Wave Packet

a propagating wave field that occupies a finite region of space at any given moment. Wave packets may occur with waves of any nature (sound, electromagnetic, and so on). Such a wave“surge” in a localized region of space may be resolved into the sum of monochromatic waves whose frequencies lie within definite limits. However, the term“wave packet” is generally used in connection with quantum mechanics.

In quantum mechanics, a plane, monochromatic de Broglie wave—that is, a wave with definite values of frequency and wavelength and occupying the entire space—corresponds to each state of a particle with certain values of momentum and energy. The coordinates of a particle having precisely defined momentum are completely indeterminate—the particle may be found with equal probability at any point of the space, since this probability is proportionate to the square of the amplitude of the de Broglie wave. This corresponds with the uncertainty principle, which states that the more definite is the particle’s momentum, the less definite is its coordinate. On the other hand, if the particle is localized in any limited region of space, its momentum no longer has a precisely defined magnitude—there is a certain spread in its possible values. The state of such a particle is represented by the sum (more accurately, by the integral, because the momentum of a free particle varies continually) of monochromatic waves with frequencies corresponding to the spread of possible values of momentum. The superposition of a group of such waves that have almost the same direction of propagation but differ slightly in frequency is the wave packet. This means that the resultant wave will be different from zero only in a certain limited region of space; in quantum mechanics it cor-responds to the fact that the probability of finding the particle in the region occupied by the wave packet is large, whereas outside this region it is practically zero.

The velocity of the wave packet (more accurately, of its center) is found to coincide with the mechanical velocity of the particle. From this it can be deduced that the wave packet describes a freely moving particle whose possible location at any given time is limited to a certain small region of coordinates (that is, the wave packet becomes the wave function of such a particle).

With the passage of time, the wave packet widens and becomes diffuse (see Figure 1). This results from the fact that the monochromatic waves forming the packet and having different frequencies propagate with different velocities even in a vacuum: some waves move faster, others more slowly, and the wave packet is deformed. This diffusion of the wave packet corresponds to the fact that the region of possible localization of the particle increases.

Figure 1. Diffusion of a wave packet with the passage of time t. At the initial moment the particle is described by wave packet Ψ₀ at time t, by wave packet Ψ_t;ǀΨ₀ǀ² and ǀΨ_t² define the probabilities of finding the particle at a certain point x; v is the velocity of the center of the packet, coinciding with the particle’s mechanical velocity. The areas encompassed by the curves and the x-axis are equal and give the total probability of finding the particle in the space at a given time.

If the particle is not free but is located near some attracting center—for example, an electron in the Coulomb field of the proton in a hydrogen atom—such a bound particle will be associated with standing waves, which retain their stability. In this case the shape of the wave packet remains invariable, which corresponds to the stationary state of the system. In a case when the system jumps into a new state owing to external influences (for example, when a particle strikes an atom), the wave packet instantly restructures itself in conformity with transition; this is called a reduction of the wave packet. Such a reduction would lead to contradictions wifli the requirements of the theory of relativity if de Broglie’s waves were ordinary material waves, such as those of the type of electromagnetic waves. Actually, in such a case the reduction of the wave packet would signify the existence of super-light (instant) signals. The probability interpretation of de Broglie waves eliminates this difficulty.