Stability of Inverted Pendulum and Heisenberg Uncertainty Principle

George T. Stamatiou, Kostas Magoutis

Rohit Chawla, Soumyabrata Paul, Jayanta K. Bhattacharjee

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.

N. K. Tyagi

Gayle Cook, Clyde S. Zaidins

The question of how long it takes an inverted point-mass pendulum to fall according to classical, semiclassical, and quantum theories is examined. The semiclassical analysis is based on the uncertainty introduced by measurement.

John Stringer