Nonrelativistic approximation

Because of the large mass of the nucleus and the low recoil velocity involved, we may use the nonrelativistic approximation... 

The kinetic energy operator in the Schrodinger equation corresponds to the quadratic term in this nonrelativistic expansion, and thus the Schrodinger equation describes only the leading nonrelativistic approximation to the hydrogen energy levels. 

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... 

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. 

In the leading nonrelativistic approximation the denominator of the photon propagator cancels the exchanged momentum squared in the numerator, and we immediately obtain the Hamiltonian for the interaction of two magnetic moments, reproducing the above result of classical electrodynamics. 

Evans showed that the existence of B is consistent with finite photon mass, m7 in Proca equation. In nonrelativistic approximation Proca equation can be written as V2A = c2A, where c is related to photon mass m,. Taking B = V x A, we see that V2B = c2B is the same as the Proca equation, because V2(V x A) = 2V x A, that is, V x 82A = V x 2A the solution is found to be... 

According to Eq. (81) with /S = 6, the three a values for H in Figure 4.14 are —3.708, 2.000, and 9.708. Thus, the asymptotically lowest hyperspherical potential supports an infinite series of Feshbach resonances in the nonrelativistic approximation, although only three lowest members remain as resonances after corrections for the relativistic and radiative effects [80, 82], as was mentioned in Section 3.1.2. Only the lowest member is indicated in the figure by a horizontal line. This resonance is supported by the diabatic potential with A = — 1 connecting from the lowest curve for large p to the middle curve for small p. 

The standard approaches for two-fermion systems like Bethe-Salpeter or Dirac-Breit satisfy this condition. But usually this property is lost in the final results, because (nonrelativistic) approximations (e.g. Ei to ) are used for one or both particles to simplify the calculation. 

In the following, an approach is presented that reproduces the main part of the recoil and hyperfine corrections without any nonrelativistic approximation and therefore with full CVT-invariance. 

Here, ag = h2/(m e2) is the reduced Bohr radius (m = M), n is a principal quantum number of the hydrogenic state i), and a = e2/(hc) 1/137 is the fine structure constant. In the following discussion, (2) will be the only restriction imposed on values of physical parameters. In particular, no distinction will be made as to whether the photon energy ho is less or greater than the ionization potential Jj of the state i). Note that = ftwi holds true only in the nonrelativistic approximation, whereas in general I) tkui. 

For a heavy element whose atomic number is beyond 50, the relativistic effects (error caused by the nonrelativistic approximation) on the valence state can not be ignored. In such a case, it is necessary to solve Dirac equation instead of nonrelativistic Schrodinger equation usually used for the electronic state calculation. The relativistic effects... 

This dipole matrix element is written in the molecular frame and includes the summation over all final rotational states. It corresponds to a spin flip and must be zero in the nonrelativistic approximation. The spin-orbit interaction mixes the FIi/2 and Si/2 states... 

Thus far we have retained the form of HpNc which is correct for arbitrary electron energy, since in heavy atoms the electrons are indeed fully relativistic near the nucleus where they contribute to HpNc- However, the nonrelativistic form of HpNc is useful for calculations in light atoms and also has some conceptual value in heavy atoms as well. Making the nonrelativistic approximations of Eq. (7) for the Dirac operators in Eq. (8) we obtain the nonrelativistic potential for a single electron interacting with a point nucleus at r = 0 ... 

Since Er is very small compared to Eq. we may write for the recoil energy (in a nonrelativistic approximation)... 

When comparing the above contributions (i) and (ii), one is reminded that the nonrelativistic approximation is never worse than for open flavours. Likewise, the electron is more relativistic in hydrogen than in positronium. If 8 + 8m < 0 when the u quark is replaced by d [contributions (iii) and (iv) being provisionally forgotten], we are in a paradoxical regime where the resulting hadron mass decreases as the mass of one of the constituents is increased. This is a warning that relativistic corrections are required [107]. 

Taking into account the isotopic invariance of the strong interaction and the nonrelativistic approximation, one can obtain from eqs. (4.3) and (4.4) the efficient T-violating nucleon-nucleon Hamiltonian... 

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