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

Define $\alpha =\hslash T/(2I)$. The free propagator acts diagonally on each angular harmonic:The kick multiplies by the Jacobi-Anger seriesMultiply the two series and collect the output harmonic $n=m+\ell$. The one-step coefficient update iswhere both sampling times are just after their respective kicks. Free evolution supplies the input-index phase before the kick mixes angular momenta.

The source uses the unnormalized basis functions $e^{im\theta }$. Thus a unit-norm wavefunction satisfies $2\pi {\sum }_m|c_m{|}^2=1$. If the basis is instead $e^{im\theta }/\sqrt{2\pi }$, its amplitudes have sum of squared moduli one and obey the same matrix update. Mixing these normalization conventions would change an energy expectation incorrectly.