Superconductivity was discovered in 1911, but its mechanism remained mysterious for decades. In 1957, Bardeen, Cooper and Schrieffer used quantum mechanics to show that electrons can feel a weak attraction through phonons—vibrations of the crystal lattice—and, at low temperatures, form “Cooper pairs.” A macroscopic number of these pairs condense into the same quantum state, creating an energy gap that prevents scattering and therefore eliminates electrical resistance. The BCS theory allows physicists to calculate critical temperatures and magnetic-field limits, providing design rules for superconducting magnets and cables. Its social impact ranges from reducing power losses to enabling powerful medical devices such as MRI.

The BCS theory introduces an effective electron–phonon interaction into the Hamiltonian and solves it via a variational approach combined with the Bogoliubov transformation. The resulting BCS wave function is a coherent superposition of momentum-opposite electron pairs, leading to a symmetric energy dispersion with a gap Δ around the Fermi level. The gap shrinks with temperature T and vanishes at the critical temperature Tc, where superconductivity disappears. The theory explains the Meissner effect, critical field Hc, heat-capacity anomalies and many other experimental facts, and has been extended to superfluid 3He and color superconductivity in QCD. Although modern high-Tc cuprates may not be phonon-mediated, the BCS framework of pair formation and condensation remains the theoretical cornerstone.

The BCS framework generalizes the Cooper problem to a many-body context with the effective low-energy Hamiltonian H = Σ_k ε_k c†_k c_k − V Σ_{k,k'} c†_{k↑} c†_{−k↓} c_{−k'↓} c_{k'↑}. Diagonalization via a Bogoliubov transformation produces the excitation spectrum E_k = √((ε_k − μ)^2 + Δ^2) and a self-consistent gap equation 1 = V Σ_k (1/2E_k) tanh(E_k/2k_B T). Gauge invariance is preserved through the Ward identity, recovering the London equations. The seminal Phys. Rev. 108, 1175 (1957) showed quantitative Tc agreement for Al, Sn, etc., paving the way to Eliashberg’s strong-coupling formalism. Modern directions include coupling to Nambu–Goldstone modes, unconventional pairing symmetries, and Bogoliubov–de Gennes analyses of topological superconductors.