Nonrelativistic limit

The weakly relativistic limit of the Hamiltonian (2.20) for fermions in external electric and magnetic fields can be derived with standard techniques, either by direct expansion or by a low order Foldy-Wouthuysen transformation. One obtains 

47) p x) is a nonrelativistic field operator of two-component structure, a are the Pauli matrices and the electron-electron interaction reduces to the Coulomb interaction, denoted by H e- As usual, the gauge term proportional to 

The density operator is defined in terms of the field operators as 

If one re-expresses the Hamiltonian (3.47) in terms of the density and current operators in order to exhibit the coupling to the external fields more explicitly, one finds 

This expression indicates that it is the sum of the paramagnetic current operator and the curl of the magnetisation density, 

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Similarides Between Potential Ruid Dynamics and Quantum Mechanics Electrons in the Dirac Theory The Nearly Nonrelativistic Limit The Lagrangean-Density Correction Term Topological Phase for Dirac Electrons What Have We Learned About Spinor Phases ... 

The second term in Eq. (154), will be shown to be smaller than the first in the near nonrelativistic limit. 

The right-hand side will be treated in a following section VI.E, where we shall see that it is small in the nearly nonrelativistic limit and that it vanishes in the absence of an elechomagnetic field. The left-hand side can be evaluated to give... 

In order to write the previously obtained equations in the nearly nonrelativistic limit, we introduce phase differences s, that remain finite in the limit c —> oo. Then... 

These terms are analogous to those on p. 265 of [7], It will be noted that the symbol c has been reinstated as in Section VI.F, so as to facilitate the order of magnitude estimation in the nearly nonrelativistic limit. We now proceed based on Eq. (168) as it stands, since the transformation of Eq. (168) to modulus and phase variables and functional derivation gives rather involved expressions and will not be set out here. 

The topological (or Berry) phase [9,11,78] has been discussed in previous sections. The physical picture for it is that when a periodic force, slowly (adiabatically) varying in time, is applied to the system then, upon a full periodic evolution, the phase of the wave function may have a part that is independent of the amplitude of the force. This part exists in addition to that part of the phase that depends on the amplitude of the force and that contributes to the usual, dynamic phase. We shall now discuss whether a relativistic electron can have a Berry phase when this is absent in the framework of the Schrddinger equation, and vice versa. (We restrict the present discussion to the nearly nonrelativistic limit, when particle velocities are much smaller than c.)... 

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... 

Hamilton-Jacobi equation, molecular systems, modulus-phase formalism, 262-265 Lagrangean density correction term, 270 nearly nonrelativistic limit, 269... 

This part of our chapter has shown that the use of the two variables, moduli and phases, leads in a direct way to the derivation of the continuity and Hamilton-Jacobi equations for both scalar and spinor wave functions. For the latter case, we show that the differential equations for each spinor component are (in the nearly nonrelativistic limit) approximately decoupled. Because of this decoupling (mutual independence) it appears that the reciprocal relations between phases and moduli derived in Section III hold to a good approximation for each spinor component separately, too. For velocities and electromagnetic field strengths that are normally below the relativistic scale, the Berry phase obtained from the Schrodinger equation (for scalar fields) will not be altered by consideration of the Dirac equation. 

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

The problem of nonrelativistic limit description for fundamental particles and their interactions may be solved in different ways. Although in all methods of nonrelativistic expansion the first terms of the Hamiltonians coincide, however the difference begins to arise at transition to the higher orders of expansion. The method of Foldy-Wouthuysen... 

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]... 

Following the common approach in relativistic field theory, which aims at a manifestly covariant representation of the dynamics inherent in the field operators, so far all quantities have been introduced in the Heisenberg picture. To develop the framework of relativistic DFT, however, it is common practice to transform to the Schrodinger picture, so that the relativistic theory can be formulated in close analogy to its nonrelativistic limit. As usual we choose the two pictures to coincide at = 0. Once the field operators in the Schroodinger-picture have been identified via j/5 (x) = tj/(x, = 0), etc, the Hamiltonians He,s, Hy s and are immediately obtained in terms of the Schrodinger-picture field operators. 

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