Past exam of the short options paper shared by Prelims and Parts A and B of the physics course of the University of Oxford / 2015 / S18 (2S98, 2S99)

Worked notes are available in Advanced Quantum Mechanics: S18 Exam 2015 Solutions. The requested Q1 amplitude sketch is supplemented below; the analytic worked parts remain available in those notes.

Alternate S18 question extract: www-thphys.physics.ox.ac.uk/people/AndreiStarinets/short_options_exam_2015.pdf. It carries the printed identifiers 2S98/2S99 and matches this option's three question cores, with wording and footer variations.

For real $x=ka$, taking the modulus squared of the amplitude gives the half-line delta-well amplitude enhancementThe supplied resonance poles have real parts $x_n\simeq n\pi (1+\lambda /2+{\lambda }^2/4)$ and negative imaginary parts $-{\gamma }_n\simeq -(\lambda n\pi /2{)}^2$ for fixed low $n$ at $\lambda \ll 1$.^[ref] Linearizing the denominator at a simple pole givesThus the sketch has narrow, tall peaks with a locally Lorentzian line shape, just above positive integer multiples of $\pi$. Their half-widths are ${\gamma }_n$, and the leading peak heights are $4/({\lambda }^2{n}^2{\pi }^2)$. In this fixed-low-mode approximation, increasing $n$ broadens and lowers the peaks. Exactly at $x=n\pi \ne 0$, the value is one. At zero it has the finite limit ${\lambda }^2/(2-\lambda {)}^2$ for $\lambda \ne 2$.

The peaks represent long-lived standing waves between the wall and the attractive delta potential. Despite the source's transmission-coefficient terminology, $|A{|}^2$ is an internal amplitude factor and can exceed one. The interior probability current is zero, and the reflected probability current equals the incident magnitude; the hard wall passes no transmitted flux.