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American Journal of Physics

The quantum pendulum: Small and large

Gregory L. Baker, James A. Blackburn, H. Smith

Published May 1, 2002• 34 citations
Publisher

Abstract

The quantum pendulum finds application in surprising contexts. We use commercially available software to numerically solve the Schrödinger equation for a microscopic pendulum subject to molecular (electromagnetic) restoring forces, and a macroscopic pendulum subject to a gravitational restoring force. The dynamics of the microscopic quantum pendulum are closely related to molecular motions known as hindered rotations. We use standard probabilistic methods to predict whether this motion is weakly or strongly hindered at ambient temperature and test the prediction against experimental data for C2H6 and K2PtCl6. For the macroscopic gravitational pendulum, we examine the uncertainty in position and find, not surprisingly, that it is too small to measure physically, but is nevertheless relatively large compared to present-day limits in computation. The latter juxtaposition of computational precision with quantum uncertainty has consequences for the study of chaotic dynamics.

DOI
10.1119/1.1456069

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The quantum pendulum finds application in surprising contexts. We use commercially available software to numerically solve the Schrödinger equation for a microscopic pendulum subject to molecular (electromagnetic) restoring forces, and a macroscopic pendulum subject to a gravitational restoring force. The dynamics of the microscopic quantum pendulum are closely related to molecular motions known as hindered rotations. We use standard probabilistic methods to predict whether this motion is weakly or strongly hindered at ambient temperature and test the prediction against experimental data for C2H6 and K2PtCl6. For the macroscopic gravitational pendulum, we examine the uncertainty in position and find, not surprisingly, that it is too small to measure physically, but is nevertheless relatively large compared to present-day limits in computation. The latter juxtaposition of computational precision with quantum uncertainty has consequences for the study of chaotic dynamics.
The behavior of the periodically kicked pendulum is determined classically and quantum mechanically. Our earlier analytic study [Phys. Rev. A 41, 2952 (1990)] is confirmed and extended by the numerical results presented here. We are able to construct minimum uncertainty, Gaussian wave packets that maintain their form, and shadow the corresponding classical motion for long times when the classical motion is nonchaotic or only weakly chaotic. However, when the classical motion is robustly chaotic, the quantum-mechanical probability distribution grows macroscopically broad exponentially fast and the quantum-classical correspondence breaks down.
The relation between determinism and predictability is the central issue in the study of 'deterministic chaos'. Much knowledge has been accumulated in the past 10 years about the chaotic dynamics of macroscopic (classical) systems. Our intention is to examine the implications of chaos in the microscopic quantum world – in other words, how to reconcile the correspondence principle with the inherent uncertainties which reflect the wave nature of quantum dynamics. Some rather surprising and counterintuitive results emerge from this effort which bear an intriguing similarity to a phenomenon associated with electric conductivity in disordered solids, namely 'Anderson localisation'. It is very rewarding that these developments do not remain in the lofty realms of abstract theoretical physics. Indeed, recent atomic physics experiments demonstrate clearly that chaos is relevant to the microscopic world. In particular, such experiments emphasise the urgent need to clarify the genuine quantum mechanism which imposes severe limitations on quantum dynamics, and renders it so very different from its classical counterpart.
We present the mechanical squeezing of a mg-scale suspended mirror (i.e. a pendulum) near quantum regimes through continuous linear position measurement. The experiment involved the pendulum interacting with photon coherent fields in a detuned optical cavity. The position uncertainty in the measured data is reduced and squeezed to 470470470 times the zero-point amplitude xzpfx_{\rm zpf}xzpf​ with a purity of about 0.0004, by means of optimal state estimation through causal Wiener filtering. The purity of the squeezed state is clearly maximized by the Wiener filter, based on precisely identified optomechanical parameters. This is the first step for measurement-based quantum control of macroscopic pendulums, e.g. generation of an entanglement state between macroscopic pendulums. Such quantum control will provide a direct insight into the quantum to classical transition and will pave the way to test semiclassical gravity and gravity sourced by macroscopic quantum oscillators.
The metric gradient force is the only purely geometric force term in the irreversible equation of motion of the Order Parameter Spacetime Theory. Its acceleration is proportional to the square of the velocity and independent of the velocity direction, making it exclusively distinguishable from all known dissipative forces (which are odd functions of velocity). We perform the first direct test of this prediction using Stern-Gerlach experimental data of Feshbach molecules from a ⁸⁷Rb Bose-Einstein condensate published by the MPQ group in 2003. By extracting the center-of-mass position data of the molecular cloud at 16 time slices from Figure 4 of the experiment, we find that the molecular cloud exhibits simple harmonic oscillation in a 100 G/cm magnetic field gradient, with an oscillation frequency of 56 Hz and an amplitude of 0.68 mm. Comparing the experimental acceleration with the theoretically predicted metric gradient force a_eff = (1/2)·(∂ ln g_ii/∂x)·ẋ², we verify the a ∝ v² scaling relation and deduce an order-parameter-space metric gradient ∂ ln g_ii/∂x ≈ 3.4 × 10³ m⁻¹. The left and right limits of the molecular cloud oscillation are approximately symmetric about the equilibrium position (deviation from center ≈ 0.68 mm), consistent with the prediction of the metric gradient force as an even function of velocity. The 2004 BEC-BCS crossover experiment by the Innsbruck group and the 2003 ⁶Li Fermi gas experiment by the ENS group provide qualitative support for this conclusion. Combined with previous tests of the theory in the cosmological weak-field (BAO and full-shape) and strong-field black hole (EHT) regimes, the Order Parameter Spacetime Theory has now passed observational tests across scales from microscopic cold-atom physics to macroscopic cosmology.
In a generalized quantum-mechanical framework with fuzzy dynamical variables, there exists a small radical length R${10}^{\mathrm{\ensuremath{-}}17}$ cm, which is related to the physical impossibility of unlimited accuracy in measuring a particle's position. It sheds light on the problem of a detailed and gradual transition, as the mass increases, from the probabilistic behavior of microscopic objects (mcR\ensuremath{\ll}\ensuremath{\Elzxh}) to the deterministic behavior of macroscopic objects (mcR\ensuremath{\gg}\ensuremath{\Elzxh}), a problem unaddressable within the framework of ordinary quantum mechanics. A generalized equation of motion for a particle with an arbitrary mass m is postulated. It possesses a new local gauge symmetry and reduces to the Schr\"odinger and the Hamilton-Jacobi equations in the limits of small m and large m, respectively. For the double-slit experiment, the theory predicts that, as the mass increases with the momentum fixed, the interference pattern will have roughly half the usual intensity and two bright spots when mcR/\ensuremath{\Elzxh}\ensuremath{\sim}1.