Past exam of the theoretical physics II course of the University of Cambridge / 2016 / Paper 1 / 1 / a / Solution

Take periodic orbital wavefunctions on the ring, and write $H_0=-({\hslash }^2/2I){\partial }_{\theta }^2$. Between kicks the evolution over one period isIntegrating through an infinitesimal kick givesThe free-motion contribution during the vanishing kick duration disappears in the ideal impulsive limit. The sign of the phase follows from the negative kick potential.

Starting just after a kick, the particle evolves freely first and is kicked at the end of the interval. Operators act right to left, so one step is $U_{\eta }{U}_T$, andThe one-period product is the Floquet operator of this quantum kicked rotor. There is no additional kick at the initial endpoint $0^{+}$. Changing the stroboscopic convention to just before each kick would change the displayed order; it is not appropriate for these endpoints.