Wave packet group velocity

The exponential in the right member of Equation 1.11 is a complex function of magnitude 1. The cosine factor forms the beat as a contour for the function y, defining what we call a wave packet. The velocity of the wave packet is given by = 8a)/8k = dra/dk. For a free electron, E = hv = ft(o = p /2m = (fik)V2m, thus 0) = ft k /2m. We obtain dra/dk = fik/m = p/m = v. The velocity of the beat (= the group velocity) for a matter wave is thus equal to the velocity of the particle. One may show that a wave packet also moves with the same acceleration as the particle. 

Thus, the wave packet P(jc, 0 represents a plane wave of wave number ko and angular frequency mo with its amplitude modulated by the factor B(x, i). This modulating function B x, i) depends on x and t through the relationship [x — (dm/dA )o/]. This situation is analogous to the case of two plane waves as expressed in equations (1.7) and (1.8). The modulating function B(x, t) moves in the positive x-direction with group velocity given by... 

In contrast to the group velocity for the two-wave case, as expressed in equation (1.9), the group velocity in (1.16) for the wave packet is not uniquely defined. The point ko is chosen arbitrarily and, therefore, the value at ko of the derivative dco/dk varies according to that choice. However, the range of k is... 

As before, the wave packet is a plane wave of wave number ko and angular frequency mo with its amplitude modulated by a factor that moves in the positive x-direction with group velocity given by equation (1.16). Following... 

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. 

Thus, the velocity v of the particle is associated with the group velocity Vg of the wave packet... 

When (mn) > 1, expansion (7.18) contains a large number of waves with different m. Interference of many such waves creates a wave packet, whose group velocity is proportional to (mn). Rotation through the angle 2tt/3 then can be related to the hopping time... 

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. 

We now contrast the behavior of the wave packet in equation (1.31) with that of the wave packet in (1.20). At any time t, the maximum amplitudes of both occur at x = vgt and travel in the positive x-direction with the same group velocity vg. However, at that time t, the value of fiTdx, /) is 1 /e of its maximum value when the exponent in equation (1.31) is unity, so that the half width or uncertainty Ax for MT,(x, /) is given by... 

The time At for a wave packet to pass a given point equals the uncertainty in its position x divided by the group velocity vg... 

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... 

In this equation, h is Planck s constant divided by 2tt, V is the crystal volume, T is temperature, fej, is Boltzmann s constant, phonon frequency, is the wave packet, or phonon group velocity, t is the effective relaxation time, n is the Bose-Einstein distribution function, and q and s are the phonon wave vector and polarization index, respectively. 

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