Current operator, matrix elements

The spinor product ipl pk represents the charge density associated with the four-spinor ipk, while c aqtpk represents the g-component of the current. The matrix elements of these operators may be reduced to elementary G-spinor integrals of the form... 

In order to compensate for the presence of such a term in the matrix element of [current operator ju(x) by stipulating... 

To lowest order in the external field, A%, the scattering is thus determined by the matrix element of the current operator ju(x) between the initial and final one-particle states, p, > and pV>. Let us consider this matrix element in greater detail. Translation invariance asserts... 

The vanishing of this matrix element is, in fact, independent of the assumption of current conservation, and can be proved using the transformation properties of the current operator and one-partic e states under space and time inversion, together with the hermiticity of jn(0). By actually generating the states q,<>, from the states in which the particle is at rest, by a Lorentz transformation along the 3 axis, and the use of the transformation properties of the current operator, essentially the entire kinematical structure of the matrix element of on q, can be obtained.15 We shall, however, not do so here. Bather, we note that the right-hand side of Eq. (11-529) implies that... 

Since the atomic nucleus consists of nucleons which themselves consist of quarks, in principle the wavefunction of the quarks within the nucleons is required in order to determine appropriate equivalent potentials for the interactions between an electron and the nucleons within the nucleus. The models currently available for the calculation of the substructure of the nucleons, however, allow only for an approximate description of the wavefunctions of the quarks (see for instance [74] for a comparison of a few of these models). One may on the other hand introduce nucleon field operators, which replace the quark field operators in the scattering matrix element (equation (66)), and relate the corresponding vector and axial coupling coefficients in the resulting equivalent potential to empirical data. 

In contrast to the vectorial term of the nnclens, the space-like axial current terms of the nucleons do not grow with increasing Z ox N since the nucleons tend to form pairs with opposite angular momentum. The total axial current term of the nuclens is proportional to the nuclear spin / and its calculation requires the matrix element of the following operator (see [31])... 

This contribution examines current approaches to Coulomb few-body problems mainly from a methodological perspective, in contrast to recent reviews which have focused on the results obtained for benchmark problems. The methods under discussion here employ wavefunctions which explicitly involve all the interparticle coordinates and use functional forms appropriate to nonadiabatic systems in which all the particles are of comparable mass. The use of such wavefunctions for states of arbitrary angular symmetry is reviewed, and the kinetic-energy operator, written in the interparticle coordinates, is presented in a convenient form. Evaluation of the resultant angular matrix elements is discussed in detail. For exponentially correlated wavefunctions, problems of integral evaluation are surveyed, the relatively new analytical procedures are summarized, and relations among matrix elements are presented. The current status of Gaussian-orbital and Hylleraas methods is also reviewed. 

The redistribution of charge in the system during the tunneling transition is described in terms of current density J (f, t) and its spatial distribution J (f), which is given by the matrix element between states A and D of the current density operator (this is the so-called transition current) ... 

Both the NDDO and the CNDO assumptions apply only to matrix elements involving the charge density operator. Other arguments have to be used for matrix elements of other operators, such as the kinetic energy and the current density. 

The four nonrelativistic operators of Eq. (4a) arise from an expansion of the lepton wave functions, while those of Eq. (4b) occur in the hadronic weak current, connecting the large and small components of nucleonic wave functions, and are relativistic. Matrix elements of different rank contribute incoherently to the decay rate. The rank R has the selection rule... 

The matrix elements of r, and a are of rank-one, r is the exact analogue of the isospin-changing El operator and the conserved-vector-current (CVC) theorem can be invoked to relate the matrix element of a to that of r. Further, [r, spin-dipole operator. The helicity operator 75 is the time-like component of the rank-zero axial current - also known as the axial charge - discussed in Section 1. We thus have five independent matrix elements, two each of rank zero (RO) and rank one (Rl) and one of rank two (R2) which we denote as... 

Let us begin with the orbital, or current, matrix element. Write p = -iftV and work with spherical vector components labelled by = 0, 1. Atomic states are labelled by 6, J, M, where 6 contains all the quantum numbers required to define a state other than J, M. The magnetic neutron operator contains the quantity K X V, and the many-electron matrix element is found to be... 

It has been pointed out (Levy 1988) that the effect linear in B also includes extraordinary contributions to the current operator. These can arise via strongly anisotropic mixing matrix elements and via spin-orbit coupling. An analogous possibility has been mentioned above in connection with structure in the thermopower S(T). In the final result for / jj(T), the extraordinary contributions also show up via an interference of the 1 = 2 and 1 = 3 channels, but with different combinations of the two phase shifts, e.g., cos(ad3 — 83) with a = 1, 2 and 3 (Levy 1988). The strong dependence of the f-channel phase shift on temperature and degeneracy of the lowest spin-orbit multiplet can in principle account for much of the rich structure of R (T). 

The application of angular-momentum theory to atomic spectroscopy is not limited to bringing eqs. (3)-(7) into play. In their book, Condon and Shortley (1935, ch. 3) developed the theory with particular attention to operators T that are vectors. They did this by specifying the commutation relations of Twith respect to J rather than by stating the transformation properties of the components of T under rotations. They considered angular momenta J built from two parts S and L) and obtained formulas for the matrix elements of operators that behave as a vector with respect to one part (say L) and a scalar with respect to the other (S, in this case). These formulas involve proportionality constants that would be called reduced matrix elements today. Condon and Shortley systematized the methods that had come into current use but which were often only hinted at, if that, by many theorists. For example. Van Vleck (1932) gave the formula... 

If it happens that the current operators are Hermitian, i.e. j = Jj, then we can also relate matrix elements of J+ and J via, for example. 

The precision of an analysis is limited by counting statistics which in turn depend on the operating conditions (counting time, probe current and voltage), the element analysed and its matrix (atomic number, line used for analysis). For light elements (boron, carbon, oxygen) there are additional uncertainties regarding the value of the correction parameters (in particular absorption coefficients). 

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