Equation time derivative

A differential equation for the time evolution of the density operator may be derived by taking the time derivative of equation (Al.6.49) and using the TDSE to replace the time derivative of the wavefiinction with the Hamiltonian operating on the wavefiinction. The result is called the Liouville equation, that is. 

Using the Heisenberg equation of motion, (AS,2,40). the connnutator in the last expression may be replaced by the time-derivative operator... 

There are tliree steps in the calculation first, solve the frill nonlinear set of hydrodynamic equations in the steady state, where the time derivatives of all quantities are zero second, linearize about the steady-state solutions third, postulate a non-equilibrium ensemble through a generalized fluctuation dissipation relation. 

If we write the time-dependent Sclnodinger equation as 5i i/(T = -(i/li)f7i i, then, after replacing the time derivative by a central difference, we obtain... 

The observable NMR signal is the imaginary part of the sum of the two steady-state magnetizations, and Mg. The steady state implies that the time derivatives are zero and a little fiirther calculation (and neglect of T2 tenns) gives the NMR spectrum of an exchanging system as equation (B2.4.6)). 

The steady-state solution without saturation to this equation is obtained by setting the time derivatives to zero and taking the tenns linear in as in equation (B2.4.11). 

Obviously, the BO or the adiabatic states only serve as a basis, albeit a useful basis if they are determined accurately, for such evolving states, and one may ask whether another, less costly, basis could be Just as useful. The electron nuclear dynamics (END) theory [1-4] treats the simultaneous dynamics of electrons and nuclei and may be characterized as a time-dependent, fully nonadiabatic approach to direct dynamics. The END equations that approximate the time-dependent Schrddinger equation are derived by employing the time-dependent variational principle (TDVP). 

Other combinations of upper- and lower-convected time derivatives of the stress tensor are also used to construct constitutive equations for viscoelastic fluids. For example, Johnson and Segalman (1977) have proposed the following equation... 

Time derivatives in expansion (2.113) can now be substituted using the differential equation (2.112) (Donea, 1984 The first order time derivative in expansion (2.113) is substituted using Equation (2.112) as... 

Repeated differentiation of Equation (2.114) with respect to the time variable also gives the higlier-order time derivatives of the unknown, for example... 

Governing flow equations, originally written in an Eulerian framework, should hence be modified to take into account the movement of the mesh. The time derivative of a variable / in a moving framework is found as... 

The selection of a time increment dependent on parameter a (i.e. carrying out Taylor series expansion at a level between successive time steps of n and n+Y) enhances the flexibility of the temporal discretizations by allowing the introduction of various amounts of smoothing in different problems. The first-order time derivatives are found from the governing equations as... 

After substitution of the first- and second-order time derivatives of the unknowns in Equations (4.132) to (4.134) from Equations (4.139) to (4.141) and spatial discretization of the resulting equations in the usual manner the working equations of the scheme are derived. In these equations, fimctions given at time level n+aAt can be interpolated as... 

Each physical quantity is expressed in one and only one unit, eg, the meter for length, the kilogram for mass, and the second for time. Derived units are defined by simple equations relating two or more base units. Some are given special names, such as newton for force and joule for work and energy. 

We denote by C the value of c(x , t) at any time. Thus, C is a function of time, and differential equations in C are ordinary differential equations. By evaluating the diffusion equation at the ith node and replacing the derivative with a finite difference equation, the following working equation is derived for each node i, i = 2,. . . , n (see Fig. 3-52). 

Unsteady material and energy balances are formulated with the conservation law, Eq. (7-68). The sink term of a material balance is and the accumulation term is the time derivative of the content of reactant in the vessel, or 3(V C )/3t, where both and depend on the time. An unsteady condition in the sense used in this section always has an accumulation term. This sense of unsteadiness excludes the batch reactor where conditions do change with time but are taken account of in the sink term. Startup and shutdown periods of batch reactors, however, are classified as unsteady their equations are developed in the Batch Reactors subsection. For a semibatch operation in which some of the reactants are preloaded and the others are fed in gradually, equations are developed in Example 11, following. 

It is worth investigating the time derivatives and demonstrating how to derive (9.1)-(9.4) from the more familiar forms of the conservation equations. The more familiar Lagrangian derivative djdt and d jdt are related by [9]... 

The conservation equations are more commonly written in the initial reference frame (Lagrangian forms). The time derivative normally used is d /dt. Equation (9.5) is used to derive (9.2) from the Lagrangian form of the conservation of mass... 

The time derivative of the momentum integral uses the faet that the mass of the element does not ehange to generate the following equation for the x-eomponent of the aeeeleration... 

The time derivative of the coefficient c ft) for the particular state 4/ (t) is found by multiplying either side of the equation by the complex conjugate of 4/°(t) and integrating. After a little manipulation we find... 

Dirac (1930a) had the idea of working with a relativistic equation that was linear in the space and time derivatives. He wrote... 

The steady state TMB model equations are obtained from the transient TMB model equations by setting the time derivatives equal to zero in Equations (25) and (26). The steady state TMB model was solved numerically by using the COLNEW software [29]. This package solves a general class of mixed-order systems of boundary value ordinary differential equations and is a modification of the COLSYS package developed by Ascher et al. [30, 31]. 

Since these assumptions are not always justifiable when applied to plastics, the classic equations cannot be used indiscriminately. Each case must be considered on its merits, with account being taken of such factors as the time under load, the mode of deformation, the service conditions, the fabrication method, the environment, and others. In particular, it should be noted that the traditional equations are derived using the relationship that stress equals modulus times strain, where the modulus is a constant. From the review in Chapter 2 it should be clear that the modulus of a plastic is generally not a constant. Several approaches have been used to allow for this condition. The drawback is that these methods can be quite complex, involving numerical techniques that are not attractive to designers. However, one method has been widely accepted, the so-called pseudo-elastic design method. 

Taking the time derivative of H(t), and using the Boltzmann equation, Eq. (1-39) for this case ... 

Hydrodynamic Equations.—Before deriving the hydro-dynamic equations, some integral theorems that are useful in the solution of the Boltzmann equation will be proved. Consider a function of velocity, G(Vx), which may also be a function of position and time let... 

Although the Klein-Gordon equation is of second order in the time derivative, for a positive energy particle the knowledge of at some given time is sufficient to determine the subsequent evolution of the particle since 8ldt is then given by Eq. (9-85). Alternatively Eq. (9-85) can be adopted as the equation of motion for a free spin zero particle of mass m. We shall do so here. 

Summarizing, we have noted that the Heisenberg operators Q+(t) obey field free equations i.e., that their time derivatives are given by the commutator of the operator with Ha+(t) = Ho+(0) and that this operator H0+(t) is equal to H(t) = H(0). The eigenstates of H0+ are, therefore, just the eigenstates of H. We can, therefore, identify the states Tn>+ with the previously defined >ln and the operator [Pg.602]

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