Wave Packet Expansion: Understanding $\omega (\vec{k})$

[ognik]

The text states:
"Let us consider a wave packet whose Fourier inverse $\phi (\vec{k})$ is appreciably different from zero only in a limited range $\Delta \vec{k}$ near the mean wave vector $\hbar \vec{\bar{k}} $. In coordinate space, the wave packet $\psi(\vec{r}, t)$ must move approximately like a classical free particle with mean momentum $\hbar \vec{\bar{k}}$. To see this behavior we expand $\omega (\vec{k})$ about $\vec{\bar{k}}$:

$ \omega (\vec{k}) = \omega(\vec{\bar{k}}) + (\vec{k} - \vec{\bar{k}}) \cdot (\vec{\nabla_k} \omega)_{k= \vec{\bar{k}}} + ...$

$=\bar{\omega} + (\vec{k} - \vec{\bar{k}})\cdot \vec{\nabla_{\bar{k}}} \bar{\omega} +... $ with obvious abbreviations"

I'm afriad not much of the above is obvious to me ...please help me understand -
What is $ \omega (\vec{k})$?
What expansion are they using?
What are the abbreviations?

 

[jvicens]

Hello,

I can help clarify the concepts mentioned in the forum post for you. First, $\omega (\vec{k})$ represents the angular frequency of a wave described by the wave packet. The wave packet itself is represented by the function $\psi(\vec{r}, t)$, which depends on position $\vec{r}$ and time $t$.

The expansion being used is a Taylor series expansion, which is a mathematical tool used to approximate a function by calculating its derivatives at a specific point. In this case, the function being expanded is $\omega (\vec{k})$ and the point of expansion is $\vec{\bar{k}}$, the mean wave vector. This expansion allows us to better understand the behavior of the wave packet in coordinate space, as mentioned in the text.

The abbreviations used, such as $\bar{\omega}$ and $\vec{\nabla_{\bar{k}}} \bar{\omega}$, are simply shorthand notations for the mean angular frequency and the gradient of the mean angular frequency, respectively. The ellipses in the equations represent higher order terms in the Taylor series expansion, which have been omitted for brevity.

I hope this helps clarify the concepts for you. Let me know if you have any further questions.