= Solution
The quantum <free electron model> treats conduction <Electrons> as independent particles moving in an approximately constant potential, with parabolic <dispersion relation> $E=\hbar^2k^2/(2m)$. <Fermi-Dirac statistics> successfully explains important features such as a <Fermi energy> and a small <electronic heat capacity>, but a constant potential gives no <band gaps> and cannot by itself explain why different band fillings produce metals, <semiconductors> and band insulators. It also misses lattice-dependent <effective masses>, detailed directional responses and the topology of real <Fermi surfaces>. Scattering and finite <electrical resistivity> require additional physics.
The <nearly-free electron model> introduces a weak periodic lattice potential. <Bloch's theorem> replaces individual free <plane waves> by <Bloch states>, and Fourier components of the potential mix plane waves whose <wavevectors> differ by a <reciprocal lattice> vector. Near degeneracy at a <Brillouin zone> boundary, this coupling opens a <band gap>; for a two-wave crossing its size is $2|V_G|$. Away from the crossing, energies remain close to free-particle values. Band curvature produces <band effective masses>, and <band filling> determines whether the <Fermi energy> lies inside an allowed band or a gap.
The improvement is the inclusion of lattice periodicity, rather than a switch from classical to quantum statistics: the quantum <free electron model> already uses <Fermi-Dirac statistics>. The strong delta barriers in the later numerical example are treated exactly by the <Kronig-Penney model>, rather than by assuming that their strength is perturbatively small.
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