Quantum manifestations of classical stochasticity. I. Energetics of some nonlinear systems

In this paper we present the results of a semiclassical investigation and a quantum mechanical study of the bound energy spectrum of the Henon–Heiles Hamiltonian (HHH) and of the Barbanis Hamiltonian (BH). We have derived a simple semiclassical formula for the energy levels E, and for their sensitivity $\mathrm{dE/d\epsilon }$ with respect to the strength ε of the nonlinear coupling for the HHH, and established general relations between E and its derivatives $d^n{\mathrm{E/d\epsilon }}^n\text{\hspace{0.17em}(n⩾1).}$ Numerical quantum mechanical computations of the energy levels were conducted for the HHH and for the BH. The nonlinear coupling constant was adjusted so that for the HHH there will be ∼150 states up to the classical critical energy $E_c$ and ∼300 states up to the dissociation energy $E_D.$ The E values were obtained by direct diagonalization using a basis containing 760 states, while the values of $\mathrm{dE/d\epsilon }$ were computed utilizing the Hellmann–Feynman theorem. Good agreement between the semiclassical and the quantum mechanical spectra was observerd well above $E_c.$ These results raise the distict possibility that the semiclassical approxmation for these nonlinear systems does not break down in the vicinity of $E_c$ and that the bound level structure does not provide a manifestation of the classical transition from quasiperiodic to chaotic motion.