Lamb shifts

These fluctuations will affect the motion of charged particles. A major part of the Lamb shift in a hydrogen atom can be understood as the contribution to the energy from the interaction of the electron with these zero point oscillations of the electromagnetic field. The qualitative explanation runs as follows the mean square of the electric and magnetic field intensities in the vacuum state is equal to... 

We have assumed the potential to be spherically symmetric.) It is precisely the perturbation -J(Aq) 0V2F(q) that gives rise to the major part of the Lamb shift in the 2s state of the hydrogenic atom. 

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The quantity L(0) = In I, where I is the mean excitation potential introduced by Bethe, which controls the stopping of fast particles (see Sect. 2.3.4) L(2) = In K, where K is the average excitation energy, which also enters into the expression for Lamb shift (Bethe, 1947). Various oscillator sum rules have been verified for He and other rare gases to a high degree of accuracy. Their validity is now believed to such an extent that doubtful measurements of photoabsorption and electron-impact cross sections are sometimes altered or corrected so as to satisfy these. 

Soon after the Schrodinger equation was introduced in 1926, several works appeared dealing with the fundamental problem of the nuclear motion in molecules. Very soon after, the relativistic equations were introduced for one-and two-electron systems. The experiments on the Lamb shift stimulated... 

As mentioned, most calculations we have done so far have concerned molecular systems. However, prior to development of the non-BO method for the diatomic systems, we performed some very accurate non-BO calculations of the electron affinities of H, D, and T [43]. The difference in the electron affinities of the three systems is a purely nonadiabatic effect resulting from different reduce masses of the pseudoelectron. The pseudoelectrons are the heaviest in the T/T system and the lightest in the H/H system. The calculated results and their comparison with the experimental results of Lineberger and coworkers [44] are shown in Table 1. The calculated results include the relativistic, relativistic recoil. Lamb shift, and finite nuclear size corrections labeled AEcorr calculated by Drake [45]. The agreement with the experiment for H and D is excellent. The 3.7-cm increase of the electron affinity in going from H to D is very well reproduced by the calculations. No experimental EA value is available for T. 

The term AEcorr contains relativistic, relativistic recoil, Lamb shift, and finite nuclear size... 

L(0) = Z In 7, where I is the mean excitation potential appearing in Bethe s stopping power equation [Eq. (4)]. 7,(2) is proportional to the logarithm of average excitation energy, which is also involved in the Lamb shift [26]. 7,(—1) has been shown to be an optical... 

Calculation of the self-energy part of the Lamb shift... 

VP vacuum polarization SE self-energy part of the Lamb shift LS = VP + SEE Lamb shift RC nucleus recoil correction, polarization Relativistic PT accounts for the main relativistic and correlation effects HOPT higher-order PT contributions. Data are from refs [1-10]. 

In the language of control theory, Tr[p(0)P] is a kinematic critical point [87] if Eq. (4.159) holds, since Tr[e p(0)e- P] = Tr[p(0)P] + Tr(7/[p(0),P]) + O(H ) for a small arbitrary system Hamiltonian H. Since we consider p in the interaction picture, Eq. (4.159) means that the score is insensitive (in first order) to a bath-induced unitary evolution (i.e., a generalized Lamb shift) [88]. The purpose of this assumption is only to simplify the expressions, but it is not essential. Physically, one may think of a fast auxiliary unitary transformation that is applied initially in order to diagonalize the initial state in the eigenbasis of P. 

Another obvious contribution to the Lamb shift of the same leading order is connected with the polarization insertion in the photon propagator (see Fig. 2.2). This correction also induces a correction to the Coulomb potential... 

It is clear now that the scale setting factor which should be used for qualitative estimates of the high order corrections to the Lamb shift is equal to Am Za) /n. Note the characteristic dependence on the principal quantum number 1/n which originates from the square of the wave function at the origin V (0) 1/n . All corrections induced at small distances (or at high... 

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. 

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 mass dependence of the correction of order a Za) beyond the reduced mass factor is properly described by the expression in (3.7) as was proved in [11, 12]. In the same way as for the case of the leading relativistic correction in (3.4), the result in (3.7) is exact in the small mass ratio m/M, since in the framework of the effective Dirac equation all corrections of order Za) are generated by the kernels with one-photon exchange. In some earlier papers the reduced mass factors in (3.7) were expanded up to first order in the small mass ratio m/M. Nowadays it is important to preserve an exact mass dependence in (3.7) because current experiments may be able to detect quadratic mass corrections (about 2 kHz for the IS level in hydrogen) to the leading nonrecoil Lamb shift contribution. 

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