Past exam of the mathematics course of the University of Cambridge / 2024 / ii / Paper 3 / 34E (Applications of Quantum Mechanics)

For two solutions of the Schrodinger equation, their Wronskian is constant because the equation has no first-derivative term. The determinant of the Floquet matrix is the ratio of the Wronskian at $x+a$ to that at $x$, hence

For real $E$ and real $V$, choose a real fundamental pair of solutions. The corresponding Floquet matrix has real entries, so $\mathrm{tr}F(E)$ is real. The trace is basis independent, so this holds for every fundamental pair.

The Floquet multipliers satisfyIf $(\mathrm{tr}F{)}^2<4$, they are complex conjugates on the unit circle, $e^{\pm ika}$, and bounded Bloch waves exist: $E$ lies in an allowed band. If $(\mathrm{tr}F{)}^2>4$, the multipliers are real reciprocal numbers, one growing and one decaying, so there is no extended bounded Bloch state and $E$ lies in a gap.

Translation preserves the alternating on-site term only when $(-1{)}^{j+\ell }=(-1{)}^j$, soThe primitive lattice spacing is therefore $2a$.

Using one even and one odd site per unit cell, Bloch theorem reduces the Hamiltonian toIts eigenvalues are

For lattice spacing $2a$, the first Brillouin zone isAt its boundaries $E_{\pm }=\pm t$, while at $k=0$ the energies are $E_{\pm }=\pm \sqrt{5},t$. Thus the lower band runs from $-\sqrt{5},t$ to $-t$, the upper from $t$ to $\sqrt{5},t$, and each has bandwidthThe requested plot consists of these two even branches, separated by a gap of size $2t$ at the zone boundaries.