frequency part, which depends also on multiphoton exchanges, is the reduced mass. 

As in the case of the Lamb shift, QED provides the framework for systematic calculation of numerous corrections to the Fermi formula for hyperfine splitting (see the scheme of muonium energy levels in Fig. 8.3). We again... 

Accuracy of the radiofrequency measurements of the classic 2S — 2P Lamb shift [15, 16, 23, 24, 25] is limited by the large (about 100 MHz) natural width of the 2P state, and cannot be significantly improved. New perspectives in reducing the experimental error bars of the classic 2S — 2P Lamb shift were opened with the development of the Doppler-free two-photon laser spectroscopy for measurements of the transitions between the energy levels with different principal quantum numbers. Narrow linewidth of such transi-... 

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