Wave-packet

We now consider the formation of a composite wave as the superposition of a continuous spectrum of plane waves with wave numbers k confined to a narrow band of values. Such a composite wave (x, /) is known as a wave packet and may be expressed as 

The weighting factor A(k) for each plane wave of wave number k is negligible except when k lies within a small interval A. For mathematical convenience we have included a factor on the right-hand side of equation (1.11). 

This factor merely changes the value of A k) and has no other effect. 

We note that the wave packet (x, t) is the inverse Fourier transform of A k). The mathematical development and properties of Fourier transforms are presented in Appendix B. Equation (1.11) has the form of equation (B.19). According to equation (B.20), the Fourier transform A k) is related to (x, t) by 

It is because of the Fourier relationships between (x, t) and A(k) that the factor (2jt) is included in equation (1.11). Although the time t appears in the integral on the right-hand side of (1.12), the function A k) does not depend on / the time dependence of (x, i) cancels the factor e ( We consider below 

In general, the angular frequency a (k) is a function of k, so that the angular frequencies in the composite wave frfrx, l), as well as the wave numbers, vary from one plane wave to another. If a (k) is a slowly varying function of k and the values of k are confined to a small range Ak, then a (k) may be expanded in a Taylor series in k about some point ko within the interval Ak 

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Scherer N F, Carlson R J, Matro A, Du M, Ruggiero A J, Romero-Rochin V, Cina J A, Fleming G R and Rice S A 1991 Fluorescence-detected wave packet interferometry time resolved molecular spectroscopy with sequences of femtosecond phase-locked pulses J. Chem. Rhys. 95 1487... 

Scherer N F, Matro A, Ziegler L D, Du M, Cina J A and Fleming G R 1992 Fluorescence-detected wave packet interferometry. 2. Role of rotations and determination of the susceptibility J. Chem. Rhys. 96 4180... 

Engel V and Metiu H 1994 2-Photon wave-packet interferometry J. Chem. Rhys. 100 5448... 

Lee S-Y 1995 Wave-packet model of dynamic dispersed and integrated pump-probe signals in femtosecond transition state spectroscopy Femtosecond Chemistry ed J Manz and L Wdste (Heidelberg VCH)... 

Tannor D J and Rice S A 1985 Control of selectivity of chemical reaction via control of wave packet evolution J. 

Zhang D H and Light J C 1996 Cumulative reaction probability via transition state wave packets J. Chem. Phys. 104 6184-91... 

In time-dependent quantum mechanics, vibrational motion may be described as the motion of the wave packet... 

I i i(q,01 in configuration space, e.g. as defined by the possible values of the position coordinates q. This motion is given by the time evolution of the wave fiinction i(q,t), defined as die projection ( q r(t)) of the time-dependent quantum state i i(t)) on configuration space. Since the quantum state is a complete description of the system, the wave packet defining the probability density can be viewed as the quantum mechanical counterpart of the classical distribution F(q- i t), p - P t)). The time dependence is obtained by solution of the time-dependent Schrodinger equation... 

In view of the foregoing discussion, one might ask what is a typical time evolution of the wave packet for the isolated molecule, what are typical tune scales and, if initial conditions are such that an entire energy shell participates, does the wave packet resulting from the coherent dynamics look like a microcanonical... 

The time evolution of is shown in figure A3.13.12 for the field free motion of wave packets for CHD2T and CHDT2 prepared by a preceding excitation along the> -axis. 

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... 

Marquardt R and Quack M 1991 The wave packet motion and intramolecular vibrational... 

Dey B D, Askar A and Rabitz H 1998 Multidimensional wave packet dynamics within the fluid dynamical formulation of the Schrddinger equation J. Chem. Phys. 109 8770-82... 

Schinke R and Huber J R 1995 Molecular dynamics in excited electronic states—time-dependent wave packet studies Femtosecond Chemistry Proc. Berlin Conf. Femtosecond Chemistry (Berlin, March 1993) (Weinheim Verlag Chemie)... 

FIdthmann H, Beck C, Schinke R, Woywod C and Domcke W 1997 Photodissociation of ozone in the Chappuis band. II. Time-dependent wave packet calculations and interpretation of diffuse vibrational... 

Loettgers A, Untch A, Keller H-M, Schinke R, Werner H-J, Bauer C and Rosmus P 1997 Ab initio study of the photodlssoclatlon of HCO In the first absorption band three-dimensional wave packet... 

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

Marquardt R and Quack M 1989 Infrared-multlphoton excitation and wave packet motion of the harmonic and anharmonic oscillators exact solutions and quasiresonant approximation J. Chem. Phys. 90 6320-7... 

Herve S, Le Quere F and Marquardt R 2001 Rotational and vibrational wave packet motion during the IR multiphoton excitation of HF J. Chem. Phys. 114 826-35... 

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