Deriving the 2D Coulomb potential from the Born approximation in quantum electrodynamics involves considering the scattering of a charged particle by a stationary charged target in two dimensions. Here's a brief outline of the derivation:
Start with the scattering of a charged particle (e.g., an electron) by a stationary charged target (e.g., a nucleus) in two dimensions. The interaction between the charged particles can be described by the Coulomb potential.
Apply the Born approximation, which assumes that the scattering process can be treated as a single scattering event, neglecting multiple scattering and higher-order effects.
The Born approximation allows us to express the scattering amplitude as a sum over all possible intermediate states. In this case, the intermediate states correspond to the different positions of the target particle.
Use the Coulomb potential to calculate the scattering amplitude for each intermediate state. The Coulomb potential in two dimensions is given by V(r) = q1*q2 / (4πε|r|), where q1 and q2 are the charges of the scattering particles, ε is the electric constant, and r is the distance between them.
Integrate the scattering amplitude over all possible intermediate states, taking into account the appropriate normalization factors.
Summing up the contributions from all intermediate states, we obtain the total scattering amplitude.
Finally, square the scattering amplitude to obtain the differential cross-section, which represents the probability of scattering into a specific angle.
It's important to note that the specifics of the derivation depend on the particular setup and assumptions made, and the mathematical details may involve integration, solving differential equations, and applying appropriate boundary conditions.
This outline provides a general overview of the process, but the actual derivation can be quite involved and requires a solid understanding of quantum electrodynamics, scattering theory, and the mathematics involved.