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A discrete memoryless channel has finite input alphabet $\mathcal{A}$, finite output alphabet $\mathcal{B}$, and transition probabilities $P(y\mid x)$; successive outputs are conditionally independent given the corresponding inputs. The Shannon second coding theorem states that rates belowadmit block codes with error probability tending to zero, whereas rates above $C$ cannot have vanishing error.

For any joint input law of $(X_1,X_2)$, conditional independence of the product channel givesChoose $X_1$ and $X_2$ independently with capacity-achieving input laws for their respective channels. Then the inequality becomes equality, so the product-channel capacity is

Let $G_n=2\pi n/a$. The free states $|k⟩$ and $|k-G_n⟩$ are degenerate at the Bragg point $k=G_n/2=n\pi /a$. Sincedegenerate perturbation theory restricts the Hamiltonian near this crossing toWriting $k=n\pi /a+\kappa$ and diagonalizing gives the nearly-free electron dispersion near a one-dimensional band gapThus the two continuous bands avoid crossing, and at $\kappa =0$ their separation isThis is the nearly-free electron model. The dispersion relation is the relation $E=E(k)$ between energy and Bloch wavevector.

For the specified potential,Henceand all other nonconstant Fourier coefficients vanish. The gaps are thereforeTheir centre energies are respectivelyThe extended-zone sketch consists of the free-particle parabolas shifted upward by $V_0$, with avoided crossings of these two widths at the listed Bragg points; elsewhere they meet to this order because the corresponding Fourier coefficients vanish.