Understanding Group Velocity for Wave Packets in a 1-D Particle in Box

[hokhani]

I have difficulty understanding the exact concept of group velocity. Consider a wave packet as a linear combination of a number of eigenstates of a 1-D particle in box. The dispersion curve(\omegaversus k) is composed of discrete points located on a parabola. Well, for each point one can calculate the phase velocity but how to calculate the group velocity? Firstly the points are discrete and it is not possible to differentiate and secondly the group velocity changes from a point to another so that we can not specify a special group velocity for that wave pocket!

 

[azntoon]

The group velocity is the velocity at which the pulse shape of the wave packet moves along the medium. It is related to the phase velocity, which is the speed at which the peaks of the wave packet move. The group velocity can be determined by taking the derivative of the dispersion curve, which is the relationship between frequency and wave number. This will give you the rate of change of the frequency with respect to the wave number. This rate of change is the group velocity. In a 1-D particle in a box, the group velocity can be calculated from the dispersion curve by finding the slope of the curve at each discrete point.