Abstract We revisit the classic “tipping pencil” instability, a rigid rod balanced upright on its tip and allowed to rotate in a vertical plane, long used as an introductory illustration of the quantum–classical boundary. In the standard semiclassical argument, the upright configuration is destabilized by minimal uncertainties in the initial angle and angular momentum,
δθ δ θ and
δL δ L , constrained by
δθδL≳ℏ/2 δ θ δ L ≳ ħ / 2 , with
δL≃Iδω δ L ≃ I δ ω . For a homogeneous cylinder of length
a=10cm a = 10 cm and mass
m=100g m = 100 g , the resulting uncertainty-based initial scales are extremely small and lead to tipping times of order a few seconds. We show, however, that quantum fluctuations are not required as the dominant physical seed for a finite tipping time. A purely classical microscopic perturbation, such as the angular-momentum transfer from a single elastic collision with a residual
O2 O 2 molecule under ultrahigh-vacuum conditions, already provides a conservative seed for the exponential instability. Using conservation of angular momentum, we estimate an imparted angular velocity
ω0,O2≃1.5×10−20s−1 ω 0 , O 2 ≃ 1.5 × 10 - 20 s - 1 , about five orders of magnitude smaller than the uncertainty-based angular-velocity scale, yet still sufficient to produce a macroscopic fall because the dependence on the seed is only logarithmic. We also include the more natural classical estimate due to thermal angular fluctuations. At room temperature, equipartition gives
θth∼3×10−10rad θ th ∼ 3 × 10 - 10 rad and
ωth∼3.5×10−9s−1 ω th ∼ 3.5 × 10 -
Ever since the advent of quantum mechanics, it has been clear that the atoms composing matter do not obey Newton's laws. Instead, their behavior is described by the Schroedinger equation. Surprisingly though, until recently, no clear explanation was given for why everyday objects, which are merely collections of atoms, are observed to obey Newton's laws. It would seem that, if quantum mechanics explains all the properties of atoms accurately, they, too, should obey quantum mechanics. This reasoning led some scientists to believe in a distinct macroscopic, or ``big and complicated,'' world in which quantum mechanics fails and classical mechanics takes over, although there has never been experimental evidence for such a failure. Even those who insisted that Newtonian mechanics would somehow emerge from the underlying quantum mechanics as the system became increasingly macroscopic were hindered by the lack of adequate experimental and theoretical tools. In the last decade, however, this quantum-to-classical transition has become accessible to experimental study and quantitative description, and the resulting insights are the subject of this article.
One of the main unsolved question in Quantum Mechanics (QM) is its compatibility with classical mechanics. The laws of QM, which describe the microscopic world, must merge into the classical ones for large enough physical systems, but it is still unknown at which point, if any, the transition occurs and if the transition is smooth or sudden as the size increases. Furthermore, a strict extension of QM to the macroscopic world leads to well known 'paradoxes', which is not straightforward to solve. This question is tightly connected with the measurement problem, since any measurement apparatus must give a response which can be described at classical level. Many experiments have been performed to answer to this question. They mainly try to find to which extent the number of particles can be increased in order to observe clear evidence of violation of standard QM. In this paper we argue that the macroscopicity parameter is not the number of particles but the number of independent degrees of freedom, as proposed in a recent model for the completion of QM. This introduces a sort of change of paradigm. To support this claim a representative set of well known experiments are analyzed from this point of view. This brings to a new interpretation of the experiments. A general scheme for the quantum-classical transition is discussed in some details.
Classical mechanics, which gives a fairly accurate description of large systems (e.g., solar system) as also of mechanical systems in our every day life, however, breaks down when applied to small (microscopic) systems such as molecules, atoms and nuclei. For example, (1) classical mechanics cannot even explain why the atoms are stable at all. A classical atom with electrons moving in circular or elliptic orbits around the nucleus would continuously radiate energy in the form of electromagnetic radiation because an accelerated charge does radiate energy. As a result the radius of the orbit would become smaller and smaller, resulting in instability of the atom. On the other hand, the atoms are found to be remarkably stable in practice. (2) Another fact of observation that classical mechanics fails to explain is wave particle duality in radiation as well as in material particles. It is well known that light exhibits the phenomena of interference, diffraction and polarization which can be easily understood on the basis of wave aspect of radiation. But light also exhibits the phenomena of photo-electric effect, Compton effect and Raman effect which can only be understood in terms of corpuscular or quantum aspect of radiation. The dual behavior of light, or radiation cannot be consistently understood on the basis of classical concepts alone or explained away by saying that light behaves as wave or particle depending on the kind of experiment we do with it (complementarity). Moreover, a beam of material particles, like electrons and neutrons, demonstrates wave-like properties (e.g., diffraction). A brief outline of phenomena that require quantum mechanics for their understanding follows.
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.