Group velocity

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The group velocity of a wave is the velocity with which the variations in the shape of the wave's amplitude (known as the modulation or envelope of the wave) propagate through space. The group velocity is defined by the equation:


v_g \ \equiv\ \frac{\partial \omega}{\partial k}

where:

vg is the group velocity
ω is the wave's angular frequency
k is the wave number

The group velocity is often thought of as the velocity at which energy or information is conveyed along a wave. In most cases this is accurate, and the group velocity can be thought of as the signal velocity of the waveform. However, if the wave is travelling through an absorptive medium, this does not always hold. For example, it is possible to design experiments where the group velocity of laser light pulses sent through specially prepared materials significantly exceeds the speed of light in vacuum. However, superluminal communication is not possible, since the signal velocity remains less than the speed of light. It is also possible to reduce the group velocity to zero, stopping the pulse, or have negative group velocity, making the pulse appear to propagate backwards.

The function ω(k), which gives ω as a function of k, is known as the dispersion relation. If ω is directly proportional to k, then the group velocity is exactly equal to the phase velocity. Otherwise, the envelope of the wave will become distorted as it propagates. This "group velocity dispersion" is an important effect in the propagation of signals through optical fibers and in the design of short pulse lasers.

The idea of a group velocity distinct from a wave's phase velocity was first proposed by W.R. Hamilton in 1839, and the first full treatment was by Rayleigh in his "Theory of Sound" in 1877.

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[edit] Matter wave group

Albert Einstein first explained the wave-particle duality of light in 1905. Louis de Broglie hypothesized that any particle should also exhibit such a duality. The velocity of a particle, he concluded, should always equal the group velocity of the corresponding wave. De Broglie deduced that if the duality equations already known for light were the same for any particle, then his hypothesis would hold. This means that

v_g = \frac{\partial \omega}{\partial k} = \frac{\partial (E/\hbar)}{\partial (p/\hbar)} = \frac{\partial E}{\partial p}

where

E is the total energy of the particle,
p is its momentum,
\hbar is Planck's constant.

Using special relativity, we find that

\begin{align} v_g &= \frac{\partial E}{\partial p} = \frac{\partial}{\partial p} \left( \sqrt{p^2c^2+m_0^2c^4} \right)\\ &= \frac{pc^2}{\sqrt{p^2c^2+m_0^2 c^4}}\\ &= \frac{\gamma m_0vc^2}{\sqrt{{\gamma}^2m_0^2v^2c^2+m_0^2c^4}} = \frac{\gamma vc}{\sqrt{{\gamma}^2v^2+c^2}}\\ &= \frac{vc}{\sqrt{v^2+ \left( c / \gamma \right)^2}} = \frac{vc}{\sqrt{v^2 + \left( \sqrt{c^2 - v^2} \right)^2}}\\ &= v. \end{align}

where

m0 is the rest mass,
c is the speed of light in a vacuum,
γ is the Lorentz factor.
and v is the velocity of the particle regardless of wave behavior.

Quantum mechanics has very accurately demonstrated this hypothesis, and the relation has been shown explicitly for particles as large as molecules.

[edit] See also

[edit] References

  • Brillouin, Léon. Wave Propagation and Group Velocity. Academic Press Inc., New York (1960).
  • Tipler, Paul A. and Ralph A. Llewellyn (2003). Modern Physics. 4th ed. New York; W. H. Freeman and Company. ISBN 0-7167-4345-0. 223 p.

[edit] External links

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