The classical and quantum two-center harmonic-like problem

In this study, the classical and quantum planar two-center system in R2 with harmonic-like interactions is considered. For the values of the parameter a∈[1,4], which defines the equilibrium configuration, the dynamics of the classical system is investigated combining time-series, Poincaré sections, and largest Lyapunov exponents as a function of the energy E. In particular, periodic, quasi-periodic, and chaotic representative trajectories are described using symmetry lines and Fourier analysis. On the plane (a,E), the heatmap of averaged largest Lyapunov exponent is calculated explicitly. The region of maximal chaos (characterized by large Lyapunov exponents) is bounded by critical threshold curves that define the underlying multi-well topology of the potential. In addition, the Harmonic Time Averaging method is employed to visualize periodic and quasiperiodic sets in the (y,py) phase space. A comparison between the numerical results and those obtained by simulating the system’s dynamics using electronic components is presented as well. For the corresponding quantum system, the ground state energy E0=E0(a) is obtained using the variational method. Specifically, for the relevant interval a∈[1,2.5] a simple and compact 2-parametric trial function is analytically constructed based on a Padé approximant. For 0≤a≲1.9 no tunneling (instantons) effects occur. Interestingly, the variational energy E0(a) develops a minimum at a≃1.5 located in the most ordered configuration of the underlying classical dynamics.