The motion of macroscopic objects such as soccer balls, vehicles and planets can be well described using classical mechanics. However, classical mechanics is inadequate to describe the movements or the structure of microscopic objects such as atoms, molecules or nuclei. To properly describe the properties of microscopic objects, the theory of quantum mechanics was developed. While quantum mechanics can be applied to macroscopic objects, it turns out that for objects of large dimensions the quantum mechanical laws evolve naturally into classical mechanics. In general, solving a problem in quantum mechanics is far more difficult than solving it in classical mechanics. Therefore, the laws of classical mechanics are employed when they can be applied safely.
Abstract A quantum description is used to find the time limit for balancing an idealized pencil-like object on end. In agreement with some prior studies, analysis for this system shows that quantum effects are not observable in any foreseeable circumstances, either by casual experimentation or by extensive laboratory preparation. A direct solution of Schrodinger’s equation avoids an a priori use of the Heisenberg uncertainty relation in the classical equations of motion, which, although very much a proper usage, may lead to misunderstanding of the results. The classical equations of motion are adequate for the pencil in all circumstances.
It is now generally accepted that quantum mechanics occupies a central place in microscopic physical theory. The importance of this discipline shows up in the research journals, and also in the graduate physics curriculum. Not only is quantum mechanics a required subject, both for experimental and theoretical students; it appears as well to be influencing the way in which classical physics is taught. Presentations of classical mechanics that slight the traditional problems of planetary motion, unsymmetrical tops and dissipative systems in favor of collision theory and Poisson brackets are becoming more and more prevalent. With the much younger subject of classical electrodynamics, the analogous tendency is less marked, although here again emphasis is being placed on those examples that bring the student closest to the formalism of quantum electrodynamics. Such trends are of course proper and reasonable, especially so long as no important classical ideas are omitted from consideration. Indeed, the study of acoustics, hydrodynamics, and elasticity probably benefits from an early introduction of field concepts.
Does quantum mechanics matter at everyday scales? We generally assume that its consequences are confined to microscopic scales. It would be very surprising if quantum effects were to be manifest in a macroscopic system. This has been claimed for the problem of balancing a pencil on its tip. The claim has also been disputed. We argue that the behaviour of a tipping pencil can be explained by the asymptotic properties of the complete elliptic function, and can be understood in purely classical terms.
I distinguish between two conceptually different kinds of physical space: a space of ordinary material bodies, which is the space of points at which I could imaginably place (say) the tip of my finger, or the center of a billiard-ball, and a space of elementary physical determinables, which is the smallest space of points such that stipulating what is happening at each one of those points, at every time, amounts to an exhaustive physical history of the universe. In all classical physical theories, these two spaces happen to coincide – and what we mean by calling a theory “classical”, and all we mean by calling a theory “classical”, is (I will argue) precisely that these two spaces coincide. But once the distinction between these two spaces in on the table, it becomes clear that there is no logical or conceptual reason why they must coincide – and it turns out (and this is the main topic of the present paper) that a very simple way of pulling them apart from one another gives us quantum mechanics.
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 -