What is a wave function?
What is the Phase Velocity and Group Velocity? Derive the relation between them.
Answer:
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 the x-direction is (wave function)
Y = A Sin (wt-kx)
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, we can derive from the wave function as well.
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 velocities and wavelengths. The amplitude and phase of the component waves superimposed and make constructive interfere and destructive interference. The Figure exhibits the superposition of two waves of slightly different frequencies and the formation of the wave packets. The velocity of a wave can be defined partly because their many kinds of waves, and partly because we can focus on different aspects or components of any 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)
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
Relation between Phase Velocity and Group Velocity
For Dispersive Medium :
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.
So the group velocity is equal to the phase velocity when medium is non-dispersive.
Leave a Reply