Field momentum

The second term represents the conserved current and the third is a momentum term. Noting that dL/dq = P, the field momentum bscomes... 

The derivation of the given whole field formulation, introducing the Dirac delta function (d/) into the surface tension force relation to maintain the discontinuous (singular) nature of this term, is to a certain extent based on physical intuition rather than first principles (i.e., in mathematical terms this approach is strictly not characterized as a continuum formulation on the differential form). Chandrasekhar [31] (pp 430-433) derived a similar model formulation and argued that to some extent the whole field momentum equation can be obtained by a formal mathematical procedure. However, the fact that the equation involves /-functions means that to interpret the equation correctly at a point of discontinuity, we must integrate the equation, across the interface, over an infinitesimal volume element including the discontinuity. 

Provided in this section are the general equations that govern the transport of heat (temperature held), electricity (electric field), momentum (flow field), and mass species (concentration field) involved in electrokinetic flow. These equations form the basis of the theoretical modeling of Joule heating in electrokinetic flow. 

Geometrically speaking, the material medium which we are about to describe is a deformable svuface, which we shall now define in mathematical terms. The physical quantities used to describe the state of the material medium will then be introduced and, when we are dealing with tensorial quantities (pressure tensor, electrical and magnetic fields, momentum), compatibility conditions wiU need to be satisfied. 

Electrons and most other fiindamental particles have two distinct spin wavefunctions that are degenerate in the absence of an external magnetic field. Associated with these are two abstract states which are eigenfiinctions of the intrinsic spin angular momentum operator S... 

Semiconductors are poor conductors of electricity at low temperatures. Since the valence band is completely occupied, an applied electric field caimot change the total momentum of the valence electrons. This is a reflection of the Pauli principle. This would not be true for an electron that is excited into the conduction band. However, for a band gap of 1 eV or more, few electrons can be themially excited into the conduction band at ambient temperatures. Conversely, the electronic properties of semiconductors at ambient temperatures can be profoundly altered by the... 

We consider an isolated molecule in field-free space with Hamiltonian //. We let Pbe the total angular momentum operator of the molecule, that is... 

We hope that by now the reader has it finnly in mind that the way molecular symmetry is defined and used is based on energy invariance and not on considerations of the geometry of molecular equilibrium structures. Synnnetry defined in this way leads to the idea of consenntion. For example, the total angular momentum of an isolated molecule m field-free space is a conserved quantity (like the total energy) since there are no tenns in the Hamiltonian that can mix states having different values of F. This point is discussed fiirther in section Al.4.3.1 and section Al.4.3.2. 

The translational linear momentum is conserved for an isolated molecule in field free space and, as we see below, this is closely related to the fact that the molecular Hamiltonian connmites with all... 

In most of the connnonly used ab initio quantum chemical methods [26], one fonns a set of configurations by placing N electrons into spin orbitals in a maimer that produces the spatial, spin and angular momentum syimnetry of the electronic state of interest. The correct wavefimction T is then written as a linear combination of tire mean-field configuration fimctions qj = example, to describe the... 

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