Wave packet dispersion

Virtually every textbook states that (all) wave packets disperse and therefore cannot serve as a serious particle model. 

Figure 23 shows an evolving wave packet on the coordinate-time grid (q, t). Due to the slight unharmonicity of the potential used, the initially compact wave packet dispersed, which means that for long enough time the wave functions fills uniformly all the available coordinate space. Two methods of propagation are compared. In the left panels... 

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)... 

Thus, the wave packet (x, i) has the same value at point x and time t that it had at point x — ct at time t = 0. The wave packet has traveled with velocity c without a change in its contour, i.e., it has traveled without dispersion. Since the phase velocity is given by mo/ko = c and the group velocity Vg is given by (djco/dk)o = c, the two velocities are the same for an undispersed wave packet. 

Since the parameter y is non-vanishing, the wave packet will disperse with time as indicated by equation (1.28). For a gaussian profile, the absolute value of the wave packet is given by equation (1.31) with y given by (1.43). We note that y is proportional to m, so that as m becomes larger, y becomes smaller. Thus, for heavy particles the wave packet spreads slowly with time. 

The matter wave function is formed as a coherent superposition of states or a state ensemble, a wave packet. As the phase relationships change the wave packet moves, and spreads, not necessarily in only one direction the localized launch configuration disperses or propagates with the wave packet. The initially localized wave packets evolve like single-molecule trajectories. 

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. 

Such a function will have a large pulse near t = to and it disperses with time. In the two-component system the pair of dispersive waves have different velocities gJiki cu2fc2) and the profile of the wave packet moves with a velocity uj — cu2)/(k — fc2), which is different from the phase velocity (oq + cu2)/(fci + fe2) of the rapidly oscillating part. Velocity of the wave packet is known as the group velocity. If the components are not too different = lo/k and vg = (cox — cu2)/(fci — fc2) = dto/dk. In terms of wavelength... 

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. 

In three dimensions a spatial wave group moves around an harmonic ellipsoid and remains compact, in contrast to the dispersive wave packets of classical optics. The distinction is ascribed to the fact that the quantum wave packet is built up from discrete harmonic components, rather than a continuum of waves. The wave mechanics of a hydrogen electron is conjectured to produce wave packets of the same kind. At small quantum numbers the wave spreads around the nucleus and becomes more particle-like, at high quantum numbers, as it approaches the ionization limit where the electron is ejected from the atom. 

Tannor and Rice s original idea, discussed above, had considerable appeal. However, it soon became apparent that for most realistic surfaces, intuiti ideas do not necessarily tell us what pulses do the best job. For example, the packet, once created, does not remain localized, but quickly disperses. An exa of this can be seen in Figure 4,2 where the wave packet in the excited state is se spread, resulting in less than ideal channel A/chaimel B branching ratios, consequence, intuition (which was based on simulations [91, 92, 100]) was doned in favor of a more systematic pulse optimization scheme. 

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