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BCS theory (Bardeen, Cooper, Schrieffer, 1957) explains superconductivity microscopically. Phonon-mediated attraction between electrons near the Fermi surface allows the formation of Cooper pairs — bound states of electrons with opposite momentum and spin (k↑, -k↓). The BCS ground state is a coherent superposition of pair-occupied and pair-empty states, described by the wavefunction |BCS> = product_k (u_k + v_k c†_{k↑} c†_{-k↓})|0>. The energy gap Delta(T) opens at E_F, with 2Delta(0) = 3.53 k_B T_c in weak coupling. The gap suppresses low-energy scattering and is responsible for zero resistance, the Meissner effect, and the exponential specific heat at low temperatures.
BCS theory, published in 1957, is one of the great triumphs of many-body quantum mechanics. It explains superconductivity from first principles in three conceptual steps: the Cooper instability, the BCS ground state, and the energy gap.
Step 1: Cooper pairs. Leon Cooper showed in 1956 that two electrons above a filled Fermi sea, interacting via even an arbitrarily weak attractive potential (provided by phonon exchange), form a bound state. This is impossible in free space in 3D — you need a minimum coupling strength for binding. But the Fermi sea, by blocking all states below E_F, effectively confines the pair to a thin shell where the problem becomes 2D-like, and any attraction produces binding. The bound state has zero total momentum (k up, -k down) and a binding energy Delta ~ hbar omega_D exp(-1/N(0)V), which is small but nonzero for any V > 0.
Step 2: The BCS ground state. Cooper's result shows that the normal Fermi sea is unstable to pairing, but a single pair does not describe superconductivity — all electrons near E_F participate. The BCS ground state is a coherent superposition: |BCS> = product_k (u_k + v_k c^dagger_{k up} c^dagger_{-k down}) |0>, where u_k and v_k are variational parameters determined by minimizing the energy. The probability |v_k|^2 that the pair (k, -k) is occupied transitions smoothly from 1 below E_F to 0 above, with the transition width set by Delta. This is a fundamentally new state of matter: it has a definite macroscopic phase (enabling supercurrents) but indefinite particle number, embodying macroscopic quantum coherence.
Step 3: The energy gap. The BCS state has a gap Delta in the quasiparticle excitation spectrum: it costs at least 2Delta to break a Cooper pair. This gap is the reason for zero resistance — there are no low-energy excitations to scatter into. The gap equation is 1/V = integral [1/(2 sqrt(xi2 + Delta2))] tanh(sqrt(xi2 + Delta2)/(2k_BT)) dxi, which determines Delta(T) self-consistently. At T = 0, the weak-coupling result is 2Delta(0) = 3.53 k_BT_c. The gap closes continuously at T_c (second-order transition) with the mean-field behavior Delta(T) proportional to (T_c - T)1/2 near T_c.
BCS theory quantitatively predicts the critical temperature, the specific heat jump at T_c (Delta C/gamma T_c = 1.43), the coherence length xi_0 = hbar v_F / (pi Delta), the penetration depth, and the nuclear spin relaxation rate (Hebel-Slichter peak). Its success established the phonon mechanism for conventional superconductors and earned Bardeen, Cooper, and Schrieffer the 1972 Nobel Prize. The theory's limitations — it applies to weak-coupling, s-wave pairing in clean materials — are precisely what makes the study of unconventional superconductors (cuprates, iron-based, heavy-fermion) so challenging and interesting.