Quantum Chaology of Double Rod Pendulum

The double rod pendulum is a well known classic chaotic system, so its quantum version is an ideal laboratory to test various diagnosis for quantum chaos. We quantise this system canonically and calculate its lowest $10^4$ eigenvalues and eigenstate wave functions with at least $10^{-4}$ relative precision by the spectral analysis method. With these eigenvalues and eigenstate wave functions, we calculate and examine the three popular diagnosis on quantum chaos. On the NNSD diagnosis, we find that, either the GOE feature of NNSD is not a necessary condition for a quantum system to be chaotic at classic limit, or the double rod pendulum is not strong chaotic at the classic level. On the OTOC diagnosis, we observed that the early time exponential growth and late time constance approaching feature of OTOC is well conformed by the double rod pendulum. On the CC diagnosis, the status is similar with NNSD. Its linear growth feature at long time limit is either not a good diagnosis for a quantum system to be chaotic at classic limit or the double rod pendulum is not a strong chaotic system at classic levels.