The charge and current densities

A subsystem is an open system, free to exchange charge and momentum with its environment. Thus the current density jg for any observable G is of particular importance in the mechanics of a subsystem, since a non-vanishing flux in this current implies a fluctuation in the subsystem average value of the property G. 

Variation of the energy functional for a subsystem is equal to an infinitesimal times the flux in the current density of the generator G through the 

The same surface integral appears in the subsystem statement of the hyper-virial theorem 

The non-vanishing of the flux of a quantum mechanical current in the absence of a magnetic field is what distinguishes the mechanics of a subsystem from that of the total system in a stationary state. The flux in the current density will vanish through any surface on which i// satisfies the natural boundary condition, Vi/ n = 0 (eqn (5.62)), a condition which is satisfied by a system with boundaries at infinity. Thus, for a total system the energy is stationary in the usual sense, 5 [i/ ] = 0, and the usual form of the hyper-virial theorem is obtained with the vanishing of the commutator average. 

As noted in the previous chapter, eqn (6.4) is a consequence of the Hermitian property of H, a property not enjoyed in general, by a subsystem. 

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According to the Dirac [36] electron theory, the relativistic wavefunction has four components in spin-space. With the Hermitian adjoint wave function , the quantum mechanical forms of the charge and current densities become [31,40]... 

This result, as well as the form of expressions (23) and (24), shows that the charge and current density relations (3), (4), and (8) of the present extended theory become consistent with and related to the Dirac theory. It also implies that this extended theory can be developed in harmony with the basis of quantum electrodynamics. 

This paper presents an account of the dynamics of electric charges coupled to electromagnetic fields. The main approximation is to use non-relativistic forms for the charge and current density. A quantum theory requires either a Lagrangian or a Hamiltonian formulation of the dynamics in atomic and molecular physics the latter is almost universal so the main thrust of the paper is the development of a general Hamiltonian. It is this Hamiltonian that provides the basis for a recent demonstration that the S-matrix on the energy shell is gauge-invariant to all orders of perturbation theory. 

Erwin Schrodinger (1926). The statement following his introduction of the charge and current densities and the quantum equation of continuity in his fourth paper on wave mechanics . 

Since a purely theoretical, quantum mechanical determination of the nuclear structure, i.e., a determination of the nuclear state functions from which the charge and current density distributions could be obtained, is neither routinely feasible nor intended within an electronic structure calculation, we have to resort to model distributions. The latter may be rather simple mathematical functions, or much more sophisticated expressions deduced from a careful analysis of experimental data. 

The components of the polarization fields can be expressed in terms of the charge and current densities by the relations... 

Harriman [59] gives a derivation which starts from the retarded potentials, Eqs. (2.142) and (2.143), for which he assumes Taylor expansions of the charge and current density of the integrand in terms of the retardation time r — r lc,... 

We next derive the relation between the dielectric function and the conductivity. Using the plane wave expressions for the charge and current densities and the electric field,... 

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