## Real space and reciprocal space Similarly to time/frequency domain conversion, real/reciprocal space conversion is also widely applied when dealing with electrodynamics. Such conversion is even more common, compared with in time, in scattering related fields, which is typical in material science. Since it is a special form of Fourier transform, same rules are applied. $\Phi(\mathbf{k}) = \int_{-\infty}^{\infty} \phi(\mathbf{r}) e^{-i\mathbf{k} \cdot \mathbf{r}} \, \mathrm{d}^3r$ $\phi(\mathbf{r}) = \frac{1}{(2\pi)^3} \int_{-\infty}^{\infty} \Phi(\mathbf{k}) e^{i\mathbf{k} \cdot \mathbf{r}} \, \mathrm{d}^3k$ Also the general rule operations $a \phi_{1}(\mathbf{r}) + b \phi_{2}(\mathbf{r}) \longleftrightarrow a \Phi_{1}(\mathbf{k}) + b \Phi_{2}(\mathbf{k})$ $(\phi_{1} * \phi_{2})(\mathbf{r}) \longleftrightarrow \Phi_{1}(\mathbf{k}) \cdot \Phi_{2}(\mathbf{k})$ The convolution is defined as $(\phi_{1} * \phi_{2})(\mathbf{r}) = \int_{\mathbb{R}^n} \phi_{1}(\mathbf{r'}) \phi_{2}(\mathbf{r} - \mathbf{r'}) \, \mathrm{d}\mathbf{r'}$. The displacement would be $\phi(\mathbf{r} - \mathbf{r}_0) \longleftrightarrow e^{-i\mathbf{k}\cdot\mathbf{r}_0}\Phi(\mathbf{k})$ and $\phi(\mathbf{r}) \cdot e^{i\mathbf{k}_0\cdot\mathbf{r}} \longleftrightarrow \Phi(\mathbf{k} - \mathbf{k}_0)$ Under Fourier transform, $\nabla$ would become $i \mathbf{k}$ For derivative, we have $\nabla^n \phi(\mathbf{r}) \longleftrightarrow (i\mathbf{k})^n \Phi(\mathbf{k}) $ The most commonly seen first and second order derivatives are $\nabla \phi(\mathbf{r}) \longleftrightarrow i\mathbf{k} \Phi(\mathbf{k}) $ $\nabla^2 \phi(\mathbf{r}) \longleftrightarrow -|\mathbf{k}|^2 \Phi(\mathbf{k})$ >[!Note] >It is worth noting that, if one component, like $z$, is a constant and does not goes into the transformation. Then its differential operations would be kept. For divergence and curl, same rule apply $\nabla \cdot \mathbf{F}(\mathbf{r}) \longleftrightarrow i\mathbf{k} \cdot \mathbf{F}(\mathbf{k})$ $\nabla \times \mathbf{F}(\mathbf{r}) \longleftrightarrow i\mathbf{k} \times \mathbf{F}(\mathbf{k})$ People may use $\mathbf{F}$ as real space expression and $\tilde{\mathbf{F}}$ as reciprocal space expression, but this is not always the case. >[!Info] >For Fourier transform in more detail, see contents in [[Index (Fourier Analysis)]]. >For real/reciprocal space conversion, see [[Frequency domain and time domain]].