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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.

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  • $\begingroup$ 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. $\endgroup$
    – Triatticus
    May 23, 2013 at 8:53
  • $\begingroup$ Why is it useful to know that? @Dan: What can you use the direction of the travel of the wave for? $\endgroup$
    – TMOTTM
    May 23, 2013 at 8:59
  • $\begingroup$ Well it is useful to know in case of reflection and transmission problems when faced with differing interfaces. Knowing the incoming direction lets you decompose the wave into components and solve for the resulting transmission and reflection coefficients. $\endgroup$
    – Triatticus
    May 23, 2013 at 9:01
  • $\begingroup$ 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} $) $\endgroup$
    – Triatticus
    May 23, 2013 at 9:06
  • $\begingroup$ 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? $\endgroup$
    – TMOTTM
    May 23, 2013 at 9:10

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