Semiclassical eigenvalues for nonseparable systems: Nonperturbative solution of the Hamilton–Jacobi equation in action-angle variables

It is shown how the Hamilton–Jacobi equation for a multidimensional nonseparable system can be efficiently solved directly in action-angle variables. This allows one to construct the total (classical) Hamiltonian as a function of the ’’good’’ action-angle variables which are the complete set of constants of the motion of the system; requiring the action variables to be integers then provides the semiclassical eigenvalues. Numerical results are presented for a two-dimensional potential well, and one sees that the semiclassical eigenvalues are in good agreement with the exact quantum mechanical values even for the case of large nonseparable coupling.