Gaussian wave

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... 

Herman M F, Kluk E and Davis H L 1986 Comparison of the propagation of semiclassical frozen Gaussian wave functions with quantum propagation for a highly excited enharmonic oscillator J. Chem. Phys. 84 326... 

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... 

Here, we shall replace Xisa single Gaussian wave function Iy ) )) as defined earlier. That is, we have used the approximation... 

In mathematics there is a large number of complete sets of one-particle functions given, and many of those may be convenient for physical applications. With the development of the modern electronic computers, there has been a trend to use such sets as render particularly simple matrix elements HKL of the energy, and the accuracy desired has then been obtained by choosing the truncated set larger and larger. Here we would like to mention the use of Gaussian wave functions (Boys 1950, Meckler 1953) and the use of the exponential radial set (Boys 1955), i.e., respectively... 

Gaussian elimination technique, 291 Gaussian wave function, 276 Gegen ions, 160... 

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... 

Figure 1.4 ( ) A gaussian wave number distribution, (b) The modulating function corresponding to the wave number distribution in Figure 1.4(n).

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

There is a close similarity with planar electromagnetic cavities (H.-J. Stockmann, 1999). The basic equations take the same form and, in particular, the Poynting vector is the analog of the quantum mechanical current. It is therefore possible to experimentally observe currents, nodal points and streamlines in microwave billiards (M. Barth et.al., 2002 Y.-H. Kim et.al., 2003). The microwave measurements have confirmed many of the predictions of the random Gaussian wave fields described above. For example wave function statistics, current flow and... 

Figure 3. Infinite resolution spectrum (sticks) generated from a Gaussian wave packet launched at the inner turning point of PO 1 in Fig. 2, and low resolution version of it (full line). 

In Fig. 3 we present the stick spectrum [corresponding to infinite resolution see eq. (12)] generated from a Gaussian wave packet of the form... 

Figure 4. Diabatic (solid lines) and adiabatic (dashed lines) potential-energy curves of Model IVa. The Gaussian wave packet indicates the initial preparation of the system at time t = 0.

We see from Fig. 2.5 that the Gaussian wave packet has its intensity, F 2, centred on x0 with a half width, W, whereas (k) 2 is centred on k0 with a half width, 1/W. Thus the wave packet, which is centred on x0 with a spread Ax — W, is a linear superposition of plane waves whose wave vectors are centred on k0 with a spread, A = jW. But from eqn (2.8), p = Hk. Therefore, this wave packet can be thought of as representing a particle that is located approximately within Ax = W of x0 with a momentum within Ap = h/W of po = hk0. If we try to localize the wave packet by decreasing W, we increase the spread in momentum about p0. Similarly, if we try to characterize the particle with a definite momentum by decreasing 1/W, we increase the uncertainty in position. 

Fig. 2.5 The relation between a Gaussian wave packet Y, and its Fourier transformation, . The quantity, ( 2, has a half width of W, where 2 has a half width of 1.

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