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[tex] f(\vec r) = \sum_{\vec k} c_\vec{k} e^{i \vec k \cdot \vec r}[/tex]

but as the solution to the equation

[tex] (\frac{\partial ^2}{\partial t^2} - \nabla^2 ) \vec A(\vec r,t) = 0[/tex]

one usually writes

[tex] \vec A(\vec r,t) = \frac{1}{\sqrt{V}}\sum_{\vec k} \vec A_{0 \vec{k}}(t) e^{i \vec k \cdot \vec r}.[/tex]

What is the reason for this? Doesn't this also screw up the dimension of the euation when [itex]\vec A_{0 \vec k}[/itex] already has the same dimension as the vector potential since the plane wave solutions are

[tex] \vec A_{0 \vec k}(t) e^{i \vec k \cdot \vec r}?[/tex]