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Canadian Journal of Physics

Long tipping times of a quantum rod

Mark B. Lundeberg, Mark R. A. Shegelski

Published Jan 1, 2006• 9 citations
Publisher

Abstract

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 Wentzel–Kramers–Brillouin 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

DOI
10.1139/p06-003

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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 Wentzel–Kramers–Brillouin 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.
We found a simple procedure for the solution of the time - independent Schrodinger equation in one dimension without making any approximation. The wave functions are always periodic. Two difficulties may be encountered: one is to solve the equation E=U(x), where E and U(x) are the total and potential energies, respectively, and the other is to calculate the integral of the square root of U(x). If these calculations cannot be made analytically, it should then be performed by numerical methods. To find the energy and the wave function of the ground state, there is no need to calculate this integral, it is sufficient to find the classical turning points, that is to solve the equation E=U(x).
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.
A more general treatment than has been available of the Wentzel-Brillouin-Kramers method of solving Schr\"odinger's equation for one degree of freedom is given. Wentzel's original energy-level condition is shown to be an asymptotic expression, good for large masses and large values of the quantum number. The connection between this method and that of Young is discussed. Finally the formulas are written in a form convenient for application to the calculation of energy levels from actual potential functions.
In this paper, the reflection and transmission of plane waves are examined from a complex potential. Such potentials have the property of absorbing wave packets incident on them and are used widely in time-dependent quantum scattering theory. The purpose of the study is to determine the optimal form of potential to be used for absorbing wave packets near the edges of finite grids in coordinate space. The best potentials for such purposes lead to the minimum possible transmission and reflection of the incident wave packet. The Jeffreys–Wentzel–Kramers–Brillouin (JWKB) theory is used to address this problem and a new form for the optimal complex potential is proposed. A scaled dimensionless form of the Schrödinger equation is also derived, so that the parameters of any optimized potential obtained for a particular collision energy and mass combination may be readily converted to apply to a new set of masses and energies.