How should I picture the Fourier transform of a fermionic creation (annihilation) operator acting on a site of a periodic, say one-dimensional, lattice? I mean, in a real-space picture, what are the operators $\hat{c}_{k}^{(\dagger)}$ occurring in the Fourier transform $$\hat{c}_{r}^{(\dagger)}=\frac{1}{\sqrt{N}}\sum_{k}e^{-(+)ikr}\hat{c}^{(\dagger)}_{k}$$ where the index $r\in\{1,2,..,N\}$ labels a specific lattice site on the chain, "creating" ("annihilating")?
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$\begingroup$ They create a single particule whose wavefunction is a plane wave with the wavevector $k$. $\endgroup$– ChristopheApr 28, 2020 at 11:00
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$\begingroup$ @Christophe which is then delocalized over the lattice, isn’t it? $\endgroup$– MilarepaApr 28, 2020 at 11:52
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1$\begingroup$ Yes. If the wavefunction of a single particle created by $c_r^+$ is $\psi_r(x)=\langle x|c_r^+|0\rangle$ then the wavefunction of a single particle created by $c_k^+={1\over\sqrt N}\sum_r e^{ikr}c_r^+$ is $\psi_k(x)=\langle x|c_k^+|0\rangle={1\over\sqrt N}\sum_r e^{ikr}\langle x|c_k^+|0\rangle={1\over\sqrt N}\sum_r e^{ikr}\psi_r(x)$. $\endgroup$– ChristopheApr 29, 2020 at 12:34
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In the example you posted, $c^{\dagger}_r$ means that you are creating a particle at one dimensional site $r$. Similarly, $c^{\dagger}_k$ means that you are creating a particle with a momentum $k$. If you want to visualise how Fourier transform actually work, we have to go back to Heisenberg Uncertainty Principle. By creating a particle at location $r$, you know the position of the particle with very high certainty, therefore the certainty of knowing the momentum is very low. Therefore, it is a liner combination of operator in momentum space which span from $-\infty$ to $\infty$ and vice-versa.
The Fourier Transformation of an operator or any quantity is useful when you have periodic constraint. If you have a periodic condition on real space i.e. $c^{\dagger}_{r+R}=c^{\dagger}_r$, then you can verify by taking Fourier transform both sides that $k=\frac{2m\pi}{R}$, where m is any integer. This quantizes the momentum space and we get our beautiful Energy momentum spectrum, Brillouin zone etc.
