Understanding Wave Packets: Exploring the Effects of Dispersion and Uncertainty

[Amith2006]

The idea of a wave packet is confusing a bit. First let me tell you what I have understood about it. A wave packet is obtained by the superposition of harmonic waves of nearby frequencies which represents a matter wave. It is well known fact that the width of a moving Gaussian wave packet in free space is time dependent due to uncertainty in momentum which leads to uncertainty in velocity. If the wave packet represented a microscopic particle its width will increase rapidly due Heisenberg’s uncertainty principle. If the same wave packet is moving in a dispersive medium, the spreading is further enhanced due to wavelength dependence of phase velocity. Is it right? Assume that time tends to infinity.

 

[Bob_for_short]

In the classical mechanics any uncertainty in the initial velocity v₀ leads to a spreading uncertainty in the final position too: ∆x(t) = ∆v₀*t = ∆p*t/m.

 

[hamster143]

It is well known fact that the width of a moving Gaussian wave packet in free space is time dependent due to uncertainty in momentum which leads to uncertainty in velocity.

Yes.

If the wave packet represented a microscopic particle its width will increase rapidly due Heisenberg’s uncertainty principle.

Not necessarily. Heisenberg's uncertainty principle relates the spatial spread of the packet with the range of frequencies in it. Unlike a point particle, a properly constructed wave packet may be quite stable.

If the same wave packet is moving in a dispersive medium, the spreading is further enhanced due to wavelength dependence of phase velocity. Is it right? Assume that time tends to infinity.

Again, not necessarily - dispersion can act to inhibit rather than enhance the spread.

 

[Amith2006]

Thanx guys for sharing your knowledge with me.