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For real gap and the particle-hole block $\left(\begin{array}{cc}{\xi }_k& -{\Delta }_k\\ -{\Delta }_k& -{\xi }_k\end{array}\right)$, the BCS coherence factors obey $u_k^2=(1+{\xi }_k/E_k)/2$, $v_k^2=(1-{\xi }_k/E_k)/2$ and $u_k{v}_k={\Delta }_k/(2E_k)$, where $E_k=\sqrt{{\xi }_k^2+{\Delta }_k^2}$. They diagonalize the Bogoliubov--de Gennes Hamiltonian and give the empty/pair amplitudes of the BCS ground state. Their relative sign depends on the pairing convention.

For an original check, write $x={\xi }_k/\Delta$ with constant real $\Delta >0$. At $x=0$, the BCS coherence factors give $u_k^2=v_k^2=1/2$. At $x=1$, they give $(u_k^2,v_k^2)=(0.853553,0.146447)$; at $x=-1$ these values exchange places. Thus the zero-temperature occupation $v_k^2$ rounds the normal Fermi sea's step over an energy interval of order $\Delta$. This mixing is strongest at the Fermi surface, where the BCS quasiparticle spectrum has its minimum.

For a particle-hole-symmetric normal density of states $g_F$ and isotropic superconducting energy gap $\Delta >0$, the electron spectral density is zero for $|E|<\Delta$ and has the displayed form outside the gap. The BCS coherence factors weight the two roots $\xi =\pm \sqrt{E^2-{\Delta }^2}$, where $\xi$ is normal-state energy relative to the Fermi energy. At positive $E$, their electron weights $(1+\xi /E)/2$ sum to one. Multiplying their sum by the Jacobian $|d\xi /dE|=E/\sqrt{E^2-{\Delta }^2}$ proves the formula. This electron spectral density differs from the unweighted count of all positive-energy quasiparticles, which receives both roots. Its gap-edge divergence is integrable and approaches the normal electronic density at large $|E|$.

The BCS quasiparticle spectrum is the positive excitation energy obtained by diagonalizing the paired particle-hole block. Here the weak coupling shell model has $\Delta \ll {\Omega }_D$: the minimum lies inside the pairing shell, and states outside it cost more. To derive it for a uniform isotropic real $\Delta >0$, measure the normal-state energy from the chemical potential, ${\xi }_k={ϵ}_k-\mu$, and use the same sign convention as the BCS coherence factors:Its eigenvalues are ${\epsilon }_{\pm }=\pm E_k$, where $E_k=\sqrt{{\xi }_k^2+{\Delta }^2}$. The positive $E_k$ is the cost of creating a quasiparticle above the BCS ground state. The lower branch $-E_k$ is its particle-hole partner in this doubled matrix description, not an excitation that lowers the ground state energy without bound.

The location and value of the minimum follow directly:The unique minimum as a function of $\xi$ is $E(0)=\Delta$, at the normal-state Fermi surface. On an isotropic three-dimensional Fermi surface it occurs for every direction of $k$, rather than at a unique point of momentum space. Near that minimum,Farther away, $E(\xi )\simeq |\xi |$ when $\Delta \ll |\xi |\ll {\Omega }_D$. For the illustrative $\lambda =0.25$ example, a state at $\xi =\Delta$ costs $\sqrt{2}\Delta \simeq 1.03644\mathrm{meV}$ rather than the minimum $0.732871\mathrm{meV}$.

For a normalized spinless paired ground state $u|0⟩+iv{c}_p^{†}{c}_{-p}^{†}|0⟩$, its mean fermion number is $2v^2$. The canonical Bogoliubov quasiparticle creation operator produces $c_p^{†}|0⟩$, containing exactly one fermion. With BCS coherence factors $v^2=(1-ϵ/E)/2$, the change in mean fermion number is $1-2v^2=ϵ/E$. This need not be an integer because the pairing Hamiltonian preserves fermion parity while failing to conserve the original fermion number operator. Every measurement of that number operator still has integer outcomes.

In a conventional isotropic spin-singlet superconductor, the nuclear spin-lattice relaxation rate can rise just below the transition before falling at lower temperature. The BCS coherence factors and enhanced near-gap density of states produce the coherence enhancement. Nuclear relaxation and ultrasonic attenuation have different coherence factors, so their contrasting temperature dependences test pairing structure rather than only the existence of a gap.^[ref]

Experimental tests of BCS theory compare several properties of the same superconducting state. In the conventional weak-coupling setting, phonon-mediated attraction pairs opposite momenta and spins, and the BCS quasiparticle spectrum is $E_k=\sqrt{{\xi }_k^2+{\Delta }^2(T)}$. Agreement with a single energy scale and its temperature dependence is more discriminating than observing zero resistance alone.

The superconducting isotope effect tests the lattice's role. Replacing one isotope by another changes ionic masses while leaving the chemical valence nearly unchanged. Phonon frequencies scale approximately as $M^{-1/2}$, so a weak-coupling phonon mechanism predicts $T_c\propto M^{-\alpha }$ with $\alpha \simeq 1/2$ when other parameters stay fixed. The shift was measured in mercury before the microscopic pairing theory was developed.^[ref] Deviations from $1/2$ can result from Coulomb, anharmonic and strong-coupling effects; the isotope observation supports lattice-mediated pairing without uniquely proving every assumption of the simplest model.

Thermodynamic measurements test the gapped excitations and condensation energy. An isotropic fully gapped superconductor has exponentially small low-temperature electronic heat capacity, rather than the normal linear law. At the transition the weak-coupling BCS thermodynamic ratios areHere $C_n$ means the electronic contribution after subtracting the lattice background. The gap closes at $T_c$, and the same framework predicts the temperature dependence of penetration depth and thermodynamic critical field. These quantities were quantitative comparisons in the original theory.^[ref]

Electron tunnelling measures the excitation spectrum more directly. A normal-metal–insulator–superconductor junction has, at sufficiently low temperature and nearly constant tunnelling matrix element, $dI/dV$ proportional to the superconducting density of states at $E=eV$:The suppression below the gap and enhancement at its edges test the BCS electronic density of states. Giaever's thin oxide-barrier junction measurements established such energy-gap structure.^[ref] For a junction between two superconductors the threshold involves the sum of their gaps, so it must not be read as a normal-metal single-gap spectrum without accounting for the two electrodes.

Nuclear spin-lattice relaxation supplies a phase-coherence-sensitive test. In the conventional s-wave case, the Hebel-Slichter coherence peak just below $T_c$ is followed by a falling rate at lower temperature. Density of states alone is not enough: the BCS coherence factors enter the transition matrix elements. The contrast between nuclear relaxation and ultrasonic attenuation was explained using the paired state, unlike a simple two-fluid population model.^[ref]

Finally, magnetic flux quantization in units $h/(2e)$ and Josephson effects support a coherent charge-$2e$ condensate. Those observations establish pairing and phase coherence more broadly; their existence alone does not specify a phonon mechanism or an isotropic weak-coupling gap. The converging isotope, spectroscopic, thermodynamic and coherence-factor tests support conventional BCS superconductivity, with explicit limits to its simplest universal ratios.

Group the nonzero modes into independent pairs $p>0$. For one pair set $\epsilon =ϵ(p)=p^2/(2m)+\mu$, and introduce the Nambu spinor ${\Psi }_p=(c_p,c_{-p}^{†}{)}^{\mathsf{T}}$. Normal ordering using $c_{-p}{c}_{-p}^{†}=1-c_{-p}^{†}{c}_{-p}$ givesThis exactly contains both kinetic terms and the pairing terms once; there is no additional factor of one half because only $p>0$ is summed.

Insert ${\Psi }_p=U_p{\mathcal{A}}_p$, where ${\mathcal{A}}_p=({\alpha }_p,{\alpha }_{-p}^{†}{)}^{\mathsf{T}}$ and $U_p$ is the Bogoliubov transformation matrix. Direct matrix multiplication yieldsChoose the diagonalizing branchThen the off-diagonal terms vanish and the upper diagonal entry is $E_p>0$. Extend the angle to negative momenta as in part b; the sine relation just written applies to the positive representative of each pair. The BCS coherence factors can be chosen asHere $u_p$ denotes $\cos {\theta }_p$, and $v_p$ denotes $\sin {\theta }_p$; these expressions are nonsingular because $\mu >0$ and $m>0$. The minus sign of $v_p$ for positive $\Delta$ is fixed by the pairing phase in this Hamiltonian.

A second use of the canonical anticommutation relations givesThe zero-momentum term was never paired and is simply $\mu {\alpha }_0^{†}{\alpha }_0$. HencewithThe constant is the ground-state energy. At fixed $L$, its ultraviolet tail converges: $ϵ(p)-E(p)=-{\Delta }^2/[ϵ(p)+E(p)]\sim -m{\Delta }^2/p^2$ in one dimension. Replacing $E(0)$ by $\sqrt{{\mu }^2+{\Delta }^2}$ would incorrectly pair a spinless mode with itself.

For real ${\Delta }_k$ and ${\xi }_{-k}={\xi }_k$, introduce the particle-hole column ${\Psi }_k=(c_{k\uparrow },c_{-k\downarrow }^{†}{)}^T$. Its quadratic block is the Bogoliubov--de Gennes HamiltonianThe BCS coherence factors can be chosen to obeyWith $u_k\ge 0$, choose the sign of $v_k$ to match ${\Delta }_k$. The rotation diagonalizes the matrix to $\mathrm{diag}(E_k,-E_k)$. Reordering ${\gamma }_{-k\downarrow }{\gamma }_{-k\downarrow }^{†}=1-{\gamma }_{-k\downarrow }^{†}{\gamma }_{-k\downarrow }$ and restoring the mean-field constant givesThe inverse transformation is ${\gamma }_{k\uparrow }=u_k{c}_{k\uparrow }-v_k{c}_{-k\downarrow }^{†}$ and ${\gamma }_{-k\downarrow }=v_k{c}_{k\uparrow }^{†}+u_k{c}_{-k\downarrow }$. Both annihilate $(u_k+v_k{b}_k^{†})|0⟩$. Since every positive-energy quasiparticle mode is empty, the normalized BCS ground state isEach momentum label refers to the pair $(k\uparrow ,-k\downarrow )$ once; distinct labels use disjoint single-particle spin states.

Use the reduced attractive reduced BCS pairing Hamiltonian with volume V, ${\xi }_k={\epsilon }_k-\mu$ and attraction $\lambda >0$:A specified pairing shell or an ultraviolet renormalization is part of the model. The trace is grand canonical. Insert fermionic coherent states into its time-sliced trace. The resulting Grassmann variables are antiperiodic, and with inverse-energy imaginary time $0\le \tau \le \beta$ the dimensionless action isThis boundary condition is essential for Fermi-Dirac statistics; replacing it by a bosonic periodic trace changes the answer. A complex Hubbard–Stratonovich transformation uses the local identitywith time-slice factors understood. The pairing field renders the fermion action quadratic. For a uniform static field the independent Nambu spinor block $({\psi }_{k\uparrow },{\bar{\psi }}_{-k\downarrow }{)}^T$ hasIts determinant is $-({\omega }_n^2+E_k^2)$. The phase-independent determinant sign cancels under consistent reference normalization. Integrating the fermions gives the BCS pairing-field effective action, ${\mathcal{S}}_{\mathrm{eff}}={\int }_0^{\beta }V|\Delta {|}^2/\lambda d\tau -\mathrm{Tr}\ln {\mathcal{G}}^{-1}$, with the normal-ordering/reference term retained. The fermionic pairing-block determinant normalization uses one independent pairing block for every k; a fully doubled spin-particle-hole trace instead needs one half. The subtracted grand potential of the reduced BCS model is particularly unambiguous:Performing the Matsubara frequency sum yieldsup to a common constant. At zero gap this reduces exactly to the two-spin free-gas result. The Matsubara derivation of the BCS saddle follows by differentiating with respect to ${\Delta }^{\ast }$ and using ${\beta }^{-1}{\sum }_n({\omega }_n^2+E^2{)}^{-1}=\tanh (\beta E/2)/(2E)$ givesThe second equation fixes the chemical potential at specified number. This stationary action is BCS mean-field theory, not an exact integration of pairing-field fluctuations. With constant single-spin density of states ${\nu }_F$ per volume and a symmetric energy cutoff $\Lambda$, weak coupling gives $\Delta (0)\simeq 2\Lambda e^{-1/(\lambda {\nu }_F)}$ and $k_B{T}_c\simeq (2e^{{\gamma }_E}/\pi )\Lambda e^{-1/(\lambda {\nu }_F)}$. The zero-temperature condensation grand-potential density is $-{\nu }_F{\Delta }^2/2$ to leading weak-coupling order; using a spin-summed density would change these displayed factors. The BCS coherence factors are $u_k^2=(1+{\xi }_k/E_k)/2$ and $v_k^2=(1-{\xi }_k/E_k)/2$, and the associated BCS ground state is the normalized product of empty and occupied pair states. ^[ref]

Allowing a slowly varying complex pairing field generates phase stiffness and, near the transition, the Ginzburg-Landau theory of superconductivity. A neutral paired system has a gapless phase sound mode; in a charged bulk superconductor electromagnetic coupling produces the Anderson-Higgs mechanism in a superconductor and the Meissner effect. These collective fluctuations are distinct from the gapped fermionic excitations. A microscopic dynamical calculation is needed to fix collective-mode dispersion and damping; the static saddle alone does not determine them. ^[ref]

Condensate depletion means particles outside a macroscopically occupied bosonic single-particle mode. Interactions can deplete that mode even at zero temperature: the Bogoliubov transformation mixes creation and annihilation operators, so its quasiparticle vacuum contains particles at nonzero momentum. A weak homogeneous three-dimensional gas with repulsive scattering length $a_s$ has leading quantum depletion of a Bose condensateThermal quasiparticles add another contribution. Depletion and normal-fluid density are distinct: a Galilean-invariant fluid can have quantum depletion while retaining its full superfluid mass response at zero temperature.

For the fermionic pairing problem, mean-field decoupling of the reduced BCS pairing Hamiltonian introduces the anomalous amplitude $\Delta$. Each paired Nambu block has diagonal entries ${\xi }_{\mathbf{k}},-{\xi }_{\mathbf{k}}$ and off-diagonal pairing entries of magnitude $\Delta$. A phase convention with negative off-diagonal entries gives eigenvalues $\pm E_{\mathbf{k}}$, where $E_{\mathbf{k}}=\sqrt{{\xi }_{\mathbf{k}}^2+{\Delta }^2}$. Its canonical Bogoliubov rotation yields the BCS coherence factorsThe partner transformation has the complementary sign required by fermionic anticommutation. At zero temperature all positive-energy BCS quasiparticle modes are empty. Expanding the supplied electron operator therefore givesSince $E_{\mathbf{k}}^2-{\xi }_{\mathbf{k}}^2={\Delta }^2$,The electron occupation is close to one far below the normal-state Fermi surface, equals one half at $\xi =0$, and decays toward zero above it. Pairing smooths the Fermi step over an energy range of order $\Delta$. The positive-energy quasiparticle occupation is instead zero everywhere at this temperature; the electron sea and the quasiparticle vacuum are different descriptions of the same state. At finite temperature, $n_{\mathbf{k}\sigma }=[1-(\xi /E)\tanh (E/(2k_{B}T))]/2$ in the same mean-field model.

For a three-dimensional parabolic normal band, the total density of states including both spins is $D(\mu )=3N/(2\mu )$. In the weak-gap limit, replace it by this constant across the narrow pairing region and use $\mu \simeq E_F$. The number of electrons above the normal-state Fermi surface isThe integral is $[\xi -\sqrt{{\xi }^2+{\Delta }^2}{]}_0^{\infty }/2=\Delta /2$. Therefore the BCS population above the normal-state Fermi surface isThe constant-density-of-states approximation and a small gap relative to the chemical potential are essential. This is the question's explicitly defined fermionic population measure; it is not the fraction of electrons outside a single bosonic condensate orbital.

For an order-of-magnitude mercury estimate, use $T_c\simeq 4.2\mathrm{K}$ and the weak-coupling scale ${\Delta }_0\simeq 1.76k_B{T}_c\simeq 0.64\mathrm{meV}$, with a representative metallic $\mu \simeq 7\mathrm{eV}$. These input approximations give $\Delta N/N\sim 7\times 10^{-5}$, of order $10^{-4}$. The use of a weak-coupling gap is an estimate, not an assertion of mercury's exact measured gap.

For a copper oxide, an illustrative gap $20\mathrm{meV}$ and effective chemical-potential scale $0.5\mathrm{eV}$ give about three per cent in this formula. Choosing $0.2$–$1\mathrm{eV}$ instead gives roughly $7.5$–$1.5$ per cent. These are explicit example inputs: the paper specifies no compound, doping or band parameters from which a unique number follows. A real cuprate's anisotropic gap, quasi-two-dimensional band and strong correlations require a band-resolved occupation integral, so the reliable contrast is the much larger gap-to-electronic-energy scale, rather than a universal cuprate percentage.

Group the nonzero modes into independent pairs $p>0$. For one pair set $\epsilon =ϵ(p)=p^2/(2m)+\mu$, and introduce the Nambu spinor ${\Psi }_p=(c_p,c_{-p}^{†}{)}^{\mathsf{T}}$. Normal ordering using $c_{-p}{c}_{-p}^{†}=1-c_{-p}^{†}{c}_{-p}$ givesThis exactly contains both kinetic terms and the pairing terms once; there is no additional factor of one half because only $p>0$ is summed.

Insert ${\Psi }_p=U_p{\mathcal{A}}_p$, where ${\mathcal{A}}_p=({\alpha }_p,{\alpha }_{-p}^{†}{)}^{\mathsf{T}}$ and $U_p$ is the Bogoliubov transformation matrix. Direct matrix multiplication yieldsChoose the diagonalizing branchThen the off-diagonal terms vanish and the upper diagonal entry is $E_p>0$. Extend the angle to negative momenta as in part b; the sine relation just written applies to the positive representative of each pair. The BCS coherence factors can be chosen asHere $u_p$ denotes $\cos {\theta }_p$, and $v_p$ denotes $\sin {\theta }_p$; these expressions are nonsingular because $\mu >0$ and $m>0$. The minus sign of $v_p$ for positive $\Delta$ is fixed by the pairing phase in this Hamiltonian.

A second use of the canonical anticommutation relations givesThe zero-momentum term was never paired and is simply $\mu {\alpha }_0^{†}{\alpha }_0$. HencewithThe constant is the ground-state energy. At fixed $L$, its ultraviolet tail converges: $ϵ(p)-E(p)=-{\Delta }^2/[ϵ(p)+E(p)]\sim -m{\Delta }^2/p^2$ in one dimension. Replacing $E(0)$ by $\sqrt{{\mu }^2+{\Delta }^2}$ would incorrectly pair a spinless mode with itself.

Put $\xi =ϵ-{ϵ}_F$ and $\Delta =|\Delta |>0$. The BCS quasiparticle spectrum has $E=\sqrt{{\xi }^2+{\Delta }^2}$, so on either inverse branchConservation of states on one branch gives $D_{\mathrm{branch}}(E)dE=g({ϵ}_F)|d\xi |$. Consequently the single-branch expression requested in the source isThere are no such states for $0<E<\Delta$ in this ideal fully gapped model. Near the edge the density of states diverges as $(E-\Delta {)}^{-1/2}$, an integrable singularity; at large $E/\Delta$ it approaches its normal-state value.

Counting qualification. The energy map in the source is two-to-one. If $g(ϵ)$ literally counts all normal-state levels both above and below ${ϵ}_F$, the unweighted positive-energy quasiparticle density from a folded spectrum must sum both preimages, and is insteadAlready at $\Delta =0$, folding $\xi$ into $|\xi |$ doubles the positive-axis density. Thus the printed formula is not the unweighted count of all states under the literal mapping without an additional normalization convention.

There is also a standard physical quantity with exactly the printed normalization: the BCS electronic density of states, measured for example by single-electron superconducting tunneling. At positive $E$, the electron spectral weights of the two roots are the BCS coherence factors $u_{\pm }^2=(1+{\xi }_{\pm }/E)/2$. They sum to one, so the weighted sum is $g({ϵ}_F)E/\sqrt{E^2-{\Delta }^2}$. This supplies a valid electronic-DOS interpretation, while keeping it distinct from counting every positive-energy excitation. Spin degeneracy must be included consistently in $g$ and the transformed density; it does not remove the branch-counting issue.