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

Journal Article · Sun Jul 01 00:00:00 EDT 1984 · Phys. Rev. A; (United States)

DOI:https://doi.org/10.1103/PhysRevA.30.5· OSTI ID:6666693

De Leon, N; Heller, E J

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 gamma-arrow-right 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 gamma-arrow-right 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 gamma-arrow-right coordinates; (2) the WKB wave function Psi/sup WKB/ is separable in gamma-arrow-right coordinates; and (3) if the transformation to gamma-arrow-right coordinates is made conformal (which we show we are allowed to do within the approximation stated above), then the Schroedinger 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.

• Research Organization: Los Alamos National Laboratory, University of California, Los Alamos, New Mexico 87540
• OSTI ID: 6666693
• Journal Information: Phys. Rev. A; (United States), Vol. 30:1
• Country of Publication: United States
• Language: English

Related Subjects

71 CLASSICAL AND QUANTUM MECHANICS
GENERAL PHYSICS
CLASSICAL MECHANICS
HAMILTON-JACOBI EQUATIONS
QUANTIZATION
BOUND STATE
PHASE SPACE
SEMICLASSICAL APPROXIMATION
VECTOR FIELDS
WKB APPROXIMATION
DIFFERENTIAL EQUATIONS
EQUATIONS
MATHEMATICAL SPACE
MECHANICS
PARTIAL DIFFERENTIAL EQUATIONS
SPACE
657001* - Theoretical Physics- General & Miscellaneous- (-1987)