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Physical Review A

Quantum dynamics of a plane pendulum

M. Leibscher, Burkhard Schmidt

Published Jul 17, 2009• 35 citations
Publisher

Abstract

A semianalytical approach to the quantum dynamics of a plane pendulum is developed, based on Mathieu functions which appear as stationary wave functions. The time-dependent Schr\"odinger equation is solved for pendular analogs of coherent and squeezed states of a harmonic oscillator, induced by instantaneous changes of the periodic potential energy function. Coherent pendular states are discussed between the harmonic limit for small displacements and the inverted pendulum limit, while squeezed pendular states are shown to interpolate between vibrational and free rotational motion. In the latter case, full and fractional revivals as well as spatiotemporal structures in the time evolution of the probability densities (quantum carpets) are quantitatively analyzed. Corresponding expressions for the mean orientation are derived in terms of Mathieu functions in time. For periodic double well potentials, different revival schemes, and different quantum carpets are found for the even and odd initial states forming the ground tunneling doublet. Time evolution of the mean alignment allows the separation of states with different parity. Implications for external (rotational) and internal (torsional) motion of molecules induced by intense laser fields are discussed.

DOI
10.1103/physreva.80.012510

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A semianalytical approach to the quantum dynamics of a plane pendulum is developed, based on Mathieu functions which appear as stationary wave functions. The time-dependent Schr"odinger equation is solved for pendular analogs of coherent and squeezed states of a harmonic oscillator, induced by instantaneous changes of the periodic potential energy function. Coherent pendular states are discussed between the harmonic limit for small displacements and the inverted pendulum limit, while squeezed pendular states are shown to interpolate between vibrational and free rotational motion. In the latter case, full and fractional revivals as well as spatiotemporal structures in the time evolution of the probability densities (quantum carpets) are quantitatively analyzed. Corresponding expressions for the mean orientation are derived in terms of Mathieu functions in time. For periodic double well potentials, different revival schemes, and different quantum carpets are found for the even and odd initial states forming the ground tunneling doublet. Time evolution of the mean alignment allows the separation of states with different parity. Implications for external (rotational) and internal (torsional) motion of molecules induced by intense laser fields are discussed.
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 (tau\\tautau) is also dramatic, having the special limiting case of tautoinfty\\tau \\to \\inftytautoinfty 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.
We explore analytically the quantum dynamics of a point mass pendulum using the Heisenberg equation of motion. Choosing as variables the mean position of the pendulum, a suitably defined generalised variance and a generalised skewness, we set up a dynamical system which reproduces the correct limits of simple harmonic oscillator like and free rotor like behaviour. We then find the unexpected result that the quantum pendulum released from and near the inverted position executes oscillatory motion around the classically unstable position provided the initial wave packet has a variance much greater than the variance of the well known coherent state of the simple harmonic oscillator. The behaviour of the dynamical system for the quantum pendulum is a higher dimensional analogue of the behaviour of the Kapitza pendulum where the point of support is vibrated vertically with a frequency higher than the critical value needed to stabilize the inverted position. A somewhat similar phenomenon has recently been observed in the non equilibrium dynamics of a spin - 1 Bose-Einstein Condensate.
A detailed quantum mechanical treatment of the motion of a plane pendulum is given. Exact solution of the Schrödinger equation shows that the energy eigenfunctions and eigenvalues go over to those of a plane rotator for vanishing acceleration due to gravity g and to those of a harmonic oscillator in the limit of infinitely massive bob. In view of this, perturbation calculations have been done using rotator and oscillator basis, and conditions for convergence of the perturbation series in both cases have been discussed.
We have studied quantum-mechanical behavior of a driven Morse oscillator coupled to a bath of harmonic oscillators. The purpose is to compare the quantum behavior of such a system with the classical solutions of a driven damped Morse oscillator. We start with the quantum Liouville equation, in which the Hamiltonian of the Morse oscillator is expressed in terms of generators of an su(2) Lie algebra. This algebra and the Markovian approximation allow us to derive the generalized master equation (for the reduced density matrix of the Morse oscillator), which contains level-dependent energy and phase relaxation terms. We have numerically integrated the differential equations of the matrix elements to obtain the time evolution of the reduced density matrix and the energy of the Morse oscillator. We show that the energy of the Morse oscillator in general varies with time and eventually reaches an asymptotic oscillatory state. The mean energy value of the asymptotic oscillatory state is studied as a function of the relaxation rates and laser frequency and amplitude. Vibrational distributions have also been found as functions of laser frequency and amplitude. The shape of the distribution changes gradually as laser amplitude increases; it first peaks at the ground state at a small field amplitude, then it peaks at some excited state at a large field amplitude, and for an amplitude in between it can have two maxima. This bimodal vibrational distribution reflects the bistability observed in the classical and semiclassical models. Thus bistability exists in quantum results, not as a hysteresis loop, but as a bimodal distribution. Finally, we show that as laser intensity increases, the time series of the oscillator energy evolves from regular to quasiperiodic behavior, and eventually a chaotic-looking series appears.
The quantum states of the Kapitza pendulum are investigated in the framework of the effective potential obtained by the method of averaging over fast oscillations. An analytical estimate of the energy spectrum of stabilized states is given using a model potential. For the lowest states of the inverted pendulum, an expression is obtained for the spectrum in the form of the energies of a harmonic oscillator, refined according to the perturbation theory. Tunneling corrections to the energies of resonance states in double-well effective potential are found. The results of calculations of the structure of vibrational and rotational spectra of the Kapitza pendulum by the semiclassical method and by the numerical Numerov’s algorithm are compared.
Abstract The wave equation of electromagnetism, the Helmholtz equation, has the same form as the Schrödinger equation, and so optical waves can be used to study quantum mechanical problems. The electromagnetic wave solutions for non-diffracting beams lead to the two-dimensional Helmholtz equation. When expressed in elliptical coordinates the solution of the angular part is the same as the Schrödinger equation for the simple pendulum. The resulting optical eigenmodes, Mathieu modes, have an optical Fourier transform with a spatial intensity distribution that is proportional to the quantum mechanical probability for the pendulum. Comparison of Fourier intensities of eigenmodes are in excellent agreement with calculated quantum mechanical probabilities of pendulum stationary states. We further investigate wave-packet superpositions of a few modes and show that they mimic the libration and the nonlinear rotation of the classical pendulum, including revivals due to the quantized nature of superpositions. The ability to ‘dial a wavefunction’ with the optical modes allows the exploration of important aspects of quantum wave-mechanics and the pendulum that may not be possible with other physical systems.