Equation stationary

Close inspection of equation (A 1.1.45) reveals that, under very special circumstances, the expectation value does not change with time for any system properties that correspond to fixed (static) operator representations. Specifically, if tlie spatial part of the time-dependent wavefiinction is the exact eigenfiinction ). of the Hamiltonian, then Cj(0) = 1 (the zero of time can be chosen arbitrarily) and all other (O) = 0. The second tenn clearly vanishes in these cases, which are known as stationary states. As the name implies, all observable properties of these states do not vary with time. In a stationary state, the energy of the system has a precise value (the corresponding eigenvalue of //) as do observables that are associated with operators that connmite with ft. For all other properties (such as the position and momentum). 

Using W2 = 17jP2, (A3.2.81 and (A3.2.9) may be used to satisfy the Smoluchowski equation, (A3.2.2). another necessary property for a stationary process. Thus u(t) is an example of a stationary Gaussian-Markov... 

The mean values of the. (t) are zero and each is assumed to be stationary Gaussian white noise. The linearity of these equations guarantees that the random process described by the a. is also a stationary Gaussian-... 

A proposal based on Onsager s theory was made by Landau and Lifshitz [27] for the fluctuations that should be added to the Navier-Stokes hydrodynamic equations. Fluctuating stress tensor and heat flux temis were postulated in analogy with the Onsager theory. Flowever, since this is a case where the variables are of mixed time reversal character, tlie derivation was not fiilly rigorous. This situation was remedied by tlie derivation by Fox and Ulilenbeck [13, H, 18] based on general stationary Gaussian-Markov processes [12]. The precise fomi of the Landau proposal is confimied by this approach [14]. 

The system of coupled differential equations that result from a compound reaction mechanism consists of several different (reversible) elementary steps. The kinetics are described by a system of coupled differential equations rather than a single rate law. This system can sometimes be decoupled by assuming that the concentrations of the intennediate species are small and quasi-stationary. The Lindemann mechanism of thermal unimolecular reactions [18,19] affords an instructive example for the application of such approximations. This mechanism is based on the idea that a molecule A has to pick up sufficient energy... 

In addition to [A ] being qiiasi-stationary the quasi-equilibrium, approximation assumes a virtually unperturbed equilibrium between activation and deactivation (equation (A3.4.125)) ... 

This leads to the quasi-stationary rate constant of equation (A3,4,133) if 4k + k +k f, which is... 

For this is identical to tire quasi-stationary result, equation (A3,4,133). although only tlie ratio [A ]/... 

A] = b/a (equation (A3.4.145)) is stationary and not [A ] itself This suggests d[A ]/dt < d[A]/dt as a more appropriate fomuilation of quasi-stationarity. Furthemiore, the general stationary state solution (equation (A3.4.144)) for the Lindemaim mechanism contams cases that are not usually retained in the Bodenstein quasi-steady-state solution. 

Diflfiision-controlled reactions between ions in solution are strongly influenced by the Coulomb interaction accelerating or retarding ion diffiision. In this case, die dififiision equation for p concerning motion of one reactant about the other stationary reactant, the Debye-Smoluchowski equation. 

For such a function, variational optimization of the spin orbitals to make the expectation value ( F // T ) stationary produces [30] the canonical FIF equations... 

As is a stationary point, the linear temis FlEZ- h/j ) q" i/iin equation B3.5.1 vanish, and higher-order temis do... 

The simplest smooth fiuictioii which has a local miiiimum is a quadratic. Such a function has only one, easily detemiinable stationary point. It is thus not surprising that most optimization methods try to model the unknown fiuictioii with a local quadratic approximation, in the fomi of equation (B3.5.1). 

Dmbining this with the constraint equation enables us to identify the stationary point, hich is at (-59/72, -23/18). 

In this chapter the general equations of laminar, non-Newtonian, non-isothermal, incompressible flow, commonly used to model polymer processing operations, are presented. Throughout this chapter, for the simplicity of presentation, vector notations are used and all of the equations are given in a fixed (stationary or Eulerian) coordinate system. 

Crouzeix, M. and Raviart, P. A., 1973. Conforming and non-conforming finite elements for solving the stationary Navier-Stokes equations. RAIRO, Seric Rouge 3, 33 -76. 

The problem of a mass suspended by a spring from another mass suspended by another spring, attached to a stationary point (Kreyszig, 1989, p. 159ff) yields the matrix equation... 

From equation 12.1 it is clear that resolution may be improved either by increasing Afr or by decreasing wa or w-q (Figure 12.9). We can increase Afr by enhancing the interaction of the solutes with the column or by increasing the column s selectivity for one of the solutes. Peak width is a kinetic effect associated with the solute s movement within and between the mobile phase and stationary phase. The effect is governed by several factors that are collectively called column efficiency. Each of these factors is considered in more detail in the following sections. 

Solving equation 12.3 for the moles of solute in the stationary phase and substituting into equation 12.2 gives... 

Note that this equation is identical to that describing the extraction of a solute in a liquid-liquid extraction (equation 7.25 in Chapter 7). Since the volumes of the stationary and mobile phase may not be known, equation 12.4 is simplified by dividing both the numerator and denominator by V thus... 

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