Tipping time of a quantum rod

TL;DR

This paper analytically investigates the quantum dynamics of a pivoted rod under gravity, calculating the tipping time using semi-classical path integrals and exploring the quantum-classical transition.

Contribution

It provides a novel analytical calculation of the quantum tipping time of a rod using semi-classical path integrals, linking quantum behavior to classical limits.

Findings

Tipping time is derived analytically.

Quantum behavior transitions to classical in the limit.

Results align with the uncertainty principle.

Abstract

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 localised 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.