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Reviews of Modern Physics

The asymmetric rotor as a model for localization

A. R. P. Rau

Published Apr 1, 1992• 16 citations
Publisher

Abstract

Together with the pendulum, the symmetric and asymmetric rotors are standard models, well studied in both classical and quantum physics. In the limit of high angular momentum $J$, the quantum problem of an asymmetric rotor approaches the classical limit, which is characterized by one unstable and two stable motions. Rotations with respect to the axes with maximum and minimum moments of inertia are stable, while the rotations about the intermediate axis are unstable. Upon diagonalization of the rotor's Hamiltonian at fixed $J$, most eigenvectors are seen to be localized around the directions of maximum and minimum moment of inertia. The asymmetric rotor displays in this regard localization similar to that in many atomic and nuclear few-body systems, wherein external interactions mix degenerate states, particularly at high excitation. Indeed, there is a complete one-to-one correspondence between the rotor and these systems, with the principal ($n$) and angular momentum ($l$) quantum numbers mapping onto $J$, and its azimuthal projection $M$, respectively, for the rotor. Localization at a maximum or a saddle point of a potential is of considerable significance in a variety of problems.

DOI
10.1103/revmodphys.64.623

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Together with the pendulum, the symmetric and asymmetric rotors are standard models, well studied in both classical and quantum physics. In the limit of high angular momentum JJJ, the quantum problem of an asymmetric rotor approaches the classical limit, which is characterized by one unstable and two stable motions. Rotations with respect to the axes with maximum and minimum moments of inertia are stable, while the rotations about the intermediate axis are unstable. Upon diagonalization of the rotor's Hamiltonian at fixed JJJ, most eigenvectors are seen to be localized around the directions of maximum and minimum moment of inertia. The asymmetric rotor displays in this regard localization similar to that in many atomic and nuclear few-body systems, wherein external interactions mix degenerate states, particularly at high excitation. Indeed, there is a complete one-to-one correspondence between the rotor and these systems, with the principal (nnn) and angular momentum (lll) quantum numbers mapping onto JJJ, and its azimuthal projection MMM, respectively, for the rotor. Localization at a maximum or a saddle point of a potential is of considerable significance in a variety of problems.
The theory of a symmetric rotator which in addition to the usual three degrees of rotational freedom has about the axis of symmetry also a degree of torsional freedom between two of its principal parts, is treated quantum mechanically. The potential energy is taken to be expressible in the form: $V=L(1\ensuremath{-}cosm{\ensuremath{\varphi}}^{\ensuremath{'}})$ where LLL is proportional to the restoring torque, ${\ensuremath{\varphi}}^{\ensuremath{'}}$ is the angle of displacement, and mmm the number of minima of the potential energy curve as ${\ensuremath{\varphi}}^{\ensuremath{'}}$ increases from zero to $2\ensuremath{\pi}$. The Schr\"odinger equation is found to be separable into two differential equations, one being the quantum mechanical equation for an ordinary symmetric rotator, and the other being of the form of Mathieu's equation: d2Mdx2+(a+16qcos2x)\frac{{d}^{2}M}{d{x}^{2}}+(a+16qcos2x)dx2d2M​+(a+16qcos2x) M=0M=0M=0 in which aaa is proportional to the energy parameter of the oscillator, qqq proportional to the restoring torque, and xxx to ${\ensuremath{\varphi}}^{\ensuremath{'}}$. It is found that the solutions to this equation must satisfy the condition: $M(x\ensuremath{-}m\ensuremath{\pi})=(\mathrm{exp}\ensuremath{-}\frac{2\ensuremath{\pi}\mathrm{iK}{{A}_{z}}^{\ensuremath{'}}}{{A}_{z}}) M(x)$ where KKK is the quantum number of angular momentum about the axis of symmetry, ${{A}_{z}}^{\ensuremath{'}}$ and Az{A}_{z}Az​ the moments of inertia of the lower part of the top and of the whole of the top respectively about the axis of symmetry. This ordinarily demands a general non-periodic solution to Mathieu's equation which, however, degenerates like the ordinary Mathieu functions to an exponential function as $q\ensuremath{\rightarrow}0$. A qualitative discussion is given about the manner in which the energy states in the limiting case where q=0q=0q=0 go over into the other limiting case where $q=\ensuremath{\infty}$, and the calculation of the intensities and the selection rules for the rotator are finally determined where q=0q=0q=0 and where $q=\ensuremath{\infty}$. These it is believed will also be valid at least in first approximation in the neighborhoods of these limiting cases where qqq no longer is quite zero and not quite equal to infinity.
Classical rotations of asymmetric rigid bodies are unstable around the axis of intermediate moment of inertia, causing a flipping of rotor orientation. This effect, known as the tennis racket effect, quickly averages to zero in classical ensembles since the flipping period varies significantly upon approaching the separatrix. Here, we explore the quantum rotations of rapidly spinning thermal asymmetric nanorotors and show that classically forbidden tunneling gives rise to persistent tennis racket dynamics, in stark contrast to the classical expectation. We characterize this effect, demonstrating that quantum coherent flipping dynamics can persist even in the regime where millions of angular momentum states are occupied. This persistent flipping offers a promising route for observing and exploiting quantum effects in rotational degrees of freedom for molecules and nanoparticles.
The consideration of the resonant interaction between a linear harmonic oscillator (LO) and a (nonlinear) angular momentum oscillator (NO) is motivated by its relation to the interaction between a field mode and a number of atomic systems, its relation to the interaction between three field modes, and the appearance of qualitatively different results quantum mechanically and classically even in cases that can be regarded as macroscopic. The analysis is carried out both classically and quantum mechanically in the rotating-wave approximation, and the time dependence of the energies of the two oscillators is examined, mainly by means of a nonlinear differential equation for the NO energy. In the classical analysis, the problem turns out to be identical to that of a spherical pendulum, where the NO energy L3{L}_{3}L3​ corresponds to the pendulum potential energy, the LO energy nnn corresponds to the pendulum kinetic energy, and the coupling energy between the two oscillators $\frac{1}{2}\frac{\ensuremath{\gamma}K}{\ensuremath{\omega}}$ is proportional to the angular momentum KKK of the pendulum about the vertical axis, all quantities being expressed in suitable units. A solution for L3(t){L}_{3}(t)L3​(t) is obtained, and is shown to be described uniquely, apart from a shift in the time origin, by the constants of motion $E(\ensuremath{\equiv}{L}_{3}+n)$ and KKK. L3{L}_{3}L3​ oscillates periodically, except for the "unstable-equilibrium" solution and the "conical-pendulum" solution. The quantum-mechanical problem is solved exactly for the cases in which the NO is a two-level system and a three-level system, and certain aspects of the exact solution are derived in the case of a four-level system. The constants EEE and KKK become a complete set of quantum numbers. It is found that unstable equilibrium, in the sense that 〈L3〉〈{L}_{3}〉〈L3​〉 is constant, does not exist, because of spontaneous emission; it is also found that 〈L3〉〈{L}_{3}〉〈L3​〉 oscillates sinusoidally for the two-level system, oscillates periodically but not sinusoidally for the three-level system, and oscillates aperiodically for a four- or higher-level system, conclusions about the higher-level systems coming from general considerations and numerical solutions found in the literature. Approximate solutions for NO' s with arbitrarily large numbers of levels are obtained. One method of approximation displays spontaneous emission but not aperiodicity. Another approximate method, which contains the statistical spread of the initial phase difference between the oscillations of the two oscillators inherent in the uncertainty principle, displays both spontaneous emission and aperiodicity. It is shown that the aperiodicity - a property which does not exist classically for precise initial conditions - does not disappear even in the macroscopic limit and is due to the averaging performed, in taking the quantum-mechanical expectation value, over an ensemble of periodic oscillations with a spread in frequencies.