# Dispersion relation

1. Oct 24, 2007

### Euclid

The group velocity of traveling wave is defined as $$v_g =\partial \omega/\partial k$$. I am confused about how to actually calculate this. For instance, in the Schrodinger equation, we find that plane waves solve the equation provided that
$$\omega = k^2 \hbar/2m$$
Does this mean that the group velocity of "Schrodinger waves" is $$k\hbar/m$$? Won't this in general depend on the amplitude of the frequency components of a given wave?
Given a specific solution to the wave equation how does one answer the question, what is the group velocity of this wave?

Edit: related question....
In elementary texts, it is shown how the superposition of two sine waves of equal amplitude and phase but slightly different frequency and speed gives rise to a "traveling envelope", the speed of which we associate with the group velocity. How do we know in general that that superposition of waves gives rise to a well defined envelope?

Last edited: Oct 24, 2007
2. Oct 24, 2007

### Gokul43201

Staff Emeritus
Yes.

Not sure what you are asking, but in general, the group velocity of a wave traveling through a dispersive medium is a function of frequency.

Just as above, take the dispersion relation in its standard form, and find its derivative.

Last edited: Oct 24, 2007
3. Oct 24, 2007

### Euclid

Ah now I've confused myself. I guess I mean to say that a given wave may be composed of many wavelength components. So for what k do I evaluate the group velocity equation $$v_g = k\hbar/m$$?

4. Oct 24, 2007

### Euclid

Ok here's an example to illustrate my confusion. Take a plane wave:
$$\Psi(x,t) = A e^{i(kx-\omega t)}$$
It's phase velocity is $$\omega/k=\hbar k/2m$$. But its "group" velocity should be the same thing, no?

5. Oct 24, 2007

### Gokul43201

Staff Emeritus
No, it's group velocity will be twice that number (and equal to the classical speed of the "free particle" described by the plane wave).

6. Oct 24, 2007

### Euclid

But that doesn't make any sense to me. What envelope is involved here?

In particular, I am really interested in finding out the details behind what's hinted at here: