Phase velocity and Group velocity | Quantum mechanics

Phase velocity and Group velocity

Phase velocity (Wave velocity)

• If a single monochromatic wave (wave of single frequency or wavelength) travels through a medium, then the velocity through which it propagate is wave velocity.
• Equation of plane progressive wave propagating in x-direction

            y = a sin (ωt - kx)

• Here a = amplitude, ω = angular frequency, and k = 2π / λ = wave vector or propagation constant
• Since in wave motion, the particles of the medium vibrates about their mean position continuously, the velocity acquired by particle during this motion is particle velocity.
• During this motion, the disturbance (crest or trough) also produced and they also move in the direction in which the wave propagate and also with the same velocity.
• The velocity of the disturbance through which it advances through the medium is phase velocity (wave velocity).

• Above equation gives the wave velocity. 
• Thus the ratio of angular frequency to the propagation constant (wave vector) is known as wave velocity.
• In equation  y = a sin (ωt - kx), (ωt - kx) is phase of wave motion
• For the planes of constant phase or wavefronts

            (ωt - kx) = constant

• Above expression represents the wave velocity.
• Since the wave velocity is the velocity with which planes of constant phase advance through the medium, and above expression is obtained for such type of wave, so it is also known as phase velocity.

                

Group velocity 

• In practical life we come across pulse or group of wave of slightly different frequencies. This group of waves are wave packets.
• A wave packet refers to the case where two or more waves exist simultaneously.
• The group velocity of a wave packet is the velocity with which the maximum amplitudes of the group advances in the medium.
• This is the velocity with which the energy in the group is transmitted.
• The group velocity is defined as the rate of change of angular velocity with respect to wave vector.

            

• Let a wave packet is made up of two harmonic waves

• Let on superposition, these waves form a wave packet, then resultant displacement of wave packet

                y = y₁ + y₂

• The amplitude of this equation is modulated by

• Here figure (a) and (b) shows the two waves of slightly different frequencies, and figure (c) shows the resultant wave of these two waves.

• If △ω and △k are very small. 

• To know more about phase velocity and group velocity please click on the link https://youtu.be/as-a0sUw1RQ  or for bilingual (Hindi/English) https://youtu.be/sLhr0cJWAew