- #36
gabbagabbahey
Homework Helper
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LocationX said:I think we are okay...
[tex] \vec{\nabla} \left( e^{-i \vec{k} \cdot \vec{r}} \right) = \frac{d}{dr_x} \left( e^{-i k_x r_x} e^{-i k_y r_y} e^{-i k_z r_z} \right) +\frac{d}{dr_y} \left( e^{-i k_x r_x} e^{-i k_y r_y} e^{-i k_z r_z} \right) +\frac{d}{dr_z} \left( e^{-i k_x r_x} e^{-i k_y r_y} e^{-i k_z r_z} \right) [/tex]
[tex]= -ik_x e^{-i \vec{k} \cdot \vec{r}} - i k_y e^{-i \vec{k} \cdot \vec{r}} - i k_z e^{-i \vec{k} \cdot \vec{r}} [/tex]
[tex] = -i \vec{k} e^{-i \vec{k} \cdot \vec{r}} [/tex]
Yes, your right...phew!...Unfortunately there is still a problem in your [tex]\vec{f_{\perp}}(\vec{k})[/tex] somewhere and I haven't been able to pinpoint it yet. Your essentially getting
[tex]\hat{k} \times \hat{k} \times \vec{f}(\vec{k})[/tex]
which would be zero. I think the answer is supposed to be :
[tex]\hat{k} \times \vec{f}(\vec{k})[/tex]
so that [tex]\vec{f_{\perp}}(\vec{k})[/tex] pulls out the component of [tex]\vec{f}(\vec{k})[/tex] perpenciular to [tex]\vec{k}[/tex].