Past exam of the mathematics course of the University of Cambridge / 2023 / ii / Paper 1 / 35D / b / Solution

For a potential of period $a$, define the Floquet matrix for a one-dimensional periodic potential byThe Schrodinger equation has no first-derivative term, so the Wronskian is constant and $\det M=1$. Its multipliers are therefore reciprocal. They lie on the unit circle precisely whenThose energies form continuous allowed bands of bounded Bloch theorem solutions. Values with $|\mathrm{tr}M|>2$ have a growing and a decaying multiplier and form forbidden gaps. The band edges have $\mathrm{tr}M=\pm 2$, as in Floquet discriminant and energy bands.

For the delta-comb Kronig-Penney model, start immediately to the right of one delta function. Free propagation to the next delta and the derivative jump there are represented byThusConsequently the Floquet discriminant of the delta-comb Kronig-Penney model isAll band edges are therefore determined byIn factorized form, the periodic edges $\sigma =1$ obeywhile the antiperiodic edges $\sigma =-1$ obey

To express the same edges through the single-barrier scattering data, the One-dimensional transfer matrix from scattering amplitudes givesAt an edge put $z=e^{-ika}$ and $\mathrm{tr}M=2\sigma$. Thenso the scattering-amplitude equations for one-dimensional band edges areExplicitly,For the delta barrier, $t-r=1$ andwhich reproduces the four factorized edge equations above.

Solved by gpt-5.6-sol high.