Tipping time of a quantum rod

Abstract

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.

Tipping time of a quantum rod

“…We define this as the 'tipping time't tip for the system. Note that this definition differs from the ones used before [2,4,5] .…”

“…We define this as the 'tipping time' t tip for the system. Note that this definition differs from the ones used before [2,4,5] 3 . We identify the motion of the rod with the trajectory of the probability density maximum in the θ -t space.…”

“…1 In [4], the WKB approximation is used to calculate energy eigenfunctions and eigenvalues, and the time evolved state is calculated in the limit that the expectation of energy of the rod is close to the potential energy at the point of unstable equilibrium. 2 In fact, the potential function used in [2,5] is of the form V 0 (cos(θ − θ 0 ) − cos(θ 0 )), consisting of a minimum at θ = 0 between two maxima. Thus, the resulting situation is a barrier tunnelling problem.…”

“…In the latter case, the uncertainty priciple causes the rod to transit out of equilibrium, i.e., tip over. Many texts on quantum mechanics, for instance [1], ask the reader to estimate this tipping time using the uncertainty principle; but a naïve application can lead to incorrect results (see [2] for an interesting discussion).…”

“…A special case of the problem we discuss has been considered, in the light of W.K.B approximation in [4] ‡. A variant of the problem has also been addressed before, where the centre of mass of the rod is localized within the base of support of the rod (in our case, the base is taken to be a point), and the tipping time is computed numerically [2,5] §. However, we seek an analytical expression for the tipping time.…”