Gaussian packet

Now we write the same Fourier of expansion for the electric field and write everything according to the magnetic field intensity H = B, and we find with the case that (e/H)Aq co the amplitude fixed to the wavelength as is the case for some solitons, for Gaussian packets, we arrive at the same cubic Schrodinger equation ... 

Exact Calculation WKB Approximation (7.9) Gaussian Packet Approximation (7.10)... 

As an example of this formulation we consider pulsed photoassociation of a coherent wave packet of cold Na atoms [345], The colliding atoms are described by an (energetically narrow) normalized Gaussian packet of J = 0 radial waves ... 

M. Oresic, D. Shalloway, Hierarchical characterization of energy landscapes using Gaussian packet states, J. Chem. Phys. 101 (1994), 9844. 

There is a fourth important point Ehrenfest s classical limit is actually reached on this time scale for molecular systems. The spreading of the wave packet turned out to be not a problem, contrary to anticipation, and we now know why [3]. This can easily be seen by considering the motion of a Gaussian packet in... 

Prom the structural point of view the situations presented by the Feynman advanced variational potentials are not as complete as that of the Gra picture. One notes that the Gaussian packets ISVP and ASVP cannot be associated with a single actual particle in the fluid. These/ packets represent the reduced mass of a two-particle subsystan in quite an involved way, which entangles all the coordinates used to describe such particles. Therefore, although pair radial structures can be defined for both approximations, the self-correlations cannot. This limits the computations in k-space when utilizing ISVP and ASVP [116,138]. The formal development of the structural issues is identical for both potentials, and for simplicity they will be referred to as SVP in what follows. 

Heather R and Metiu H 1985 Some remarks concerning the propagation of a Gaussian wave packet trapped in a Morse potential Chem. Phys. Lett. 118 558-63... 

Sawada S and Metiu H 1986 A multiple trajectory theory for curve crossing problems obtained by using a Gaussian wave packet representation of the nuclear motion J. Chem. Phys. 84 227-38... 

Braun M, Metlu H and Engel V 1998 Molecular femtosecond excitation described within the Gaussian wave packet approximation J. Chem. Phys. 108 8983-8... 

The big advantage of the Gaussian wavepacket method over the swarm of trajectory approach is that a wave function is being used, which can be easily manipulated to obtain quantum mechanical information such as the spechum, or reaction cross-sections. The initial Gaussian wave packet is chosen so that it... 

To deal with the problem of using a superposition of functions, Heller also tried using Gaussian wave packets with a fixed width as a time-dependent basis set for the representation of the evolving nuclear wave function [23]. Each frozen Gaussian function evolves under classical equations of motion, and the phase is provided by the classical action along the path... 

A typical initial condition in ordinary wave packet dynamics is an incoming Gaussian wave packet consistent with particular diatomic vibrational and rotational quantum numbers. In the present case, of course, one has two diatomics and with the rotational basis representation of Eq. (30) one would have, for the full complex wave packet. 

The wave packets <()( ) and x(0 to be propagated forward and backward, respectively, are expanded in terms of the frozen Gaussian wave packets as (see also Section II.B)... 

The similar expansion applies to x(f). The frozen Gaussian wave packets gy,q p, are explicitly given by... 

Without loss of generality y = y can be assumed. If the dipole moment can be assumed to be a linear function of coordinate within the spread of the frozen Gaussian wave packet, the matrix element (gy,q,p, Pjt(r) Y,q, p ) can be evaluated analytically. Since the integrand in Eq. (201) has distinct maxima usually, we can introduce the linearization approximation around these maxima. Namely, the Taylor expansion with respect to bqp = Qq — Qo and 8po = Po — Po is made, where qj, and pj, represent the maximum positions. The classical action >5qj, p , ( is expanded up to the second order, the final phase-space point (q, p,) to the first order, and the Herman-Kluk preexponential factor Cy pj to the zeroth order. This approximation is the same as the ceUularization procedure used in Ref. [18]. Under the above assumptions, various integrations in U/i(y, q, p ) can be carried out analytically and we have... 

In order to obtain a specific mathematical expression for the wave packet, we need to select some form for the function A k). In our first example we choose A(k) to be the gaussian function... 

Figure 1.5 The real part of a wave packet for a gaussian wave number distribution.

As time increases from —oo to 0, the half width of the wave packet y(x, t) continuously decreases and the maximum amplitude continuously increases. At t = 0 the half width attains its lowest value of flja and the maximum amplitude attains its highest value of 1 /a/2, and both values are in agreement with the wave packet in equation (1.20). As time increases from 0 to oo, the half width continuously increases and the maximum amplitude continuously decreases. Thus, as f- increases, the wave packet y(x, t) remains gaussian in shape, but broadens and flattens out in such a way that the area under the square y(x, t) of the wave packet remains constant over time at a value of (2-y/ a), in agreement with ParsevaTs theorem (1.18). 

---