Lamb shifts, calculated formulae

This expression fits the exact potential in the interval 0.01 < x < 1.0 with an accuracy of about 1%, and due to an exponential decrease of the wave functions and smallness of the potential at large distances, it may be safely used for calculations at all x. After numerical calculation with this interpolation formula we obtain for the 2P — 2S Lamb shift [29]... 

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... 

Theoretical calculation of the fLe+ Lamb shift is straightforward with all the formulae given above. It is only necessary to recall that all contributions scale with the power of Z, and the terms with high power of Z are enhanced in comparison with the hydrogen case. Theoretical uncertainty is estimated by scaling with Z the uncertainty of the hydrogen formulae. After calculation we obtain Lth(2S — 2P,He ) = 14 041.46 (3) MHz. We see that the theoretical and experimental results for the classic Lamb shift in helium differ by about two standard deviations of the experimental result. Clearly, further reduction of the experimental error is desirable, and the reason for this mild discrepancy should be clarified. 

In the nonrelativistic calculation the Dirac equation is replaced by the Schrodinger one. The formula that is obtained (see [2]), which is convergent, is, if the dipole approximation is applied (i.e. Tj-(k) are replaced by Tj-(O)), the formula used in [4] for the first calculation proposed for the explanation of the Lamb shift. But this last formula is divergent and its use implies that the integration upon k is cut off for a k = kmax. In [23] the value of kmax = ante2 has been proposed and was used in the following calculations of the Lamb shift. 

It is to be emphasized that these formulae have been obtained by the use of second-order perturbation theory taken to the first significant terms. The Lamb shifts obtained from them have been refined by the evaluation of smaller terms in the second order, and by the calculation of higher orders in the perturbation expansion also by corrections for the finite mass and volume of the nucleus, The papers [119], [la], [52] (see also [9]), summarize the position at the time of writing. 

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