Dispersion of a wave packet

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

In this section we investigate the change in contour of a wave packet as it propagates with time. 

The general expression for a wave packet lF(x, t) is given by equation (1.11). The weighting factor A( k) in (1.11) is the inverse Fourier transform of (x, t) and is given by (1.12). Since the function A(k) is independent of time, we may set t equal to any arbitrary value in the integral on the right-hand side of equation (1.12). If we let t equal zero in (1.12), then that equation becomes 

Since the limits of integration do not depend on the variables and k, the order of integration over these variables may be interchanged. 

Equation (1.27) relates the wave packet Tfix, t) at time t to the wave packet Tfrx, 0) at time t = 0. However, the angular frequency x (k) is dependent on k and the functional form must be known before we can evaluate the integral over k. 

The general expression for a wave packet Ifix, z) is given by equation 

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As follows from the second dispersion-theory approximation, the distance Tcompr at which the duration of a chirped laser pulse reaches its minimum is defined by the dispersion-spreading length L i p of a wave packet ... 

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/... 

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... 

An important characteristic of solitons is their non-dispersive (shape-conserving) motion. Conventional wave packets will lose their shape because the Fourier components of the packet propagate at different velocities. In a non-linear medium the velocity depends not only on the frequency of a wave but also on its amplitude. In favourable circumstances the effect of the amplitude dependence can compensate that of the frequency dependence, resulting in a stable solitary wave. A technical application of this idea is the propagation of soliton-like pulses in fibre optics, which considerably increases the bit rate in data transmission. 

The group velocity describes the velocity of electron wave packets formed from states with wave vectors grouped around a mean vector t. Formally, it is calculated from the energy dispersion e(k) as Vg = /h... 

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