Implicit time dependence

Electron Nuclear Dynamics (48) departs from a variational form where the state vector is both explicitly and implicitly time-dependent. A coherent state formulation for electron and nuclear motion is given and the relevant parameters are determined as functions of time from the Euler equations that define the stationary point of the functional. Yngve and his group have currently implemented the method for a determinantal electronic wave function and products of wave packets for the nuclei in the limit of zero width, a "classical" limit. Results are coming forth protons on methane (49), diatoms in laser fields (50), protons on water (51), and charge transfer (52) between oxygen and protons. 

Sg denotes the adjoint electron spin operator. One should notice that the expression [exp(—iLtx)Sp Do i p(f2ML)] results in the S-operators and the ml being (implicitly) time-dependent. In order to continue any further, we need to specify the lattice and its Liouvillian. 

The form of the principal part 02 (6) with separation into three terms helps to understand the geometrical characteristics (Melchior, ). The first term of 02 in square brackets is independent of 2 it is implicitly time dependent on the declination of the sun or moon, which have periods of 14 days and 6 months, respectively. The second term describes the diurnal tides, amplitude-modulated by sin 2d. The third term is periodic with 12 h describing semidiurnal tides, amplitude-modulated by cos. ... 

We also assume that H explicitly depends on time. This means that in general there will be an explicit dependence on time, t, in the potential energy operator U(r, t) = Ux (r) -I- U (r, t), and that there will be an implicit time dependence in V( l>) being function of time. 

At large distances from the fiber such that QRp v, we can replace the Hankel function by its asymptotic form of Eq. (37-87). If we include the implicit time dependence, Eq. (24-7) is proportional to... 

The spatial dependence of the electric field E(x, y, z) and the magnetic field H(x, y, z) of an optical waveguide is determined by Maxwell s equations. We assume an implicit time dependence exp( — iwt) in the field vectors, current density J and charge density a. The dielectric constant s(x, y, z) is related to the refractive index n(x, y, z) by e = n CQ, where Eq is the dielectric constant of free space. For the nonmagnetic materials which normally constitute an optical waveguide, the magnetic permeability p is very nearly equal to the free-space value Pq. Thus for convenience we assume p = Po throughout this book unless otherwise stated. Under these conditions. Maxwell s equations are expressible in the form[l]... 

The derivation of the conjugated or unconjugated reciprocity theorems requires two distinct electromagnetic situations one characterized by refractive-index profile n, current density J and electric and magnetic fields E and H while the other is characterized by n, J, E and H. All vector quantities contain the implicit time dependence exp( — iwt), where (o is the angular frequency. The refractive-index profiles n and n may depend on all three spatial variables, but the permeability is taken to have the free-space value Po unless otherwise stated. 

To obtain the group velocity, we need a generalized reciprocity theorem which allows for variations in the frequency w of the implicit time dependence of the fields. We let E and H be the fields of the jth mode at wavelength L These fields satisfy the source-free Maxwell equations, and if we allow for variations in the permeability p, they have the form... 

Consider two distinct electromagnetic situations within a fiber. In the first situation, an arbitrary current distribution J gives rise to electric and magnetic fields Eand H, and in the second situation, a point current dipole J results in fields E and H. All six electromagnetic quantities include the implicit time dependence exp (—tot). The arbitrary location and orientation of the dipole is expressed by... 

An aspect in which circumstellar environments differ from interstellar is the net mass advection through the medium. Abundances become time dependent— and hence space dependent— in the envelope, due both to the implicit time dependence of the reactions and to the transport of matter through different radii via stellar wind flow. The atomic abundances are fixed at stellar photosphere, rather than having to be assumed for some mixture of physical parameters of temperature and pressure as they must for molecular clouds. It is then essentially an initial-value problem to compute the abundances which will be a function of radius in the envelope. For a steady-state wind, the abundances become strictly a function of radius. Also, unlike a molecular cloud, the density profile of the envelope is specified from the assumption of steady mass loss at the terminal velocity for the wind, so that p(r) = M/(47Tr Voo), where M is the mass loss rate and Woo is the terminal velocity of the wind. 

---