Electron Charge and Current Density

According to Max Born s interpretation of the wavefunction the square of the absolute value of the wavefunction of an electron, = V (a) gives the probability 

Making use of the indistinguishabihty of the electrons again we can alternatively write the electron density as an expectation value 

The subscript 1 that indicates coordinates of electron 1 is therefore dropped in the following. 

This expression will be used later in Part III, when we want to calculate the electron density for various approximate wavefunctions. 

Sometimes, it is also necessary to generalize the definition of the electron density P r) by defining a reduced one-electron density matrix 

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Electronic Charge and Current Densities for iUlany Eiectrons... 

All solutions of this Hamiltonian are thereby electronic, whether they are of positive or negative energy and contrary to what is often stated in the literature. Positronic solutions are obtained by charge conjugation. From the expectation value of the Dirac Hamiltonian (23) and from consideration of the interaction Lagrangian (16) relativistic charge and current density are readily identified as... 

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]... 

In order to derive Rosenfeld s equation from Eq. (3.14) we must make some such argument as the following. Let us assume that the self-consistent quantum field theory (Sect. 2) has been worked through and has yielded composite -particle elementary excitations that we identify with molecules. We can then define a charge and current density for a molecule containing n electrons and nuclei,... 

The matter field was originally postulated by Louis de Broglie, and discovered in the electron diffraction studies of Davisson and Germer [30] and of G. P. Thomson [31]. From Schrodinger s understanding of the matter field of, say, an electron, it must be represented in the source terms (charge and current density) of Maxwell s equations, as the moduli of these waves. 

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. 

It is common to treat the semiconductor-electrolyte interface in terms of charge and current density boundary conditions. The total charge held within the electrolytic solution and the interfacial states, which balances the charge held in the semiconductor, is assumed to be constant. This provides a derivative boundary condition for the potential at the interface. The fluxes of electrons and holes are constrained by kinetic expressions at the interface. The assumption that the charge is constant in the space charge region is valid in the absence of kinetic and mass-transfer limitations to the electrochemical reactions. Treatment of the influence of kinetic or mass transfer limitations requires solution of the equations governing the coupled phenomena associated with the semiconductor, the electrolyte, and the semiconductor-electrolyte interface. 

Now, to go to the experimental situation, what happens as we insert a metal electrode into an electrolyte solution without connecting it to an external electron source As we have discussed before (p. 22), an El is built up and hence a certain potential is established across the interface region. At this potential, charge transfer between electrode and electroactive species takes place, but, since no net current flows, the rates of electronation and de-electronation are identical. The system has reached the equilibrium potential at which the current density z for electronation is equal to the current density of de-electronation i. This current density is designated i0, the equilibrium exchange current density (cf. Table 6), given by the expression ... 

These processes govern all relaxation phenomena. Charge transport and the relaxation rates of the carrier populations to re-establish equilibrium conditions are normally described using the continuity and current density relations. For example, the one-dimensional (1-D) case for electrons under low injection is given by... 

This implies that at steady state under a constant force the electron moves with a constant speed v = rf. Using f = — e where is the electric field and — e is the electron charge, and the expression for the electric current density in terms of the electron density zi, charge —e and speed v. 

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