Coupled field equations

S, I, P, L, one obtains a set of coupled field equations for the determination of the following set of unknown field quantities ... 

The field- and time-dependent cluster operator is defined as T t, ) = nd HF) is the SCF wavefunction of the unperturbed molecule. By keeping the Hartree-Fock reference fixed in the presence of the external perturbation, a two step approach, which would introduce into the coupled cluster wavefunction an artificial pole structure form the response of the Hartree Fock orbitals, is circumvented. The quasienergy W and the time-dependent coupled cluster equations are determined by projecting the time-dependent Schrodinger equation onto the Hartree-Fock reference and onto the bra states (HF f[[exp(—T) ... 

Equation (28) is the set of exact coupled differential equations that must be solved for the nuclear wave functions in the presence of the time-varying electric field. In the spirit of the Born-Oppenheimer approximation, the ENBO approximation assumes that the electronic wave functions can respond immediately to changes in the nuclear geometry and to changes in the electric field and that we can consequently ignore the coupling terms containing... 

Simple chemical systems with several components (HCl, KOH, KCl in hydrogel) were used for modeling mass and charge balances coupled with equations for electric field, transport processes and equilibrium reactions [146]. This served for demonstrating the chemical systems function as electrolyte diodes and transistors, so-called electrolyte-microelectronics . 

The gauge potentials A now have 4x4 traceless representations. The scalar field theory that describes the vacuum will now satisfy field equations that involve all 16 components of the gauge potential. By selectively coupling these fields to the fermions, it should be possible to formulate a theory that recovers a low-energy theory that is the standard model with the 0(3)p gauge theory of electromagnetism. 

However, these two Lagrangian densities do not couple the two fields together. This requires that the free-field equation for the gauge field becomes... 

Equations (2), (4), and (5) can be combined with the deviatoric elasticity equation and the equilibrium equations to form a set of field equations consisting of a Navier-type equation and two coupled diffusion equations. For the class of problems characterized by an irrotational displacement field with chemical and hydraulic loadings only, the two coupled diffusion equations simplify to... 

Figure 7. Formal representation of the interaction of an external EM field with the dissipative subsystem in the Frohlich vibrational model (n is the nonlinear coupling parameter (Equation 17) the other quantities have the same meaning as...

Fig. 7.9 Computer simulations of a neuronal network (lOx 10 neurons) with nearest neighbor gap-junction coupling (see equations). The diagrams show the local mean field potentials (LFP) which are calculated as the mean potential values V of all neurons during continuously increasing coupling strength gc- Insets illustrate the different types of patterns in which the neurons originally operate (with randomly set initial values), (a) The network of tonic firing...

These two equations resemble the pair of coupled differential field equations of hydrodynamics, which describe the irrotational flow of a compressible fluid by [40] ... 

Now we can insert expressions for the modal fields and normalization constant into coupled mode equation (13.15)... 

An example for one of the first applications of the coupled-channel equations y with quantized fields for photodissociation problems is shown in Figure 12.1, where, the dissociation of the IBr molecule by a two-photon (visible + IR) process wasJS studied [388], The results of the calculations, shown in Figure 12.2, demonstrate 3 how the strong IR photon broadens the transition ( power broadening ) allowing th system to be dissociated even if the first photon is tuned substantially away from, 7 resonance. This illustrates how multiphoton transitions induced by strong fields [392] are less restricted insofar as they need not be very close to an intermediate T resonance, the situation described in Section 3.3. 

Figure 15.1 shows a pair of protons in the magnetic field B0. Because each spin possesses a magnetic moment, each is surrounded by a magnetic field that is experienced by the other. This is the direct, or through-space, dipole-dipole (or dipolar) coupling. The equation that describes the direct splitting Av (in hertz) in the spectrum is... 

It has been shown that under some geometric restrictions that involve conditions in distances and dimensions of the complete experimental device that is, lamps, reflectors, and reactors, the radiation field produced by the tubular lamp, and the parabolic reflector can be modeled by a onedimensional representation (Alfano et al., 1986). These limitations were imposed on the equipment design of this work. Since is a function of the radiation-absorption species concentration, in this case. Equation (34) is coupled with Equation (32). 

ESR spectroscopy is a sensitive detection method designed to provide information on radicals. This method, also known as electron paramagnetic resonance (EPR) spectroscopy, is naturally a good choice for coupling to electrochemistry as radicals can easily be electrogenerated. ESR involves the detection of unpaired electrons which are affected by nearby nuclei. Radical spin states are modified by interaction with a magnetic field, equation (17). ... 

---