# Derivation of the BCS Hamiltonian

$$\hat{H}_{\mathrm{BCS}}=\sum_{k \sigma} \varepsilon_{k} c_{k \sigma}^{\dagger} c_{k \sigma}-\sum_{k k^{\prime}} G_{k k^{\prime}} c_{k \uparrow}^{\dagger} c_{-k \downarrow}^{\dagger} c_{-k^{\prime} \downarrow} c_{k^{\prime} \uparrow}$$

Any books on how to derive it?

I get that the first part is the Hamiltonian of an electron and the second describes the creation of Cooper pairs, but the text I'm reading simply gives it and I was wondering how was it derived, I can't seem to find a satisfactory online text.

• Try searching for Fröhlich Hamiltonian. The interaction term is an effective potential induced by lattice phonons. – eranreches Jun 14 at 9:12