Time-dependent coefficients

We can find the time-dependent coefficient for being in state 2 by multiplying from the left by and integrating over spatial coordinates ... 

The eigenfunctions fa form a complete set thus, the wavefunctaon V (q t 92, 0 can be expanded in terms of the wavjefunetions fa, with file use of the time-dependent coefficients b (r). The resulting expression is then... 

Let us assume a spherical mineral with radius R which initially contains a gas with concentration C0(r), r being the radial distance from the center. Upon incremental heating, this gas is lost to the extraction line and at the ith heating step when time is tf, the fraction of initial gas remaining is/(tf). Loss takes place by radial diffusion with temperature-dependent, hence time-dependent, coefficient 3>(t). We assume that the total amount of gas held by the mineral at t=0 is equal to one, i.e., that... 

An ansatz is then made for the time-dependent wavefunction l o (t)) depending on time-dependent coefficients which are expanded in the orders of the perturbations Ef or The wavefunction coefficients in each order of the perturbation have to be determined from the time-dependent Schrddinger equation or an equivalent time-dependent variation principle. Expressions... 

To this end, we resort to a novel general approach to the control of arbitrary multidimensional quantum operations in open systems described by the reduced density matrix p(t) if the desired operation is disturbed by linear couplings to a bath, via operators S B (where S is the traceless system operator and B is the bath operator), one can choose controls to maximize the operation fidelity according to the following recipe, which holds to second order in the system-bath coupling (i) The control (modulation) transforms the system-bath coupling operators to the time-dependent form S t) (S) B(t) in the interaction picture, via the rotation matrix e,(t) a set of time-dependent coefficients in the operator basis, (Pauli matrices in the case of a qubit), such that ... 

Each component of the perturbations has been separated into two terms a time-dependent amplitude An and Tm, and a time-dependent spatial term cos (nnx). If the uniform state is stable, all the time-dependent coefficients will tend in time to zero. If the uniform state is temporally unstable even in the well-stirred case, but stable to spatial patterning, then the coefficients A0 and T0 will grow but the other amplitudes Ax-Ax and 7 1-7 0O will again tend to zero. If the uniform state becomes unstable to pattern formation, at least some of the higher coefficients will grow. This may all sound rather technical but is really only a generalization of the local stability analysis of chapter 3. 

Equations of motion for the time-dependent coefficients Aj time-dependent single particle functions, and time-dependent Gaussian parameters A K s) = aj c s), f r]jK s can be derived via the Dirac-Frenkel variational principle [1], leading to... 

The Gaussian (6.11) with this E and with the averages (6.15) constitutes the solution of the linear Fokker-Planck equation (6.4) with time-dependent coefficients. 0... 

In the usual derivations of the Klein-Kramers equation, the moments of the velocity increments, Eq. (68), are taken as expansion coefficients in the Chapman-Kolmogorov equation [9]. Generalizations of this procedure start off with the assumption of a memory integral in the Langevin equation to finally produce a Fokker-Planck equation with time-dependent coefficients [67]. We are now going to describe an alternative approach based on the Langevin equation (67) which leads to a fractional IGein-Kramers equation— that is, a temporally nonlocal behavior. 

The first term is the initial state with energy Ei before the light beam is switched on and the second term represents the total wavefunction in the upper electronic state. The initial conditions for the corresponding time-dependent coefficients are aj(0) = 1 and af(0 Ef,n) = 0 for all energies Ef and all vibrational channels n. The sum over a in (2.9) is replaced in (2.64) by an integral over Ef and a sum over n. The integral over Ef reflects the fact that the spectrum in the upper electronic state is continuous and the summation over all open vibrational channels n accounts for the degeneracy of the continuum wavefunctions. 

In this context, Berry [277] studied the enzyme reaction using Monte Carlo simulations in 2-dimensional lattices with varying obstacle densities as models of biological membranes. That author found that the fractal characteristics of the kinetics are increasingly pronounced as obstacle density and initial concentration increase. In addition, the rate constant controlling the rate of the complex formation was found to be, in essence, a time-dependent coefficient since segregation effects arise due to the fractal structure of the reaction medium. In a similar vein, Fuite et al. [278] proposed that the fractal structure of the liver with attendant kinetic properties of drug elimination can explain the unusual... 

This form is very similar to the model often used when the molecules move across fractal media, e.g., the dissolution rate using a time-dependent coefficient given by (5.12) to describe phenomena that take place under dimensional constraints or understirred conditions [16]. The previous differential equation has the solution given by (9.9). 

In conclusion, the solutions E Qt (f)] for the expected values for such stochastic models are the same as the solutions qT (t) for the corresponding deterministic models, and the transfer-intensity matrix H is analogous to the fractional flow rates matrix K of the deterministic model. If the hazard rates are constant in time, we have the stochastic analogues of linear deterministic systems with constant coefficients. If the hazard rates depend on time, we have the stochastic analogues of linear deterministic systems with time-dependent coefficients. 

Eckerman, K., Leggett, R., and Williams, L., An elementary method for solving compartmental models with time-dependent coefficients, Radiation Protection Dosimetry, Vol. 41, 1992, pp. 257—263. 

In the next step, the time-dependent coefficients Cjn Roo, i), calculated by (f)jn r,j) (Roo,r,T,t)), are Fourier transformed to give the partial cross sections [308] according to... 

The matrix elements H j contain the coupling between the two modes. Taking this approach, the time-dependent coefficients Cj and c j in Eqn. (29) can be expressed as linear combinations of formal position and momentum coordinates qjandpi [57]... 

The equations of motion (14a) and (14b) contain time-dependent coefficients. Nevertheless, surprisingly, it is possible to find their analytical solution in the semiclassical approach for an arbitrary but real modulation amplitude... 

As an alternative to Master Equations with memory, many authors suggest memory-less equations with time dependent coefficients. This approach seems... 

This is a linear system for e(r), but it has a chaotic time-dependent coefficient u(t)... 

What is the quantum mechanical analog of this approach Consider the simple example that describes the decay of a single level coupled to a continuum. Fig. 9.1 and Eq. (9.2). The time-dependent wavefunction for this model is (Z) = Ci (t) 11)+ Ci(t ) l, where the time-dependent coefficients satisfy (cf. Eqs (9.6) and (9.7))... 

To find the coefficients zfjf and, one has to solve an infinite set of coupled equations (18) (k = 1,2,...) with time-dependent coefficients moreover, each equation also contains an infinite number of terms. However, the problem can be essentially simplified, if the walls perform small oscillations at the frequency [Pg.323]

We suppose that initially the system was in the ground state with the only nonzero coefficient Cq (0) = 1, and that the frequency of wall s vibrations is close to twice the frequency of the unperturbed mode (i v 2 v. Looking for the solution at 0 in the same form (216)—(218), but with time-dependent coefficients, and neglecting the rapidly oscillating terms containing exp (2/7), we get the following equation for the coefficient c(t) ... 

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