Lamb shift approximations

The main contribution to the Lamb shift was first estimated in the nonrela-tivistic approximation by Bethe [8], and calculated by Kroll and Lamb [9], and by French and Weisskopf [10]. We have already discussed above qualitatively the nature of this contribution. In the effective Dirac equation framework the... 

The Pauli form factor also generates a small contribution to the Lamb shift. This form factor does not produce any contribution if one neglects the lower components of the unperturbed wave functions, since the respective matrix element is identically zero between the upper components in the standard representation for the Dirac matrices which we use everywhere. Taking into account lower components in the nonrelativistic approximation we easily obtain an explicit expression for the respective perturbation... 

Table 4. Lamb shift contribution for the ground state of 238U91+ ion (in eV). Here Ro denotes the nuclear radius, M is nuclear mass and ao is the Bohr radius. The finite nuclear-size correction is calculated for a Fermi distribution with (r2)1 /2 = 5.860 0.002 fm. The corrections VPVP (f) and S(VP)E are known only in Uehling approximation. The inaccuracies assigned to these rather small corrections are estimated as the average of the inaccuracies of the Uehling approximation deduced from exact results for the corrections VPVP (e) and SEVP (g),(h),(i)...

In all experiments the velocity was measured by observing the decay of the 2p atoms produced from the 2s atoms under the action of a nonadiabatic field. To determine the velocity from the experimentally obtained decay length 0 = vr, we must know t, i.e. the lifetime of the 2p state. The value of t was calculated the error was estimated to be of the order of 1 ppm, which is acceptable for the determination of the Lamb shift with approximately the same accuracy. 

The term Lamb shift of a single atomic level usually refers to the difference between the Dirac energy for point-like nuclei and its observable value shifted by nuclear and QED effects. Nuclear effects include energy shifts due to static nuclear properties such as the size and shape of the nuclear charge density distribution and due to nuclear dynamics, i.e. recoil correction and nuclear polarization. To a zeroth approximation, the energy levels of a hydrogen-like atom are determined by the Dirac equation. For point-like nuclei the eigenvalues of the Dirac equation can be found analytically. In the case of extended nuclei, this equation can be solved either numerically or by means of successive analytical approximation (see Rose 1961 Shabaev 1993). 

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. 

Table 1 Comparison of Za-expansion values to numerical calculcations including all orders in Za for a -order QED contributions to the Lamb shift of the lsi/2-state in lead and uranium. The Kalldn-Sabry contribution VPVP b) c) and the S(VP)E contribution are considered in the Uehling approximation only.

Much worse is the situation with the calculations going beyond the Dirac approach. The first estimation for molecules of relativistic effects beyond the Dirac approximation was carried out by Ladik, and then by Jeziorski and Kotos. Besides the computation of the Lamb shift for the water molecule, not much has been computed in this area for years. 

Above we have proposed to measure the the small difference frequency df = f(1S-2S) - 3 f(2S-nS). This frequency depends critically on the Lamb shifts of the participating levels, and can provide a stringent test of QED. For n=100, the theoretical uncertainty of df is dominated by the nuclear size effect ( 70 kHz) and by approximations in the computation of electron structure corrections and uncalculated higher order QED corrections (65 kHz). The contribution of the Rydberg constant (1 kHz), the electron mass (0.05 kHz), and the fine structure constant (4 kHz) are negligible by comparison. The QED computations can be improved, and if theory is correct, a precision measurement of df can provide accurate new values for the charge radii of the proton and deuteron. 

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