Gap Δ: 0.000 meV
Coherence length ξ₀: 0 nm
DOS peak: 0.00
T/Tc: 0.00
BCS theory introduces the Bogoliubov-Valatin transformation that diagonalizes the pairing Hamiltonian. Quasiparticle operators γ_k = u_k c_k↑ − v_k c†_{-k↓} with |u_k|² + |v_k|² = 1 and v_k²/u_k² = (E_k − ξ_k)/Δ, where ξ_k = ε_k − μ and E_k = √(ξ_k² + Δ²) is the quasiparticle energy.
The gap equation: 1/V = Σ_k 1/(2E_k) tanh(E_k/2T) determines Δ self-consistently. The gap vanishes at T_c = (2γ/π)ωD·exp(−1/VN₀) ≈ 1.13 ωD exp(−1/VN₀) (BCS formula, γ = Euler-Mascheroni constant).
The density of states N(E)/N₀ = |E|/√(E²−Δ²) diverges at E = ±Δ (coherence peaks) — directly observed in tunneling spectroscopy. The coherence factors u_k, v_k show the mixing of electron and hole character that enables Cooper pairing.