Upper components

In Eq. (168), the first, magnetic-field term admixes different components of the spinors both in the continuity equation and in the Hamilton-Jacobi equation. However, with the z axis chosen as the direction of H, the magnetic-field temi does not contain phases and does not mix component amplitudes. Therefore, there is no contribution from this term in the continuity equations and no amplitude mixing in the Hamilton-Jacobi equations. The second, electric-field term is nondiagonal between the large and small spinor components, which fact reduces its magnitude by a further small factor of 0 particle velocityjc). This term is therefore of the same small order 0(l/c ), as those terms in the second line in Eqs. (164) and (166) that refer to the upper components. 

The effective value of B, for the lower components of the doubled levels, can be obtained from the P and R branches by the same method of combination differences used for a type of band and, for the upper components, from the Q branch. From these two quantities and may be calculated. 

The method of combination differences applied to the P and R branches gives the lower state rotational constants B", or B" and D", just as in a A transition, from Equation (6.29) or Equation (6.32). These branches also give rotational constants B, or B and D, relating to the upper components of the 77 state, from Equation (6.30) or Equation (6.33). The constants B, or B and D, relating to the lower components of the state, may be obtained from the Q branch. The value of q can be obtained from B and B. ... 

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Composite Upper component prevent (FML) Lower component minimize (clay) — —... 

A composite hydraulic barrier, consisting of (a) a 0.5-mm-thick GM upper component and (b) a 0.6-m-thick CCL lower component, with the CCL having a minimum hydraulic conductivity of 1 x 10 7 cm/s. 

The lower 2-spinor components, f3 = —1, are related to the upper components by the kinetic matching relation. 

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... 

As the reflux ratio decreases, so does the slope of the upper component balance line, The effect of reflux ratio on the component balance lines is illustrated in Fig, 2,11, using the benzene-toluene system in Exampla 2.1. 

In the 75%-NC-blend curve, the interpretation is quite different. The broad relaxation seems to have components at 63° and 95°C. Possibly the upper component of the E" relaxation represents the Tg of a nearly pure NC phase. The peak at 63°C would then represent a second phase containing both NC and PCL. However, both peaks could represent NC-rich microphases of differing compositions. 

In performing the similarity transformation above, we have changed our representation of the particle from a Dirac representation to the so-called Foldy-Wouthuysen representation. This new representation provides a very simple link with the non-relativistic Schrodinger-Pauli representation. The latter, which is a two-component representation, just corresponds to the two upper components of the Foldy-Wouthuysen representation (3.125). It must be noted that under a similarity transformation such as (3.105), the operators which represent physical observables are also transformed. For example, the position observable whose operator is R in the Dirac representation is transformed to R " in the Foldy-Wouthuysen representation where... 

Let U be the exact FW transformation defined by Eq. (35) that decouples the Dirac Hamiltonian. The transformed wavefunction for class I (electronic) solutions has thus only a non-vanishing upper component and is given by... 

We start this chapter with a discussion of the non-relativistic limit (nrl) of relativistic quantum theory (section 2). The Levy-Leblond equation will play a central role. We also discuss the nrl of electrodynamics and study how properties differ at their nrl from the respective results of standard non-relativistic quantum theory. We then present (section 3) the Foldy-Wouthuysen (FW) transformation, which still deserves some interest, although it is obsolete as a starting point for a perturbation theory of relativistic corrections. In this context we discuss the operator X, which relates the lower to the upper component of a Dirac bispinor, and give its perturbation expansion. The presentation of direct perturbation theory (DPT) is the central part of this chapter (section 4). We discuss the... 

For electronic states (+ sign in (12)) the upper components

lower components ip are smaller by an order of for... 

This is an equation for a 4-component spinor field o> analogous to the Dirac 4-component spinor with an upper component lower component xoi both of which are two-component spinors. 

We know that for a positive-energy bound state (or an unbound state with W close to me ) the lower component is (except possibly very close to a nucleus, for atoms depending on the sign of k) smaller than the upper component by a factor of 0(c ) (hence the alternative names large and small components for p> and respectively - which are only valid for this kind of state). In order to deal with quantities of the same order of magnitude we introduce [33]... 

Then we use that for negative-energy states the upper component becomes small, and the lower component large, we hence introduce... 

We see that for small r, (f>2 differs from 2 by be. for H-like ions by —Zip jr. Similarly one finds that like ions behaves for small r as... 

The eigenfunctions of the effective Hamiltonian are the upper components ifk of the Dirac spinors, but L,- is not hermitean. 

Rather than start from a single nonrelativistic eigenfunction (po and improve it towards the upper component

Dirac spinor ijj, we now start from a d-dimensional model space of nonrelativistic functions (f = 1,... d) and we search for an effective Hamiltonian L as a. matrix representation in this model space, such that the eigenvalues of L are equal to a subset of eigenvalues of the Dirac operator. 

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