(9637) Fermi-sea energy including Coulomb interaction

(1) Write the Hamiltonian \(H_0 \) of free electron in 3D box of volume \( L^3 \) using field operators \( a_{k,s} \) and \( a_{k,s}^{\dagger} \) that destroy and create electrons in momentum orbitals.

(2) Assuming two body interaction \( u(r) \), write the interaction term \(H_{int} \) using the same field operators. In particular specify the result for Coulumb interaction \( u(r) = e^2/r \).

Calculate the energy \( E = \langle (H_0+H_{int}) \rangle \) of an unperturbed Fermi sea, given the Fermi momentum \( k_F \).

Based on [C08.1]