Group velocity

Japanese: 群速度 - ぐんそくど

The speed at which a wave packet travels through a dispersive medium. A wave packet here refers to a continuous wave packet that spreads over a finite length from front to back, and can be considered as a superposition of sine waves over a finite frequency range. A dispersive medium is one in which the speed at which a sine wave front travels (phase velocity) varies with frequency, and therefore the refractive index varies with frequency. The speed at which a sine wave front travels (phase velocity) is v = ν λ = ν/ k . Here, ν is the frequency, λ is the wavelength, and k is the wave number, or the reciprocal of the wavelength. _The group velocity vg is vg = dν ( k )/ dk . In other words, it is equal to the differential coefficient obtained by differentiating the frequency ν (ν = ν( k )) as a function of wave number k with respect to k . If the phase velocity v is written as _v = v ( k ) as a function of wave number k , then the group velocity vg _is
vg = v + k dv ( k ) _/ dk
In a dispersive medium, _the group velocity vg has a different value than the phase velocity v .

In the case of matter waves such as electrons, the wave packets of matter waves correspond to particles. The Einstein-de Broglie relationship E = h ν, p = hk exists between the energy E and momentum p of a particle and the frequency ν and wave number k of a matter wave.
Here, h is the Planck constant. Using the relationship between E and p of a particle, E = p ² /(2 m ), where m is the mass of the particle, we get
v _g = d ν/ dk = dE / dp = p / m
This result shows that the momentum p of a particle divided by its mass, i.e., the velocity of the particle, is equal to _the group velocity vg of the matter waves.

[Yoshiro Kainuma]

Group velocity diagram
The wavefront of a sine wave travels one wavelength during one period. As shown in Figure (1), in a dispersive medium, the distance traveled by a wave packet during one period is not one wavelength. As shown in Figure (2), if there is no wave dispersion, the wave packet would also travel one wavelength during one period .

Group velocity diagram

Source: Shogakukan Encyclopedia Nipponica About Encyclopedia Nipponica Information | Legend

Japanese:

波束が分散性媒質内を伝わる速度。ここに波束というのは、先端から後尾まで有限の長さに広がる一つながりの波のかたまりであり、これは有限の振動数領域にわたる正弦波の重ね合わせとみなすことができる。分散性媒質というのは、その媒質内では正弦波の波面の進行速度（位相速度）が振動数によって異なり、したがって屈折率が振動数によって異なるような媒質である。正弦波の波面が進行する速度（位相速度）vは、v＝ν・λ＝ν/kである。ここに、νは振動数、λは波長、kは波数すなわち波長の逆数である。群速度v_gはv_g＝dν(k)/dkである。すなわち、波数kの関数としての振動数ν(ν＝ν(k))をkで微分した微係数に等しい。位相速度vを波数kの関数としてv＝v(k)と書くと、群速度v_gは、
　v_g＝v+k・dv(k)/dk
とも書ける。分散性媒質では、群速度v_gは、位相速度vと異なる値をもつ。

電子のような物質波においては、物質波の波束が粒子に対応する。粒子のエネルギーE、運動量pと、物質波の振動数ν、波数kの間には、アインシュタイン‐ド・ブローイの関係
　E＝hν, p＝hk
が成立する。ここに、hはプランク定数である。粒子のEとpとの関係E＝p²/(2m)(ここにmは粒子の質量)を用いると、
　v_g＝dν/dk＝dE/dp＝p/m
となる。この結果は粒子の運動量pをその質量で割った商、すなわち粒子の速度が物質波の群速度v_gに等しいことを示している。