Dispersion Relation of a De Broglie Wave

[Silly Sausage]

Homework Statement

Use the dispersion relation to find the group velocity v_group and phase velocity v_phase.

Homework Equations

omega(k) = [(hbar)k^2]/2m

The Attempt at a Solution

v_group = domega(k)/dk = [hbar]k/m = h/m lambda = p/m = v

This isn't right.

v_phase = omega / k = 2pi f/(2pi/lambda) = f lamda = v

again... is this right?, doesn't look it but as far as I can see there is no mistake in the calc as I have doing it several times already, am I goign wrong in my physics?

 

[wdednam]

Hi there,

Your first answer is definitely right.

But then, why did you substitute omega = 2pi f and k = 2pi/lambda in the expression for the phase velocity? When I substitute the expression you were given for omega and divide it by k I get v_phase = 1/2 v.

I don't think f lambda equals v here, because if omega = (hbar*k^2)/2m I get f = h/(2m*lambda^2) and if you multiply that by lambda you'll get h/(2m*lambda) = p/2m which again gives you 1/2 v.

Relations like p = h/lambda, omega = 2pi f and k = 2pi/lambda are always true, but in a dispersive medium f lambda = v = v_group is not always true, only when omega = v * k, then v = v_group = v_phase.

Hope that helps.

Wynand.