= Solution
In <BCS theory>, pairing mixes an <electron> state with a hole in its opposite-momentum partner. For a uniform spin-singlet gap, the uniform <Bogoliubov--de Gennes Hamiltonian> gives the momentum-pair eigenproblem
$$
\begin{pmatrix}\xi_{\mathbf k}&\Delta\\\Delta^*&-\xi_{\mathbf k}\end{pmatrix}
\binom{u_{\mathbf k}}{v_{\mathbf k}}=E_{\mathbf k}\binom{u_{\mathbf k}}{v_{\mathbf k}},\qquad
\boxed{E_{\mathbf k}=\sqrt{\xi_{\mathbf k}^2+|\Delta|^2}.}
$$
Here $\xi_{\mathbf k}=\varepsilon_{\mathbf k}-\mu$. A <fermionic Bogoliubov quasiparticle> is a coherent particle–hole combination, with
$$
|u_{\mathbf k}|^2=\frac12\left(1+\frac{\xi_{\mathbf k}}{E_{\mathbf k}}\right),\qquad
|v_{\mathbf k}|^2=\frac12\left(1-\frac{\xi_{\mathbf k}}{E_{\mathbf k}}\right).
$$
Far above the <Fermi surface> the excitation is mostly particle-like; far below it is mostly hole-like; at $\xi_{\mathbf k}=0$ the two weights agree. The transformation preserves the fermionic <canonical anticommutation relations>. Its mean-field particle-number mixing does not mean that the full physical theory violates charge conservation.
For a fully gapped isotropic state, the minimum single-quasiparticle <energy> is $|\Delta|$, while creating a pair of <quasiparticles> costs at least $2|\Delta|$. This produces suppressed low-temperature thermal populations, an activated electronic <heat capacity> and a gap feature in superconducting tunneling. Gap nodes in an unconventional state instead allow low-energy excitations, changing the <temperature> dependence. These fermionic excitations are distinct from collective phase and amplitude oscillations of the paired condensate.
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