Wave function: Phase Velocity and Group Velocity

What is wave function? What is the Phase Velocity and Group Velocity? Derive relation between them.


wave function

Consider a wave of moving particle whose angular frequency w = 2piv, A is amplitude, wave number or propagation constant is k = 2pi/ lemda; then the displacement of wave in x-direction is (wave function)

Y = A Sin (wt-kx)

wave packet

So the wave function in quantum mechanics describes the quantum state of an isolated system of one or more particles. The wave function containing all the information’s about the entire system like position, frequency, amplitude, wavelength, direction etc. Quantities associated with measurements, such as the average momentum of a particle, can be derived from the wave function also.

Phase Velocity and Group Velocity

Phase velocity 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.

So the wave packet is a combination of n number of waves with slightly different velocity and wavelengths. The amplitude and phase of the component waves superimposed and make constructive interfere as well as destructive interference. The Figure exhibits the superposition of two waves of slightly different frequency and formation of wave packet. The velocity of a wave can be defined in many different ways, partly because there many different kinds of waves, and partly because we can focus on different aspects or components of any given wave. So, the velocity of component waves of a wave packet is called the phase velocity and the velocity of the wave packet is called the group velocity.

Phase velocity

The velocity of component waves of a wave packet is known as the phase velocity of those waves.

If a wave traveling in x direction represented by

Y = A Sin (wt-kx)                                        (1)

Phase velocity

This is the expression for the phase velocity.

Group velocity

The velocity of wave packet or the wave packet obtained due to superposition of waves traveling in a group is known as the group velocity.

If we consider two waves of same amplitude, but of slightly different

Group velocity

Group velocity

Group velocity

Relation between Phase Velocity and Group Velocity

For Dispersive Medium  :

relation phase velocity and Group velocity

Relation between Phase Velocity and Group Velocity: For non-dispersive medium

If the wave velocity depends only on the physical properties of the medium then the wave velocity is constant, independent of frequency; such medium is known as non-dispersive medium.

relation phase velocity and Group velocity non despersive

So the group velocity is equal to the phase velocity when medium is non-dispersive.

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