Wave-packet states

On a single-molecule basis, a unidirectional molecular rotation requires careful preparation of the non-stationary quantum state of the rotor. The motive power of such rotating wave-packet states is weak and requires... 

Equation (3.3) is the bridge between the deep underworld where the forces are at work and the apparent world where observations are performed. In Eq. (3.3) a> and fe> are wave-packet states, which belong to the Fock space 0> and [ (x)] is a functional of (normally ordered) products of the (x) fields. 

An inhomogeneous equation, whose solution is analogous to that of the time-independent Schrodinger equation. The inhomogeneity reflects the preparation of the system at time t = 0 in the wave packet state x(t = 0)). 

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)... 

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

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... 

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)... 

Mowrey R C and Kouri D J 1987 Application of the close coupling wave packet method to long lived resonance states in molecule-surface scattering J. Chem. Phys. 86 6140... 

Neuhauser D 1990 Bound state eigenfunctions from wave packets—time -> energy resolution J. Chem. Phys. 932611... 

Beck M H and Meyer H D 1998 Extracting accurate bound-state spectra from approximate wave packet propagation using the filter-diagonalization method J. Chem. Phys. 109 3730... 

We present state-to-state transition probabilities on the ground adiabatic state where calculations were performed by using the extended BO equation for the N = 3 case and a time-dependent wave-packet approach. We have already discussed this approach in the N = 2 case. Here, we have shown results at four energies and all of them are far below the point of Cl, that is, E = 3.0 eV. 

The Schrddinger Wave Packet.—Let the system at time t be in a state represented by the vector f> in This state may be specified by giving its components with respect to every coordinate eigenvector q> in, namely <(q <>. This is the wave function as ordinarily understood in the Schrodinger theory, see Eq. (8-45). On the other hand we may also specify the state > by giving its components with respect to every momentum eigenvector in namely... 

Electron Nuclear Dynamics (48) departs from a variational form where the state vector is both explicitly and implicitly time-dependent. A coherent state formulation for electron and nuclear motion is given and the relevant parameters are determined as functions of time from the Euler equations that define the stationary point of the functional. Yngve and his group have currently implemented the method for a determinantal electronic wave function and products of wave packets for the nuclei in the limit of zero width, a "classical" limit. Results are coming forth protons on methane (49), diatoms in laser fields (50), protons on water (51), and charge transfer (52) between oxygen and protons. 

The ability to create and observe coherent dynamics in heterostructures offers the intriguing possibility to control the dynamics of the charge carriers. Recent experiments have shown that control in such systems is indeed possible. For example, phase-locked laser pulses can be used to coherently amplify or suppress THz radiation in a coupled quantum well [5]. The direction of a photocurrent can be controlled by exciting a structure with a laser field and its second harmonic, and then varying the phase difference between the two fields [8,9]. Phase-locked pulses tuned to excitonic resonances allow population control and coherent destruction of heavy hole wave packets [10]. Complex filters can be designed to enhance specific characteristics of the THz emission [11,12]. These experiments are impressive demonstrations of the ability to control the microscopic and macroscopic dynamics of solid-state systems. 

In the DC-biased structures considered here, the dynamics are dominated by electronic states in the conduction band [1]. A simplified version of the theory assumes that the excitation occurs only at zone center. This reduces the problem to an n-level system (where n is approximately equal to the number of wells in the structure), which can be solved using conventional first-order perturbation theory and wave-packet methods. A more advanced version of the theory includes all of the hole states and electron states subsumed by the bandwidth of the excitation laser, as well as the perpendicular k states. In this case, a density-matrix picture must be used, which requires a solution of the time-dependent Liouville equation. Substituting the Hamiltonian into the Liouville equation leads to a modified version of the optical Bloch equations [13,15]. These equations can be solved readily, if the k states are not coupled (i.e., in the absence of Coulomb interactions). 

As the number of eigenstates available for coherent coupling increases, the dynamical behavior of the system becomes considerably more complex, and issues such as Coulomb interactions become more important. For example, over how many wells can the wave packet survive, if the holes remain locked in place If the holes become mobile, how will that affect the wave packet and, correspondingly, its controllability The contribution of excitons to the experimental signal must also be included [34], as well as the effects of the superposition of hole states created during the excitation process. These questions are currently under active investigation. 

The theory behind body-fixed representations and the associated angular momentum function expansions of the wavefunction (or wave packet) in terms of bases parameterized by the relevant constants of the motion and approximate constants of the motion is highly technical. Some pertinent results will simply be stated. The two good constants of the motion are total angular momentum, J, and parity, p = +1 or 1. An approximate constant of the motion is K, the body-fixed projection of total angular momentum on the body-fixed axis. For simplicity, we will restrict attention to the helicity-decoupled or centrifugal sudden (CS) approximation in which K can be assumed to be a constant of the motion. In terms of aU its components, and the iteration number k, the real wave packet is taken to be [21]... 

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