Wave packet properties

Since the wave packet property is crucial in maintaining a high level of accuracy, during a simulation of a molecular encounter all measures should be employed to preserve this property. A counterexample should stress the point. Coulombic wave functions are not wave packets because of the singularity of the potential. For this reason pseudospectral methods lose much of their advantage. This accounts for the popularity of the semilocal FD and finite element grid methods for simulating encounters under the influence the Coulomb potential. For other applications the accuracy of the FD methods is not sufficient. 

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... 

This uncertainty relation is also a property of Fourier transforms and is valid for all wave packets. 

Again, this relation arises from the representation of a particle by a wave packet and is a property of Fourier transforms. 

By way of contrast, recall that in treating the free particle as a wave packet in Chapter 1, we required that the weighting factor A(p) be independent of time and we needed to specify a functional form for A(p) in order to study some of the properties of the wave packet. 

In this section we state the postulates of quantum mechanics in terms of the properties of linear operators. By way of an introduction to quantum theory, the basic principles have already been presented in Chapters 1 and 2. The purpose of that introduction is to provide a rationale for the quantum concepts by showing how the particle-wave duality leads to the postulate of a wave function based on the properties of a wave packet. Although this approach, based in part on historical development, helps to explain why certain quantum concepts were proposed, the basic principles of quantum mechanics cannot be obtained by any process of deduction. They must be stated as postulates to be accepted because the conclusions drawn from them agree with experiment without exception. 

In the recent work reviewed in this chapter, we have shown the creation and properties of molecular superrotors, illustrated by application of the optical centrifuge to simple diatomic molecules. The technique is certainly not limited to diatomics, and to date we have created superrotor wave packets in a range of molecules. All that is required is an anisotropic polarizability such that the molecule can be... 

If we wish to follow the time-dependent, dynamical properties of the wave packet, we must solve the TDSE with the ground state fo(x) of the unperturbed system as an initial condition ... 

We turn now to study the properties of the metastable state in more detail. We, therefore, concentrate on the long-time behavior, i.e., t > f0, and defer the discussion of the short-time dynamics to a later section. Figure 1.3 shows snapshots of the probability density of the evolving wave packet at different... 

Let us recap the properties of a wave packet populating a resonant state we observed in Section 2 ... 

The first volume contained nine state-of-the-art chapters on fundamental aspects, on formalism, and on a variety of applications. The various discussions employ both stationary and time-dependent frameworks, with Hermitian and non-Hermitian Hamiltonian constructions. A variety of formal and computational results address themes from quantum and statistical mechanics to the detailed analysis of time evolution of material or photon wave packets, from the difficult problem of combining advanced many-electron methods with properties of field-free and field-induced resonances to the dynamics of molecular processes and coherence effects in strong electromagnetic fields and strong laser pulses, from portrayals of novel phase space approaches of quantum reactive scattering to aspects of recent developments related to quantum information processing. 

Coherent control Control of the motion of a microscopic object by using the coherent properties of an electromagnetic held. Coherent phase control uses a pair of lasers with long pulse durations and a well-defined relative phase to excite the target by two independent paths. Wave packet control uses tailored ultrashort pulses to prepare a wave packet at a desired position at a given time. 

The phase and amplitude spectrum of the laser pulse are tailored to create a wave packet with selected properties. The various eigenstates that comprise the wave packet are populated by different frequency components of the laser pulse, each with its specified amplitude and phase. For example, rovibrational wave packets of Li2 in the El E+ state were created, consisting of vibrational levels v = 12-16 and rotational levels J = 11, 19. The phases and amplitudes of the pump pulse shown in Fig. 20 were generated with a 128-pixel liquid crystal SLM. The pulse was tailored to optimize the ionization signal at a delay time of 7 ps. The phases used to maximize or minimize the ionization signal are shown by solid and dashed lines, respectively, and the intensities at the eigenfrequencies of the wave packet are indicated by circles. 

Another popular approach to wave-particle duality, which originated with Schrodinger, was to view the quantum particle as a wave structure or wave packet. This model goes a long way towards the rationalization of particle-like and wave-like properties in a single construct. However, the simplified textbook discussion, which is unsuitable for the definition of quantum wave packets, relies on the superposition of many waves with a continuous spread of wavelengths, defines a dispersive wave packet, and therefore fails in modelling an electron as a stable particle. 

The group velocity of de Broglie matter waves are seen to be identical with particle velocity. In this instance it is the wave model that seems not to need the particle concept. However, this result has been considered of academic interest only because of the dispersion of wave packets. Still, it cannot be accidental that wave packets have so many properties in common with quantum-mechanical particles and maybe the concept was abandoned prematurely. What it lacks is a mechanism to account for the appearance of mass, charge and spin, but this may not be an insurmountable problem. It is tempting to associate the rapidly oscillating component with the Compton wavelength and relativistic motion within the electronic wave packet. 

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