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.
We study the tipping time of a quantum mechanical rod that is constrained to move in a plane in a gravitational potential. The initial state of the center of mass of the rod is localized within the base of support of the rod. The tipping time is found to increase exponentially with the height H of the rod as C1t0 exp[C2(H/H0)9], where C1 and C2 are dimensionless constants of order one, t0 is the time scale of the motion, and H0 is the length scale of the rod. We show that the tipping time cannot be obtained using the uncertainty principle alone, and compare our result to that obtained using the Wentzel–Kramers–Brillouin approximation.
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
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 -
The path integral formulation of quantum mechanics in the semiclassical or WKB approximation provides a physically intuitive way of relating a classical system to its quantum analog. A fruitful way of studying quantum chaos is based upon applying the Gutzwiller periodic orbit sum rule, a result derived by the path integral method in the WKB approximation. This provides some motivation for learning about path integral techniques. In this paper a pedagogical example of the path integral formalism is presented in the hope of conveying the basic physical and mathematical concepts. The ‘‘quantum bouncer’’ is studied—the quantum version of a particle moving in one dimension above a perfectly reflecting surface while subject to a constant force directed toward the surface. The classical counterpart of this system is a ball bouncing on a floor in a constant gravitational field, collisions with the floor being assumed to be elastic. Path integration is used to derive the energy eigenvalues and eigenfunctions of the quantum bouncer in the WKB or semiclassical approximation. The results are shown to be the same as those obtained by solving the Schrödinger equation in the same approximation.
We explore analytically the quantum dynamics of a point mass pendulum using the Heisenberg equation of motion. Choosing as variables the mean position of the pendulum, a suitably defined generalised variance and a generalised skewness, we set up a dynamical system which reproduces the correct limits of simple harmonic oscillator like and free rotor like behaviour. We then find the unexpected result that the quantum pendulum released from and near the inverted position executes oscillatory motion around the classically unstable position provided the initial wave packet has a variance much greater than the variance of the well known coherent state of the simple harmonic oscillator. The behaviour of the dynamical system for the quantum pendulum is a higher dimensional analogue of the behaviour of the Kapitza pendulum where the point of support is vibrated vertically with a frequency higher than the critical value needed to stabilize the inverted position. A somewhat similar phenomenon has recently been observed in the non equilibrium dynamics of a spin - 1 Bose-Einstein Condensate.
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.