Long tipping times of a quantum rod

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

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] .…”

“…is satisfied. But the left hand side of the above equation with the form of propagator suggested by (5), is certainly not a delta function; the reason being that not all classical paths have been incorporated in (5). Nevertheless it is plausible that one recovers a delta function-approximately for small enough t, and exactly in the limit t → 0.…”

“…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.…”

“…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.…”