Phase and Group Velocity

Introduction

Phase and group velocity are two important and related concepts in wave mechanics. They arise in quantum mechanics in the time development of the state function for the continuous case, i.e. wave packets.

Discussion

Harmonic Waves and Phase Velocity

A one-dimensional harmonic wave (Figure 1) is described by the equation,

where A₀ is the wave amplitude, w is the circular frequency; k is the wave number; and j is an initial, constant phase. The argument for the sine function, q (x, t) = wt - kx + j is called the phase. Sometimes the wave number is referred to as the spatial frequency or propagation constant.

Figure 1: Harmonic Wave

This is a monochromatic wave (one frequency). There are no strictly monochromatic waves in nature. For example, the generating source of the wave may move slightly, introducing spurious frequencies.

In general, these waves propagate without warping. That is, the phase q (x, t) is a constant:

v_phase is the phase velocity for a wave.

From the point of view of sending information, these waves are not useful. They are the same throughout time and space. Something must therefore be modulated, such as frequency or amplitude, in order to convey information. The resulting wave may be a perturbation that acts over a short distance, i.e. a wave packet. This wave packet can be considered to be a superposition of a number of harmonic waves, in other words a Fourier series or integral.

Group Velocity

In order to convey information, something more than a simple harmonic wave is needed. However, the superposition of many such waves of varying frequencies can result in an "envelope" wave and a carrier wave within the envelope. The envelope can transmit data. A simple example is the superposition of two harmonic waves with frequencies that are very close (w₁ ~ w₂) and of the same amplitude. The equations for the motion are,

The plot of such a wave is shown in Figure 2.

Figure 2: Group Velocity

The envelope (the green line) is given by u₁ and travels at the group velocity. The carrier wave (the blue line) travels at the phase velocity and is given by u₂. The wave packet moves at the group velocity. It is the envelope which carries information. Group velocity and phase velocity are not necessarily the same. Group velocity is given by,

Phase and group velocity are related through Rayleigh's formula,

If the derivative term is zero, group velocity equals phase velocity. In this case, there is no dispersion. Dispersion is when the distinct phase velocities of the components of the envelope cause the wave packet to "spread out" over time. The components of the wave packet (or envelope) move apart to the degree where they no longer combine to complete the envelope.

Links

Wave Packet Explorer Add waves to get a wave packet.

Demonstration of Group Velocity Applet and description of the difference between group and phase velocity.

Group Velocity and Phase Velocity Another demonstration of phase and group velocity.

Superposition Principle of Waves Applet demonstrating the addition of two waves.

References

Liboff, R. L. Introductory Quantum Mechanics, Fourth Edition. San Francisco, CA: Addison Wesley, pp. 156-157 2003.

Ostrovsky, L.; Potapov, A. Modulated Waves: Theory and Applications. Baltimore and London: The Johns Hopkins University Press, pp. 1-9, 1999.

Menzel, D.; Mathematical Physics. New York, NY: Dover Publications Inc,pp.349-351, 1961.