Semiclassical eigenvalues for non-separable bound systems from classical trajectories

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Abstract
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.