The BCS Ground State

We assume a generalized BCS Hamiltonian which contains the kinetic energies of “electrons” and “holes”, and the pairing Hamiltonian arising from the phonon-exchange attraction and the Coulomb repulsion. We follow the original BCS theory to construct a many-pairon ground state and find a ground state energy: $W=(1/2)N_{0}(w_{1}+w_{2})$, where $N_{0}$ is the total number of the pairons, and $w_{1}$ and $w_{2}$ are respectively the ground state energies of “electron” ($j=1$) and “hole” ($j=2$) pairons. Energy gaps $\Delta_{j}$ are found in the quasi-electron excitation spectra: $E_{k}^{(j)}\equiv (\epsilon_{k}^{2}+\Delta_{j}^{2})^{1/2}$.