= BCS coherence factors
{c}
{title2=$u_k^2+v_k^2=1$}
For real gap and the particle-hole block $\left(\begin{smallmatrix}\xi_k&-\Delta_k\\-\Delta_k&-\xi_k\end{smallmatrix}\right)$, the BCS coherence factors obey $u_k^2=(1+\xi_k/E_k)/2$, $v_k^2=(1-\xi_k/E_k)/2$ and $u_kv_k=\Delta_k/(2E_k)$, 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.
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