Oscillatory states of quantum Kapitza pendulum

Pavel A. Golovinski, Vladimir A. Dubinkin

Thai M. Hoang, Corey Gerving, Ben Land +3 more

J.R. Choi

Pavel A. Golovinski, Vladimir A. Dubinkin

The quantum states of the Kapitza pendulum are investigated within the framework of the effective potential obtained by the method of averaging over fast oscillations. An analytical estimate of the energy spectrum of stabilized states is given using a model potential. For the lowest states of the inverted pendulum, an expression is obtained for the spectrum in the form of the energies of a harmonic oscillator, refined according to the perturbation theory. Tunneling corrections to the energies of resonance states in two wells of the effective potential are found. The results of calculations of the structure of vibrational and rotational spectra of the Kapitza pendulum by the semiclassical method and by Numerov’s numerical algorithm are compared.

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.

J. J. Duistermaat