Vector fields, line integrals, and the Hamilton-Jacobi equation: Semiclassical quantization of bound states

In 1972, P. Pechukas [J. Chem. Phys. 57, 5577 (1972)] proposed "classical states" associated with Miller-Good transformations, semiclassical quantization, and the Hamilton-Jacobi equation. In this paper we use some of the concepts established by Pechukas and extend them to define generalized curvilinear coordinates →γ associated with nonlinear dynamical systems with Hamiltonians of the form H=T(→p)+V(→q). For reasons discussed in the paper we call these coordinates "nodal" coordinates. We show that a transformation to nodal coordinates is to a very good approximation dependent only on Cartesian variables. Using this approximation we demonstrate that, within the regular regime of phase space, canonical transformation to →γ coordinates simplifies the analysis of the classical and quantum mechanics of dynamical systems. Our fundamental conclusions are as follows: (1) The solution of the Hamilton-Jacobi equation is separable in →γ coordinates; (2) the WKB wave function ΨWKB is separable in →γ coordinates; and (3) if the transformation to →γ coordinates is made conformal (which we show we are allowed to do within the approximation stated above), then the Schr\"odinger equation and wave functions become separable. Finally, the concepts in this paper are discussed for systems of two degrees of freedom but can be generalized to more degrees of freedom.