Time-dependent amplitudes

In the basis set formulation, we need to evaluate matrix elements over the G-H basis functions. We can avoid this by introducing a discrete variable representation method. We can obtain the DVR expressions by expanding the time-dependent amplitudes a (t) in the following manner ... 

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. 

The decay of Nal can be described in an alternative way [K.B. Mpller, N.E. Henriksen, and A.H. Zewail, J. Chem. Phys. 113, 10477 (2000)]. In the bound region of the excited-state potential energy surface, one can define a discrete set of quasi-stationary states that are (weakly) coupled to the continuum states in the dissociation channel Na + I. These quasi-stationary states are also called resonance states and they have a finite lifetime due to the coupling to the continuum. Each quasi-stationary state has a time-dependent amplitude with a time evolution that can be expressed in terms of an effective (complex, non-Hermitian) Hamiltonian. 

In this section we outline a theoretical framework enabling the 2 + 1 resonant photoionization of a hydrogenic atom to be analyzed. Generally, the process is assumed to be induced by a non-monochromatic laser field with a time-dependent amplitude and taking place in the presence of intermediate resonance. It is shown... 

Analysis shows that the system of equations, which governs the time evolution of the given 3-level system under the action of the laser field with a time-dependent amplitude and frequency, resembles its nonrelativistic counterpart [19] and has the form ... 

The last term in Eq. 4 represents the time dependent RF irradiation field in the rotating frame, which can be written in terms of two perpendicular time dependent amplitudes as... 

The characteristics of the two sets of trajectories and their weights differ considerably. Trajectories riding initially on the lower left diabat, the aa fifi ) = (fill) class, will climb the wall in this surface and, as they enter the coupling region, they will be additionally accelerated and decelerated by the off-diagonal forces whose effects are modulated by the time dependent amplitude term pistPat + / p pt + For this class of trajectories,... 

We recall that qlt) is just the time-dependent amplitude of the vector potential, and by (3.47) qlt) is related to the electric field. On the other hand, Eq. (3.56) has... 

Double jump techniques reveal CTI by time-dependent amplitudes of folding phases. 

The expression for the explicit time dependence of the CARS amplitude can also be simplified. This time-dependent amplitude is given by... 

The time evolution of the nonlinear O-H stretching absorption shows pronounced oscillatory signals for all types of dimers studied. In Fig. 15.5, data for OD/OD dimers are presented which were recorded at 3 different spectral positions in the O-D stretching band. For positive delay times, one finds rate-like kinetics which is due to population and thermal relaxation of the excited dimers and, more importantly, superimposed by very strong oscillatory absorption changes. In contrast to the intramolecular hydrogen bonds discussed above, the time-dependent amplitude of the oscillations displays a slow modulation with an increase and a decrease on a time scale of several hundreds of femtoseconds. 

Other variants are due to Fano [76], Anderson [77], Lee [78], and Friedrichs [79] and have been successfully applied to study, for example, autoionization, photon emission, or cavities coupled to waveguides. The dynamics can be solved in several ways, using coupled differential equations for the time-dependent amplitudes and Laplace transforms or finding the eigenstates with Feshbach s (P,Q) projector formalism [80], which allows separation of the inner (discrete) and outer (continuum) spaces and provides explicit expressions ready for exact calculation or phenomenological approaches. For modern treatments with emphasis on decay, see Refs. [31, 81]. Writing the eigenvector as [31, 76]... 

As the molecule vibrates (undergoes atom displacements)) the electronic charge distribution and, hence, the polarizability (a) varies in time. The polarizability is related to the electron density of the molecule and is often visualized in three dimensions as an ellipsoid and represented mathematically as a symmetric second-rank tensor. The time-dependent amplitude (Q ) of a normal vibrational mode executing simple harmonic motion is written in terms of the equilibrium amplitude Q , the normal mode frequency o), and time t). 

Equations 8.40 and 8.41 show that the time-dependent amplitudes and phases of the laser pulses define the temporal structure of the photoassociated waveform. This is true in the multichannel as well as in the single-channel AREA. We now assume that all the laser pulses have similar simple time profiles, and concentrate on using PA for determining the multichannel structure of the input wavepacket. The measurement is based on controlling the interference of quantum pathways during AREA, each pathway corresponding to the adiabatic passage via one of the intermediate 1)... n) states. 

If the evolving shape itself must be determined over the course of time as part of the solution, the underlying boundary value problem is no longer linear. Analysis of surface evolution in such situations must rely on approximate methods, in general. The two most common approaches are (i) to determine approximate solutions to the governing partial differential equations by numerical methods and (ii) to express the evolving shape in terms of a small number of modes, each with its own time-dependent amplitude. In the former approach, the solution proceeds incrementally. An elasticity... 

The time-dependent amplitudes, U t) and are then determined from the... 

Having determined the time dependence of the coupled cluster and A wave-functions to first order, i.e. having derived expressions for the Fourier components and of the first-order time-dependent amplitudes in Eqs. (11.78)... 

As in Sect.3.1, we describe the damped oscillator by its time-dependent amplitude... 

Co being the average concentration, and Sc(t) being a time-dependent amplitude. The temporal evolution is found from Pick s law of diffusion and the continuity equation, which in one dimension are... 

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