Kramers equation numerical solution

Two coupled first order differential equations derived for the atomic central field problem within the relativistic framework are transformed to integral equations through the use of approximate Wentzel-Kramers-Brillouin solutions. It is shown that a finite charge density can be derived for a relativistic form of the Fermi-Thomas atomic model by appropriate attention to the boundary conditions. A numerical solution for the effective nuclear charge in the Xenon atom is calculated and fitted to a rational expression. 

VTST has also been applied to systems with two degrees of freedom coupled to a dissipative bath." Previous results of Berezhkovskii and Zitserman which predicted strong deviations from the Kramers-Grote-Hynes expression in the presence of anisotropic friction for the two degrees of freedom " were well accounted for. Subsequent numerically exact solution of the Fokker-Planck equation further verified these results. 

Numerical methods for finding bound-state solutions of Equation 1.15 are described in Section 1.3.1.3. However, in conceptual terms a considerable amount may be understood in terms of semiclassical arguments [7]. In semiclassical methods, the Schrodinger equation is expanded semiclassically in powers of h. The resulting first-order JWKB (Jeffreys-Wentzel-Kramers-Brillouin) quantization condition gives remarkably accurate results for the vibration-rotation energies E l of diatomic molecules ... 

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