Cooper showed that two electrons above a filled Fermi sea with an arbitrarily weak attractive interaction form a bound state. Why does the Fermi sea play an essential role?
AThe Fermi sea provides a background potential that strengthens the attraction
BThe Fermi sea blocks the low-momentum states via the Pauli exclusion principle, restricting the pair to a thin shell near E_F where the density of states is high — this effective confinement to 2D in energy space allows a bound state for arbitrarily weak attraction, unlike the 3D free-space case which requires a minimum coupling strength
CThe Fermi sea contributes additional attractive interactions between the pair
DWithout the Fermi sea, the electrons would not have opposite momenta
In three-dimensional free space, a bound state requires the attractive potential to exceed a threshold. But at the Fermi surface, the Pauli exclusion principle confines the pair to a narrow energy shell of width ~ħω_D above E_F, effectively making the problem two-dimensional in energy. In 2D, any attractive potential — no matter how weak — supports a bound state. The Cooper pair binding energy is Δ ~ ħω_D exp(-1/N(0)V), which is nonzero for any V > 0 but exponentially small for weak coupling. This is why superconductivity is a weak-coupling instability of the Fermi sea.
Question 2 Multiple Choice
The BCS ground state is not a state with a definite number of particles — it is a coherent superposition of states with different numbers of Cooper pairs. Why is this number uncertainty essential?
AIt is a mathematical convenience with no physical significance
BThe number-phase uncertainty relation (ΔN Δφ ≥ 1) means that a state with a well-defined macroscopic phase (needed for coherent supercurrent and the Josephson effect) must have uncertainty in particle number. The BCS state has a definite phase and indefinite particle number — this is the essence of off-diagonal long-range order and macroscopic quantum coherence
CIt accounts for electrons entering and leaving the superconductor
DIt corrects for the fact that electrons are indistinguishable
The BCS wavefunction |BCS> = Π_k(u_k + v_k c†_{k↑}c†_{-k↓})|0> is a product of terms, each of which is a superposition of pair-present (amplitude v_k) and pair-absent (amplitude u_k). This gives a definite phase relationship between different pair-number sectors. The macroscopic phase φ of the order parameter Δ = |Δ|e^{iφ} enables phenomena like the Josephson effect and flux quantization. A state with definite particle number would have completely uncertain phase and no supercurrent.
Question 3 Short Answer
The BCS energy gap Δ(T) closes continuously at T_c and the transition is second-order. Below T_c, what physical quantity does Δ measure?
Think about your answer, then reveal below.
Model answer: The energy gap Δ is the minimum energy required to break a Cooper pair into two individual quasiparticle excitations. It costs 2Δ to create a quasiparticle pair (one electron-like excitation above E_F and one hole-like excitation below). At T = 0, the gap is maximum: 2Δ(0) = 3.53 k_BT_c in weak-coupling BCS theory. As T → T_c, thermal fluctuations break pairs faster than they can form, Δ → 0, and the system transitions continuously to the normal state. The gap's existence is directly observable in tunneling experiments (sharp onset of current at voltage eV = Δ), infrared absorption (photons must exceed 2Δ to break pairs), and the exponential low-temperature specific heat C ∝ exp(-Δ/k_BT).
The exponential specific heat is one of the clearest signatures of the gap: with no low-energy excitations available, the number of thermally excited quasiparticles drops exponentially as T → 0. This is qualitatively different from the linear-T specific heat of a normal metal.
Question 4 Short Answer
Why do Cooper pairs form with zero total momentum (k↑, -k↓) rather than with finite center-of-mass momentum?
Think about your answer, then reveal below.
Model answer: Zero total momentum maximizes the phase space for pairing. The phonon-mediated attraction operates between electrons in a thin shell of width ~ħω_D around the Fermi surface. For a pair with zero total momentum, both electrons sit on the Fermi surface, and ALL electrons in the shell can pair. If the pair had finite center-of-mass momentum Q, one electron would be at k and the other at -k+Q, and only electrons on the intersection of two displaced Fermi surface shells could pair — a much smaller phase space. Since the pairing energy depends exponentially on the available density of states, even a small reduction in phase space dramatically weakens pairing. This is why the BCS state strongly favors Q = 0 pairing.
The FFLO (Fulde-Ferrell-Larkin-Ovchinnikov) state is a rare exception where finite-Q pairing occurs under extreme conditions (large magnetic field with mismatched Fermi surfaces), but it requires very specific material properties and has been definitively observed in only a few systems.