# What does the wavevector $\textbf{k}$ mean?

In Ashcroft, Mermin Solid State Physics, Eq. 17.43 is

$$\epsilon(\textbf{k}) = \frac{\hbar^2 k^2}{2m} - e\phi(\textbf{r})$$

where $\textbf{k}$ is the wavevector and all other symbols have their usual meaning.

What does the wavevector tell me and why do we use it instead of position $\textbf{r}$?

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Well $\vec{r}$ is actually a position vector it points out a particles particular position with reference to some origin, the wave vector $\vec{k}$ actually tells us more as it is related to momentum $\vec{p}$ which gives us a sense of direction of travel of the wave.
Imagine a plane wave, $\vec{r}$ will point from the origin to a spot on the plane wave not necessarily in the direction of travel, whereas the wave vector points from the wave front outwards. –  Dan May 23 '13 at 8:53
We can also then solve various problems using this wave solution like the transfer of energy of a wave via the poynting vector which requires knowledge of the direction of the electric field waves (the magnetic field is related to the direction of the electric field by $\vec{B} = \frac{1}{c} \hat{k} \times \vec{E}$) –  Dan May 23 '13 at 9:06
I'm coming from a chemistry background, so I think in terms of discrete eigenfunctions. But I know in solids, the eigenfunctions start form continuous bands. Is it fair to say that $\textbf{k}$ is actually what points out an eigenfunction in a band? –  TMOTTM May 23 '13 at 9:10