Calc group velocity and wave velocity for a wave

[leonne]

Homework Statement

a wave function $$\Psi$$=Ae^i(px-Et)/h where p is momentum and E=p²/2m calc group and wave velocity

Homework Equations

group vel=dw/dk
wave v=w/k

The Attempt at a Solution

I just have no idea what they did here to solve it.
i(px-Et)/h=i(kx-wt) than somehow got k=p/h than w=E/h than just used these to solve it but i don't get how they got it. Are these formulas? didn't seem them in the book. I kind of see how they got it. Did they just make it so the right side matches the left? Funny, as soon as i finish typing a problem here, i seem to figure it out.

 

[ideasrule]

No, the formula wasn't just made up. e^(ipx/h) is the general equation for the momentum eigenstates, and since p can be any real number, it's also the general equation for a plane wave. According to Schrödinger's time-independent equation, the eigenstates' time evolution should be e^(-iEt/h), so the time-dependent wavefunction is e^(ipx/h)*e^(-iEt/h).

 

[leonne]

o ok thanks for the help