The Bethe, Brown, and Stehn Lamb Shift Calculation Revisited

As applied by Bethe, Brown, and Stehn in 1950, the first theory of the Lamb shift, although it contains many physical and mathematical approximations, is known to be in agreement with experiment (tens of MHz discrepancies). In the present work, thanks to new analytic formulas and computer means, we perform their original method of calculation again, without two approximations in the contribution ∆E< of the low radiation energies:(i) a mathematical approximation within the Bethe logarithm, and (ii) the neglect of retardation exponentials eik·x in the impulsion matrix elements < m|eik·xp|n >. Applied to the Lamb shift of the 1s level of hydrogen, our calculation gives a value of 8526 MHz, which is 352 MHz above the 8174 MHz value deduced from experiments. None of the improvements published in the modern literature, which at most are of the order of tens of MHz, can compensate this huge new discrepancy. The precision of our integration over Concepts of Physics, Vol. II (2005) 145 the discrete and continuous spectra was tested by a sum rule, and found to be 0.9999991. 146 Concepts of Physics, Vol. II (2005) The Bethe, Brown, and Stehn Lamb shift calculation revisited