Wave packet revivals and the energy eigenvalue spectrum of the quantum pendulum

Abstract

The rigid pendulum, both as a classical and as a quantum problem, is an interesting system as it has the exactly soluble harmonic oscillator and the rigid rotor systems as limiting cases in the low-and high-energy limits respectively. The energy variation of the classical periodicity (τ ) is also dramatic, having the special limiting case of τ → ∞ at the 'top' of the classical motion (i.e. the separatrix.) We study the time-dependence of the quantum pendulum problem, focusing on the behavior of both the (approximate) classical periodicity and especially the quantum revival and superrevival times, as encoded in the energy eigenvalue spectrum of the system. We provide approximate expressions for the energy eigenvalues in both the small and large quantum number limits, up to 4th order in perturbation theory, comparing these to existing handbook expansions for the characteristic values of the related Mathieu equation, obtained by other methods. We then use these approximations to probe the classical periodicity, as well as to extract information on the quantum revival and superrevival times. We find that while both the classical and quantum periodicities increase monotonically as one approaches the 'top' in energy, from either above or below, the revival times decrease from their low-and high-energy values until very near the separatrix where they increase to a large, but finite value.

Statistical Thermodynamics of Polymer Quantum Systems

“…Following the comparison with the quantum rigid rotor [52], we recall that the spectrum of a quantum rigid rotor in two dimensions is n 2 2 2I , with I = mR 2 2 being the moment of inertia and R the radius of the rotor. Notice that the second term in (2.12) has the same functional dependence on n as for the rotor with effective moment of inertia…”

Classical versus quantum dynamics of the atomic Josephson junction

“…The striking difference between the quantum and classical predictions is that the quantum prediction undergoes periodic collapses and revivals due to the discrete energy spectrum. In general, the time scale for revivals of the quantum wave function is given by [30] T…”

Supersymmetry and eigensurface topology of the planar quantum pendulum

“…First tackled by Condon in 1928 [1] in its planar variety, the quantum pendulum has since turned up in a number of research areas of atomic, molecular and optical physics, ranging from spectroscopy to the stereodynamics of molecular collisions to the manipulation of matter by external electric, magnetic and optical fields. Although both the planar and the full-fledged 3D spherical pendular varieties possess analytic asymptotic states [2][3][4][5][6][7][8][9][10], the planar case has been explored with particular tenacity [11][12][13][14][15][16][17][18][19], apparently because of its prototypical character, dwarfed only by that of few other systems such as of the harmonic oscillator.…”