Boundary time-independent

For the particular case that A% is time-independent, we could obtain the boundary conditions that 6 is to satisfy by inserting into the defining relation, Eq. (10-241), a complete set of states that are eigenstates of H. Thus, when x0 > x 0, we can write... 

Solve the time-independent Schrodinger equation for this particle to obtain the energy levels and the normalized wave functions. (Note that the boundary conditions are different from those in Section 2.5.)... 

Equation (64) provides just one example of a phenomenon that may easily occur whenever species with different migration characteristics equilibrate sluggishly with each other and especially when they obey different boundary conditions. Namely, in uniform material subjected to time-independent boundary conditions, a steady state can be approached at long times that is not spatially uniform. We shall note a possible manifestation of such an effect in Section 5 of III. 

Since each input of mass to a perfect plug flow unit is independent of what has been input previously, its condition as it moves along the reactor will be determined solely by its initial condition and its residence time, independently of what comes before or after. Practically, of course, some interaction will occur at the boundary between successive inputs of different compositions or temperatures. This is governed by diffusional behaviors which are beyond the scope of the present work. 

Diffusive crystal growth at a fixed temperature would not result in a constant crystal growth rate (see below). However, under some specific conditions, such as continuous slow cooling, or in the presence of convection with diffusion across the boundary layer, time-independent growth rate may be achieved. Similarly, time-independent dissolution rate may also be achieved. 

The theories of transient processes leading to steady detonation waves have been concerned on the one hand with the prediction of the shape of pressure waves which will initiate, described in Section VI, A of Ref 66, and on the other hand with the pressure leading to the formation of such.an initiating pulse, described in Section VI, B. In Section V it was shown that the time-independent side boundary conditions are important in determining the characteristics of steady, three-dimensional waves. It now becomes necessary to take into consideration time-dependent rear boundary conditions. For one-dimensional waves, the side boundary conditions are not involved... 

Stationary electro-convection at an electrically inhomogeneous permselective membrane.7 Once again the time-independent version of (6.4.45)-(6.4.49) with the boundary conditions (6.4.54a,b) at x — 0. 

The terms in delta functions will be integrated to get the invariant 5 , which is still space- and time-independent. Hence, these two Green s functions can be combined as they can have the same value at the moment of particle creation (t = 0) and at this time only these terms are non-zero. This expression needs a little further manipulation. Following Lebedev et al. [506] on the variational principle, drop the term V ZJe- 17 VGG and include a new term of the form V (Ae fJUG G — e u4>G — e u4tG )r where f is a unit radial vector and

scalar functions of r which are related directly to the boundary conditions which both G and G satisfy. Multiply the delta function term by 2. 

Equation (316) should be compared with eqn. (44). It is second order because it involves the second space derivative V2, partial because of the three space dimensions and time (independent variables), inhomogeneous because the term J (r, t) is taken to be independent of p(r, t), linear because only first powers of the density p appear, and self adjoint in efic/p(r, t), the importance of which we shall see in the next section [491, 499]. The homogeneous equation corresponding to eqn. (316) has a solution p0 (r, t), which satisfies the same boundary conditions as p... 

Because of interelectronic repulsions, the Schrodinger equation for many-electron atoms and molecules cannot be solved exactly. The two main approximation methods used are the variation method and perturbation theory. The variation method is based on the following theorem. Given a system with time-independent Hamiltonian //, then if

well-behaved function that satisfies the boundary conditions of the problem, one can show (by expanding

[Pg.271]

Finally, we mention that the (Mg,Co)0 crystal surface with the lower oxygen potential is morphologically unstable. Therefore, in a strict sense the boundary conditions (which have been tacitly assumed to be stationary) are not time-independent. This phenomenon will be discussed further in Chapter 11. 

If a steady state exists, both the stationary thicknesses A (v) and the chemical potential differences Afjt( established in reservoirs R, and R2 are time-independent (v = a,/9,...). It follows that (v)) for the other components i n in all phases v are also time-independent as long as the phase boundaries are morphologically stable. Furthermore, the fluxes are constant and, therefore, the velocity... 

These assumptions, however, oversimplify the problem. The parent (A,B)0 phase between the surface and the reaction front coexists with the precipitated (A, B)304 particles. These particles are thus located within the oxygen potential gradient. They vary in composition as a function of ( ) since they coexist with (A,B)0 (AT0<1 see Fig. 9-3). In the Af region, the point defect thermodynamics therefore become very complex [F. Schneider, H. Schmalzried (1990)]. Furthermore, Dv is not constant since it is the chemical diffusion coefficient and as such it contains the thermodynamic factor /v = (0/iV/01ncv). In most cases, one cannot quantify these considerations because the point defect thermodynamics are not available. A parabolic rate law for the internal oxidation processes of oxide solid solutions is expected, however, if the boundary conditions at the surface (reaction front ( F) become time-independent. This expectation is often verified by experimental observations [K. Ostyn, et al. (1984) H. Schmalzried, M. Backhaus-Ricoult (1993)]. 

In Section 2.5 we have constructed the degenerate continuum wavefunc-tions 4/ f(R, r Ef, n), which describe the dissociation of the ABC complex into A+BC(n). They solve the time-independent Schrodinger equation for fixed energy Ef subject to the boundary conditions (2.59). Furthermore, the 4/f(R,r Ef,n) are orthogonal and complete and thus they form a basis in the corresponding Hilbert space, i.e., any function can be represented as a linear combination of them. 

The time-independent and time-dependent approaches merely provide different views of the dissociation process and different numerical tools for the calculation of photodissociation cross sections. The time-independent approach is a boundary value problem, i.e., the stationary wavefunction... 

The radial concentration scans obtained from the UV spectrophotometer of the analytical ultracentrifuge can be either converted to a radial derivative of the concentrations at a given instant of time (dc/dr)t or to the time derivative of the concentrations at fixed radial position (dc/dt)r (Stafford, 1992). The dcf dt method, as the name implies, uses the temporal derivative which results in elimination of time independent (random) sources of noise in the data, thereby greatly increasing the precision of sedimentation boundary analysis (Stafford, 1992). Numerically, this process is implemented by subtracting pairs of radial concentration scans obtained at uniformly and closely spaced time intervals c2 — G)/( 2 — h)]. The values are then plotted as a function of radius to obtain (dc/dt) f versus r curves (Stafford, 1994). It can be shown that the apparent sedimentation coefficient s ... 

The rationale behind this approach is the variational principle. This principle states that for an arbitrary, well-behaved function of the coordinates of the system (e.g., the coordinates of all electrons in case of the electronic Schrodinger equation) that is in accord with its boundary conditions (e.g., molecular dimension, time-independent state, etc.), the expectation value of its energy is an upper bound to the respective energy of the true (but possibly unkown) wavefunction. As such, the variational principle provides a simple and powerful criterion for evaluating the quality of trial wavefunctions the lower the energetic expectation value, the better the associated wavefunction. 

At variance from Xe, the presented properties for Kr require more computional efforts. In order to reach the small-4 range of S(q), large-scale molecular dynamics have been carried out in the microcanonical ensemble (NVE) with the usual periodic boundary conditions. The equations of motion are integrated in the same discrete form as for Xe. The time step At is the same as for Xe and g r) is extracted over a sample of 8000 time-independent configurations every lOAf. 

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