where H' is the perturbation Hamiltonian representing the collision and E and E′ are respectively the initial and final energies of the system constituted of both the carrier and the electron and phonon gas. The Dirac δ δ -function stands for the conservation of energy. In addition, the term ⟨ ⟨ k | H ′ | k ′ ⟩ ⟩ , generally referred to as the matrix element, mathematically represents an inner product of the initial and final wave functions of the carrier: