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Journal of Physics A Mathematical and Theoretical

On the quantum dynamics of the rigid rotor

K. Kowalski, Jakub Rembieliński, J Zawadzki

Published Dec 17, 2014• 3 citations
Publisher

Abstract

The dynamics is investigated of a free particle on a sphere (rigid rotor or rotator) that is initially in a coherent state. The instability of coherent states with respect to the free evolution leads to nontrivial temporal development of averages of observables representing the position of a particle on a sphere that can be interpreted as quantum beats of the rotor. The beats are related to occuring quantum coherent state wave packet revivals on a sphere.

DOI
10.1088/1751-8113/48/3/035301

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The dynamics is investigated of a free particle on a sphere (rigid rotor or rotator) that is initially in a coherent state. The instability of coherent states with respect to the free evolution leads to nontrivial temporal development of averages of observables representing the position of a particle on a sphere that can be interpreted as quantum beats of the rotor. The beats are related to occuring quantum coherent state wave packet revivals on a sphere.
Recently, the stochastic nonlinear wave equation describing the dynamics of an observed quantum system has been solved for a free particle [Phys. Rev. A 45, 1347 (1992)]. In this paper, transient effects that occur during a continuous observation of the particle position are discussed. It is shown that the dispersion of the Gaussian wave packet of the particle always decreases in the beginning of the observation, then takes the form of oscillations which, after a short time, decay rapidly to the asymptotic value.
We analyze the time-behaviour of the Gaussian wave packet of a quantum particle moving in the field of a linear force, undergoing a continuous nondemolition observation of its position. In particular, the time-dependence of position and momentum dispersions, as well as position-momentum uncertainty is investigated. The discussion includes the case of a free particle and initially coherent state of a harmonic oscillator. It is found that in the beginning of observation the momentum dispersion and position-momentum uncertainty always increase in contrast to the position dispersion. It is also proved that all considered functions asymptotically take the form of harmonic oscillations, decaying to finite values. For initially coherent states, the asymptotic bounds for all considered functions are given.
Analytical eigenfunctions and eigenvalues for the Morse oscillator were applied to investigate the quantum resonant beats and revivals of wave packet propagation. A concise way for exact prediction of the complete revival period of the Morse oscillator was given for the first time. It was suggested that any complete period was made of integer numbers of the minimum or fundamental period. Within the fundamental period, the anharmonicity of this oscillator appeared to cause interesting space-time phenomena that include relatively simple Farey-sum revival structures. In addition, a simple sum of two Morse oscillators led to a double-Morse well whose geometric symmetry provided analytical eigenfunctions and eigenvalues for certain low-lying energy levels. The quantum tunneling between the double-Morse well significantly affected the resonant beats and revivals local to each well, and gave rise to interesting tsunami-like waves in the middle of the double well. Furthermore, quantum rotor wave functions based upon Wigner-D matrix were applied to investigate the quantum resonant beats and revivals that occur in experimentally accessible spin systems. Interesting physical effects in quantum rotors between half-integer spin and integer spin systems were observed to show effects of symmetry. Essentially, the quantum revivals in these quantum systems exhibited number-information aspects of surprisingly simple Farey-sum and Ford circles geometry. Such quantum dynamics will provide a physical insight to further develop matter wave packet technology, and might have applications for quantum information processing and quantum computing.
Physical consequences of the theorem of equivalence among harmonic oscillators of different frequencies, including a free particle as a special case, are explored. A new pictorial way of understanding time-dependent coherent states and squeezed states are presented and contrasted with the standard view. Three-dimensional spherical analogues of squeezed coherent states are constructed. It is also shown as a corollary of the theorem that a free wave packet which coincides initially with any one of the eigenfunctions of a harmonic oscillator maintains its shape, that is, the probability density of position does not change apart from a time-dependent scaling of spatial coordinates. Most of the results are derived without use of any sophisticated techniques but with the knowledge of free wave packets alone.
Physical consequences of the theorem of equivalence among harmonic oscillators of different frequencies, including a free particle as a special case, are explored. A new pictorial way of understanding time-dependent coherent states and squeezed states are presented and contrasted with the standard view. Three-dimensional spherical analogues of squeezed coherent states are constructed. It is also shown as a corollary of the theorem that a free wave packet which coincides initially with any one of the eigenfunctions of a harmonic oscillator maintains its shape, that is, the probability density of position does not change apart from a time-dependent scaling of spatial coordinates. Most of the results are derived without use of any sophisticated techniques but with the knowledge of free wave packets alone.