= Solution
Both models respect <Bloch's theorem> and describe <Electrons> in a periodic <lattice>, but they start from different useful limits.
In the <nearly-free electron model>, start with extended plane waves and free-electron energies $\varepsilon^0_{\mathbf k}=\hbar^2k^2/(2m)$. A weak <periodic potential> has <Fourier coefficients> $V_{\mathbf G}$ at <reciprocal lattice> vectors and couples $\mathbf k$ to $\mathbf k-\mathbf G$. Away from degeneracies its corrections are small. Near a <Bragg scattering>, retain the two resonant plane waves and diagonalize
$$
H_{\rm pair}=\begin{pmatrix}\varepsilon^0_{\mathbf k}+V_0&V_{\mathbf G}\\V_{\mathbf G}^*&\varepsilon^0_{\mathbf k-\mathbf G}+V_0\end{pmatrix}.
$$
The resulting <eigenvalues> are
$$
E_\pm=V_0+\frac{\varepsilon^0_{\mathbf k}+\varepsilon^0_{\mathbf k-\mathbf G}}2\pm\sqrt{\left(\frac{\varepsilon^0_{\mathbf k}-\varepsilon^0_{\mathbf k-\mathbf G}}2\right)^2+|V_{\mathbf G}|^2}.
$$
At the crossing the gap is $\boxed{2|V_{\mathbf G}|}$. The two standing-wave combinations place electronic density differently relative to the <periodic potential> and therefore acquire different energies. Folding the many free parabolas into the first <Brillouin zone> produces the repeated band labels; the nonzero potential opens avoided crossings between coupled states. A Fourier component forbidden by <symmetry> need not open its corresponding first-order gap.
In the <tight-binding model>, instead start with localized orbitals on well-separated atoms or sites, with discrete on-site energies. Their overlap permits tunneling represented by <hopping integrals>. Bloch sums of these orbitals diagonalize the <lattice translation> <symmetry>. For one orthonormal orbital per site in a one-dimensional nearest-neighbor chain,
$$
\boxed{E(k)=E_0+2t\cos(ka),\qquad \text{bandwidth}=4|t|.}
$$
The many initially degenerate copies of one <atomic orbital> broaden into a band. Other atomic orbitals or multiple sites per primitive cell produce additional bands; their hybridization creates bonding and antibonding combinations and may open gaps. For two orbitals per cell, one illustrative <Hamiltonian> is
$$
H(k)=\begin{pmatrix}E_A&t_1+t_2e^{-ika}\\t_1+t_2e^{ika}&E_B\end{pmatrix},
$$
with <eigenvalues> $(E_A+E_B)/2\pm\sqrt{(E_A-E_B)^2/4+|t_1+t_2e^{-ika}|^2}$. This shows explicitly how two bands arise from the two cell degrees of freedom, even when the isolated sites have equal on-site energies.
\Image[/past-exam-of-the-quantum-condensed-matter-course-of-the-university-of-cambridge/2014-band-models.png]
{title=Original nearly-free-electron and two-orbital tight-binding band examples}
For each orbital degree of freedom per primitive cell there is one band per <spin>; <band filling> and the presence of a gap at the <Fermi level> determine metallic or insulating behavior. The nearly-free limit emphasizes weak <Bragg scattering> of extended states, while <tight binding> emphasizes hopping between localized states. They are complementary approximations to electronic band structure, rather than two incompatible explanations of why periodic solids have bands.
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