Stochastic behavior of a quantum pendulum under a periodic perturbation

Abstract

This paper discusses a numerical technique for computing the quantum solutions of a driver pendulum governed by the Hamiltonian

$$H = (p_\theta ^2 /2m\ell ^2 ) - [m\ell ^2 \omega _o ^2 \cos \theta ] \delta _p (t/T) ,$$

where p_e is angular momentum, ϑ is angular displacement, m is pendulum mass, l is pendulum length, ω_O ² = g/l is the small displacement natural frequency, and where δ_p (t/T) is a periodic delta function of period T. The virtue of this rather singular Hamiltonian system is that both its classical and quantum equations of motion can be reduced to mappings which can be iterated numerically and that, under suitable circumstances, the motion for this system can be wildly chaotic. Indeed, the classical version of this model is known to exhibit certain types of stochastic behavior, and we here seek to verify that similar behavior occurs in the quantum description. In particular, we present evidence that the quantum motion can yield a linear (diffusive-like) growth of average pendulum energy with time and an angular momentum probability distribution which is a time-dependent Gaussian just as does the classical motion. However, there are several surprising distinctions between the classical and quantum motions which are discussed herein.