Wave packet phase velocity

Plane waves with phase velocity form wave packets with group velocity Vg, such that v Vg = (= 1/eoMo). The phase and group velocities are the same only... 

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. 

This expansion is exact there are no terms of higher order than quadratic. From equation (1.40) we see that the phase velocity Uph of the wave packet is given by... 

Thus, both the angular frequency u> k) and the phase velocity Uph are dependent on the choice of the zero-level of the potential energy and are therefore arbitrary neither has a physical meaning for a wave packet representing a particle. 

The velocity c, of an individual wave component is called the phase velocity. There is also another velocity. It can be argued that the major contribution to a wave packet originates from the neighbourhood ko(x,t) found by setting... 

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 mysterious phase velocity of the de Broglie wave and the group velocity of the amplitude wave, c2/ > c, refer to the, by now familiar superluminal motion in the interior of the electron. As many authors noted and Molski(1998) recently reviewed [86] an attractive mechanism for construction of dispersion-free wave packets is provided in terms of a free bradyon4 and a free tachyon that trap each other in a relativistically invariant way. It is demonstrated in particular how an electromagnetic spherical cavity may be... 

The picture of a single wave packet is in any case inappropriate for the description of a continuous source because it implies a well-defined internal phase structure. This is not provided in our experiment, and the beam can only be regarded as a statistical, and therefore incoherent, mixture of the various momenta. Nevertheless, the beam can operationally be characterized by a coherence length, which is the length that measures the fall-off of the interference visibility when the difference between two interfering paths increases. The longitudinal coherence length is determined by the Fourier transformation of the velocity distribution and it is of the order Lc A2/A A = Xv/ Av. 

The use of a wave packet instead of a plane wave provides a description of the motion of a single particle which bears a close similarity to the classical description It should be noted, however, that the width ax of a wave packet in the general case increases with time /46c,47/f which is a consequence of the dispersion of the plane waves in the pactet, having different velocities of propagation (phase velocities) Therefore, a localization of the particle is possible only for relatively short time intervals for which the condition /47/... 

Let us return to Figure 2.2. In Appendix B we expand the concepts of wave packet and group and phase velocities. 

There are two velocities associated with the moving wave packet, the phase velocity and the group velocity. The phase velocity of a wave Vp is the rate at which the phase of the wave propagates in space. This is the speed at which the phase of... 

Establishing that an electron moving through space must be represented by a wave packet also resolves the paradox that the velocity of the waves seems to be different from the velocity of the electron. The point is that the electron waves have differing phase and group velocities. 

With the advent of femtosecond lasers, it became possible to observe in real time the actual motion of nuclei and to study the elementary mechanisms pictured by Bodenstein in his description of gas-phase reactions. In all branches of femto-chemistry, this study of elementarity is basic and is due to the inherent resolution achieved in femtochemical studies. Since the velocity of atoms in reactions is 1 km/sec, with 10 fs resolution the distance scale reached is 0,1 A, the atomic scale of motion. As discussed below, this ability to create such localized, coherent wave packets with the atomic scale of distance resolution was part of the development of quantum mechanics as a theoretical construct, but was not an experimental reality until the development of the required time resolution of motion in atoms, molecules, and reactions. 

In three dimensions and for central potentials, the Schrodinger equation separates in spherical coordinates analogously to corresponding classical equations of motion. The incoming particle is described by a wavefunction with a local wavenumber ic(r). Repulsive potentials decrease the velocity of the wave packet and the resulting scattered wavefunction has fewer nodes, resulting in a negative phase shift. Conversely, attractive potentials result in an increase in velocity the wavefunction acquires more nodes and this results in a positive phase shift. The overall phase shift for scattered particles is... 

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