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

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.

Put $x=r\cos \theta$. The function $e^{ikx}$ solves the two-dimensional Helmholtz equation, $({\nabla }^2+k^2)e^{ikx}=0$. Expand its angular dependence as a Fourier series,Substitution into the polar-coordinate Laplacian,givesWith $\rho =kr$, this is the Bessel differential equation. Regularity of the original plane wave at the origin excludes the singular second-kind solution. Thus $F_m(r)$ is a constant multiple of the regular first-kind function, andFor integer negative orders, $J_{-m}=(-1{)}^m{J}_m$, so this indexing still supplies the regular radial solution. The radial equation fixes the form of each term; the small-radius plane-wave expansion fixes its coefficient.

For a nonnegative integer $m$, expand the plane wave in powers of $\rho$:The first contribution to the positive angular harmonic $e^{im\theta }$ occurs at $n=m$. Its coefficient is $(i\rho {)}^m/(2^{m}m!)$. Comparing this with $a_m{J}_m(\rho )\sim a_m{\rho }^m/(2^{m}m!)$ gives $a_m=i^m$ for $m\ge 0$.

The plane wave is even in $\theta$, so its $m$ and $-m$ Fourier coefficients agree. Therefore $a_{-m}{J}_{-m}=a_m{J}_m$, implying $a_{-m}=(-1{)}^m{i}^m=i^{-m}$. HenceThis is the cosine form of the Jacobi-Anger expansion. The factorial asymptotic stated in the source is used for nonnegative order; applying it literally to negative integers would introduce an undefined factorial. The Integer-order Bessel reflection identity is the required extension, not a separate set of divergent radial solutions.

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.

On periodic ring states the angular-momentum index $m$ is an integer. At the stated resonance,so $U_T$ is the identity on every orbital state. After $N$ steps from the normalized ground state,Its phase varies in space but its position probability density remains uniform. Periodicity permits integration by parts, givingConsequentlyEquivalently, the angular-momentum probabilities are $P_m=J_m(N\eta {)}^2$, with ${\sum }_m{m}^2{J}_m(z{)}^2=z^2/2$. The mean angular momentum remains zero while its variance grows as $N^2$. This quantum resonance of the kicked rotor gives coherent ballistic energy growth. The Bessel second-moment identity connects the momentum spreading to that quadratic energy law; an incoherent random walk would instead give linear energy growth.