- #1

confused_man

- 16

- 1

\begin{align*}

v_p &= \frac{\omega}{k} \\

v_g &= \frac{d\omega}{dk} .

\end{align*}

Suppose that I'm looking at a free quantum mechanical particle, the dispersion relation is

\begin{align*}

E &= \hbar\omega =\frac{\hbar^2 k^2}{2m}\\

\omega &= \frac{\hbar k^2}{2m}

\end{align*}

Since ##\omega## depends the wavenumber in a non-linear way, I understand that a wavepacket made of a combination of these plane waves will experience dispersion and will spread out.

If I'm talking about a pure plane wave with frequency ##k##, then I wouldn't use the group velocity to describe how fast it's moving, I'd just use the phase velocity.

My question is if I'm looking at a wave packet built up from a number of different plane waves and I want to calculate its group velocity, what value of ##k## should I use? The group velocity equation only has a single ##k## in it.