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roshan2004
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How can we show that the group velocity is equal to the particle velocity?
Meir Achuz said:The derivation that the group velocity, so defined, is given by v_g=dw/dk is in many advanced textbooks.
Claude Bile said:Indeed, though the group velocity is actually a Taylor series of which dw/dk is the first term. Higher order terms govern the dispersion of the wave.
It is worth keeping in mind (perhaps not specifically for this thread, but in general) that v_g=dw/dk is a first order approximation.
Group velocity and particle velocity are terms commonly used in physics and engineering to describe the movement of waves or particles. Group velocity refers to the speed at which the overall shape or envelope of a wave or particle moves, while particle velocity refers to the speed at which individual particles or particles within a wave move.
Group velocity is equal to the particle velocity when waves or particles are traveling at a constant speed with no change in direction. In this case, the overall movement of the wave or particle and the movement of individual particles within it are equivalent.
There are several factors that can affect the relationship between group velocity and particle velocity, including the medium through which the wave or particle is traveling, the frequency of the wave, and any external forces or interactions.
Yes, group velocity can be greater than particle velocity in certain situations. For example, in a dispersive medium where the speed of a wave depends on its frequency, the group velocity can be greater than the particle velocity at certain frequencies.
The concept of group velocity and particle velocity is commonly used in fields such as optics, acoustics, and quantum mechanics to analyze and understand the behavior of waves and particles. This understanding is crucial in designing and optimizing technologies such as fiber optics, lasers, and electronic devices.