Semiclassical eigenvalues for non-separable bound systems from classical trajectories

A new form of the semiclassical quantum conditions in non-separable systems is proposed. In two dimensions (2D) it has the form (ħ = 1) where CΣ is the path of a classical trajectory closed in phase space, Nx and Ny are the number of circuits in the x and y ‘senses’ on the invariant toroid and Jx and Jy are the ‘good’ action variables on the toroid; these action variables, Jx and Jy , must have the values 2π(n 1 + ½) and 2π(n 2 + ½) respectively where n 1 and n 2 are the integer quantum numbers. Closed classical trajectories occur only for the exceptional toroids with rational frequency ratios. In the general case we imply that the trajectory has closed on itself to some arbitrary accuracy. Results for the 2D potentials studied are in agreement with previously published work. It is shown how the method may be extended to 3D systems.

[1]  B. C. Garrett,et al.  Semiclassical eigenvalues for nonseparable systems: Nonperturbative solution of the Hamilton–Jacobi equation in action‐angle variables , 1976 .

[2]  I. Percival,et al.  Vibrational quantization of polyatomic molecules , 1976 .

[3]  D. W. Noid,et al.  Semiclassical calculation of bound states in a multidimensional system. Use of Poincaré’s surface of section , 1975 .

[4]  R. Marcus,et al.  Semiclassical calculation of bound states of a multidimensional system , 1974 .

[5]  Neil Pomphrey,et al.  Numerical identification of regular and irregular spectra , 1974 .

[6]  I. Percival Variational principles for the invariant toroids of classical dynamics , 1974 .

[7]  I. C. Percival,et al.  Regular and irregular spectra , 1973 .

[8]  E. Amaldi The unity of physics , 1973 .

[9]  R. Marcus Semiclassical theory for collisions involving complexes (compound state resonances) and for bound state systems , 1973 .

[10]  J. Leray,et al.  Théorie des perturbations et méthodes asymptotiques , 1972 .

[11]  W. Miller CLASSICAL-LIMIT GREEN'S FUNCTION (FIXED-ENERGY PROPAGATOR) AND CLASSICAL QUANTIZATION OF NONSEPARABLE SYSTEMS. , 1972 .

[12]  Joseph Ford,et al.  Amplitude Instability and Ergodic Behavior for Conservative Nonlinear Oscillator Systems , 1969 .

[13]  Carl Erik Fröberg,et al.  Introduction to Numerical Analysis , 1969 .

[14]  Ralph Abraham,et al.  Foundations Of Mechanics , 2019 .

[15]  M. Gutzwiller Phase-Integral Approximation in Momentum Space and the Bound States of an Atom , 1967 .

[16]  M. Hénon,et al.  The applicability of the third integral of motion: Some numerical experiments , 1964 .

[17]  G. Contopoulos A Classification of the Integrals of Motion. , 1963 .

[18]  V. I. Arnol'd,et al.  PROOF OF A THEOREM OF A.?N.?KOLMOGOROV ON THE INVARIANCE OF QUASI-PERIODIC MOTIONS UNDER SMALL PERTURBATIONS OF THE HAMILTONIAN , 1963 .

[19]  J. Gillis,et al.  Numerical Solution of Ordinary and Partial Differential Equations , 1963 .

[20]  Joseph B. Keller,et al.  Asymptotic solution of eigenvalue problems , 1960 .

[21]  Joseph B. Keller,et al.  Corrected bohr-sommerfeld quantum conditions for nonseparable systems , 1958 .