Calculating Group Velocity of Schrodinger Waves: Dispersion Relation

[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 Schrödinger equation, we find that plane waves solve the equation provided that
$$\omega = k^2 \hbar/2m$$
Does this mean that the group velocity of "Schrödinger 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?

 

[Gokul43201]

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 Schrödinger equation, we find that plane waves solve the equation provided that
$$\omega = k^2 \hbar/2m$$
Does this mean that the group velocity of "Schrödinger waves" is $$k\hbar/m$$?

Yes.

Won't this in general depend on the amplitude of the frequency components of a given wave?

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.

Given a specific solution to the wave equation how does one answer the question, what is the group velocity of this wave?

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

 

[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 $$v_g = k\hbar/m$$?

 

[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?

 

[Gokul43201]

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).

 

[Euclid]

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).

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:
https://www.physicsforums.com/showthread.php?t=173138
Where does the Fourier transform come into this?

 

[Claude Bile]

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 $$v_g = k\hbar/m$$?

Typically the centre value or the mean value for k is used, keeping in mind that the expression for the group velocity is only valid where the spread of values for k is small compared to the central/mean value of k.

Regarding the Fourier variables, I'm not exactly sure what Meir Achuz was alluding to but I suspect that it might be linked to the fact that w and t are Fourier conjugate variables, as are k and x. The Fourier relationship between these variables is the key between obtaining an expression for velocity (i.e. x/t) in terms of w and k.

Claude.