The Nonrelativistic Limit

For c — 00 (this means instant action of forces) the radial Dirac equations take the form of the radial Schrodinger equation for a particle moving in a central field V r). If we write Eqs. (6.69) and (6.70) — after subtraction of —meC, division of Eq. (6.70) by c and introduction of = E, — mgC — as 

In the derivation presented above we subtracted nteC from both radial Dirac equations in order to remove any c-dependence from the upper equation, which then allowed us to easily evaluate the limit c — oo. A similar effect was also induced in the first discussion on the nonrelativistic limit in section 5.4.3. 

The consideration of the nonrelativistic limit of the Dirac energy eigenvalue for the hydrogen-like atom with a Coulombic potential for the electron-nucleus attraction, Eq. (6.3), demonstrates the effect of subtracting the rest energy mgC and leads us to a discussion of the reference energy in the following section 6.7. 

The energy eigenvalue in Eq. (6.122) in Hartree atomic units h = e = ntg = 4 7t o = lO niay be expanded in a Taylor series. 

From this equation we understand that the nonielativistic Schrodinger energy eigenvalue for hydrogen-like atoms with point-like Coulomb nucleus, Eq. (6.19), is obtained only after subtraction of the rest energy, 

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In the nonrelativistic limit (at c = 10 °) the band contribution to the total energy does not depend on the SDW polarization. This is apparent from Table 2 in which the numerical values of Eb for a four-atom unit cell are listed. The table also gives the values of the Fermi energy Ep and the density of states at the Fermi level N Ef). 

When is a one component scalar function, one can take the square root of Eq. (9-237) and one thus obtains the relativistic equation describing a spin 0 particle discussed in Section 9.4. This procedure, however, does not work for a spin particle since we know that in the present situation the amplitude must be a multicomponent object, because in the nonrelativistic limit the amplitude must go over into the 2-component nonrelativistic wave function describing a spin particle. Dirac, therefore, argued that the square root operator in the present case must involve something operating on these components. 

Since p = 1 in the nonrelativistic limit where the contribution vanishes, we can replace p by (p — 1) in the expression for H. In this case the residual EDM interaction of an electron with the internal electric field reduces to... 

It is clear from these equations that in the nonrelativistic limit (n, r irrational number in this case. 

We would like to point out some steps of derivation of the nonrelativistic limit Hamiltonians by means of the Foldy-Wouthuyisen transformation (Bjorken and Drell, 1964). The method is based on the transformation of a relativistic equation of motion to the Schrodinger equation form. 

For the nitrogen hyperfine tensors, there is no satisfactory empirical scheme for estimating the various contributions, so that Table II compares the total observed tensor to the DSW result. The tensors are given in their principal axis system, with perpendicular to the plane of the heme and along the Cu-N bond. The small values (0.1 - 0.2 MHz) found for A O in the nonrelativistic limit are not a consequence of orbital motion (which must vanish in this limit) but are the result of inaccuracies in the decomposition of the total tensor into its components, as described above. 

In the nonrelativistic limit (c °o), the small component is related to the large component by [20]... 

Under electron impact ionization, when the energy transfer greatly exceeds the orbital ionization energy, the process resembles a Rutherford-type collision between two nearly free electrons. On this Mott [17] imposed the condition of indistinguishability of the outgoing electrons and obtained in the nonrelativistic limit... 

To reach the nonrelativistic limit of this equation, the right-hand side is expanded as... 

Figure 2. Envelope of the absorption edge part of the differential cross section dajdo scaled in 0 3 a.u. as a function of energy transfer scaled in units of 1000 for an electrical field strength of F = 1 a.u or vector potential A = 3186 a.u.. The initial election energy is W = 100 a.u.. The solid line denotes the result for electrons, the short dashed one the differential cross section for spinless particles and the long dashed one the result for the nonrelativistic limit.

If desired, the two-fold integral here can be further reduced to a one-fold one by introducing the hypergeometric function F of two variables [67]. Expression (58) is already convenient, however, both for the investigation of general properties as well as for the accurate numerical evaluation. In particular, the nonrelativistic limit of (58) reads ... 

The factor tC (not to be confused with the reduced coupling constant K) yields the relativistic effect. The nonrelativistic limit, which is given by taking c oo, yields /C = 1 and eq. (4.8) leads to the nonrelativistic Hamiltonian in the electromagnetic fields. We may insert w = p + A into the right-hand side of eq. (4.8) to obtain... 

The nonrelativistic limit of this operator yields the paramagnetic spin-orbit (PSO) contribution of Ramsey s theory. The remaining terms in eq. (4.10b) result in the ZORA relativistic spin-orbit Hamiltonian,... 

V operates solely on the inside of the parentheses. The nonrelativistic limit of this operator corresponds to the sum of the Fermi-contact (FC) and spin-dipolar (SD) terms of Ramsey s theory. Second-order differentiation of eq. (4.12) with respect to ptAj and psk gives... 

In the nonrelativistic limit, c —> oo, eq. (4.16c) yields the well-known Fermi-contact operator and the spin-dipolar interaction operator because... 

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