We calculate the tipping time of a quantum rod that has a height several times that of the edge length of its square base. We use an expression for the tipping time that has heuristic value, and gives the average time at which, upon measurement, the initially balanced rod is found to tip. We use two methods to calculate the tipping time. One method is to examine the "late time" behaviour of the quantum state of the center of mass of the rod by using an equation that has the form of the time-independent Schrödinger equation except that it involves a "complex energy." The other method uses energy resonances in the eigenstates of the Hamiltonian to determine the tipping time. We use the well-known WentzelKramersBrillouin approximation to calculate the energy eigenstates. With these methods, we obtain expressions for the tipping time that are valid for very long tipping times. PACS Nos.: 03.65.w, 03.65.Xp
The behaviour of a quantum rod, pivoted at its lower end on an impenetrable floor and restricted to moving in the vertical plane under the gravitational potential, is studied analytically under the approximation that the rod is initially localized to a 'small-enough' neighbourhood around the point of classical unstable equilibrium. It is shown that the rod evolves out of this neighbourhood. The time required for this to happen, i.e. the tipping time, is calculated using the semi-classical path integral. It is shown that equilibrium is recovered in the classical limit, and that our calculations are consistent with the uncertainty principle.
The classical and quantum mechanical Hamiltonians for a cylinder subject to holonomic constraints are derived. The quantum mechanical Hamiltonian is simplified and cast into a dimensionless form. The tipping time of a quantum mechanical cylinder subject to gravity is calculated. Numerical solutions for an appropriate initial wave function are obtained. We find that the tipping time is given by 〈t〉tip = t0C1 exp [C2(r/r0)9], where t0 is the time scale, C1 and C2 are constants of order unity, r is the radius of the cylinder, and r0 is the length scale for the tipping. We compare our results with those found in previous works.
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 -
Abstract: It is a well-known fact in physics that classical mechanics describes the macro-world, and quantum mechanics describes the atomic and sub-atomic world. However, principles of quantum mechanics, such as Heisenberg’s Uncertainty Principle, can create visible real-life effects. One of the most commonly known of those effects is the stability problem, whereby a one-dimensional point base object in a gravity environment cannot remain stable beyond a time frame. This paper expands the stability question from 1- dimensional rod to 2-dimensional highly symmetrical structures, such as an even-sided polygon. Using principles of classical mechanics, and Heisenberg’s uncertainty principle, a stability equation is derived. The stability problem is discussed both quantitatively as well as qualitatively. Using the graphical analysis of the result, the relation between stability time and the number of sides of polygon is determined. In an environment with gravity forces only existing, it is determined that stability increases with the number of sides of a polygon. Using the equation to find results for circles, it was found that a circle has the highest degree of stability. These results and the numerical calculation can be utilized for architectural purposes and high-precision experiments. The result is also helpful for minimizing the perception that quantum mechanical effects have no visible effects other than in the atomic, and subatomic world. Keywords: Quantum mechanics, Heisenberg Uncertainty principle, degree of stability, polygon, the highest degree of stability
Abstract Most approaches towards a quantum theory of gravitation indicate the existence of a minimal length scale of the order of the Planck length. Quantum mechanical models incorporating such an intrinsic length scale call for a deformation of Heisenberg’s algebra resulting in a generalised uncertainty principle and constitute what is called gravitational quantum mechanics. Utilising the position representation of this deformed algebra, we study various models of gravitational quantum mechanics. The free time evolution of a Gaussian wave packet is investigated as well as the spectral properties of a particle bound by an external attractive potential. Here the cases of a box with infinite walls and an attractive potential well of finite depth are considered.