Schrodinger Coulomb wave function

EDE in the external Coulomb field in Fig. 1.6. The eigenfunctions of this equation may be found exactly in the form of the Dirac-Coulomb wave functions (see, e.g, [10]). For practical purposes it is often sufficient to approximate these exact wave functions by the product of the Schrodinger-Coulomb wave functions with the reduced mass and the free electron spinors which depend on the electron mass and not on the reduced mass. These functions are very convenient for calculation of the high order corrections, and while below we will often skip some steps in the derivation of one or another high order contribution from the EDE, we advise the reader to keep in mind that almost all calculations below are done with these unperturbed wave functions. 

The Bethe logarithm is formally defined as a certain normalized infinite sum of matrix elements of the coordinate operator over the Schrodinger-Coulomb wave functions. It is a pure number which can in principle be calculated with arbitrary accuracy, and high accuracy results for the Bethe logarithm can be found in the literature (see, e.g. [13, 14] and references therein). For convenience we have collected some values for the Bethe logarithms [14] in Table 3.1. 

The Darwin potential generates the logarithmic correction to the nonrela-tivistic Schrodinger-Coulomb wave function in (3.65), and the result in (3.97) could be obtained by taking into account this correction to the wave function in calculation of the contribution to the Lamb shift of order a Za.ym. This logarithmic correction is numerically equal 14.43 kHz for the IS -level in hydrogen, and 1.80 kHz for the 2S level. 

One could insist that the mass dependence in (5.9) is natural because the calculations leading to it are done without expansion over the mass ratio, and are therefore exact. On the other hand, the factor m / mM) symmetric with respect to the masses naturally arises in all apparently nonrecoil calculations just from the Schrodinger-Coulomb wave function squared. According to the tradition we preserve the coefficient before the logarithm squared term in the form given in (3.53). Then the new contribution contained in (5.9) has the form [12, 13]... 

In the Schrodinger-Coulomb approximation the expression in (6.33) reduces to the leading nuclear size correction in (6.3). New results arise if we take into account Dirac corrections to the Schrodinger-Coulomb wave functions of relative order (Za). For the nS states the product of the wave functions in (6.33) has the form (see, e.g, [17])... 

The leading electron polarization contribution in (7.7) was calculated in the nonrelativistic approximation between the Schrodinger-Coulomb wave functions. Relativistic corrections of relative order (Za) to this contribution may easily be obtained in the nonrecoil limit. To this end one has to calculate the expectation value of the radiatively corrected potential in (7.1) between the relativistic Coulomb-Dirac wave functions instead of averaging it with the nonrelativistic Coulomb-Schrodinger wave functions. 

The simple calculation of the matrix element of this Hamiltonian between the nonrelativistic Schrodinger-Coulomb wave functions gives the Fermi result [2] for the splitting between the l S i and states ... 

The logarithmic nuclear size correction of order Za) EF may simply be obtained from the Zemach correction if one takes into account the Dirac correction to the Schrodinger-Coulomb wave function in (3.65) [7]... 

The nuclear effects (both for the hyperline structure and the Lamb shift) are a result of short-distance contributions and in the leading order are proportional to the Schrodinger-Coulomb wave function at the origin,... 

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