Waves, dispersion, group speed

[quasar987]

Where did you pull that dispersion relation from?

And what about the other dispersion relations? For exemple, how was it established that the dispersion relation for waves in deep water is what it is? was it experiemntally?

 

[Galileo]

That plane wave will only be a solution to the Schrodinger equation if that relation between w and k hold. So that's where it comes from. Just plug $\Psi(\vec r)=A\exp\left(\vec k \cdot \vec r - \omega t\right)$ into the Schrodinger equation and see for yourself.

I honestly don't know how the dispersion relation is derived for waves in water. I`m sure sSomeone else may be able to help you there.



[tex]

 

[Hans de Vries]

The group velocity is very important in QM (in my opinion at least) since the group velocity of a wave packet allows us to recover the classical expression of a free particle's kinetic energy.

The relation between the group and phase speed of the de Broglie wave is
very different from that of classical waves, see my write-up here:

Relativistic kinematics of the wave packet:
http://www.chip-architect.com/physics/deBroglie.pdf

Section 1: The the broglie-wave is purely the result of the non-simultaneity of SR
Section 2: The wave packet at rest
Section 3: The moving wave packet
Section 4: The >c phase speed
Section 5: The group speed (recovering the correct time-dilation)
Section 6: The relativistic rotated wave front.

The last part shows that a significant part of both Special Relativity
and Quantum mechanics can be derived from the rule that the wave front
is always at 90 degrees angles with the (physical) group speed, for light
waves as matter waves.

Now illustrated with simulation images.



Regards, Hans