JWKB method

The validity of equation (6.379) is restricted to classically accessible regions, where p(R) is real. It breaks down catastrophically at any classical turning points where p(R) is zero but methods are available to correct JWKB solutions around such singularities. Proper real combinations of i/f (R) in the region R < R < R2 may be expressed as [79, 80]... 

A numerical method for solving the Schrodinger radial equation, which makes use of JWKB wave functions, was described by Cooley [86], This method is readily available as a computer program called level from Le Roy [87] a helpful summary of the method is given by Cashion [88]. 

We share the opinion expressed by Farrelly and Reinhardt (1983) that discrepancies between Stark effect results obtained by the use of the Carlini (JWKB) approximation and by accurate numerical calculations cannot be attributed to the break-down of the approximation, but are due to a failure to use the approximation in a correct way. An appropriate approach based on the phase-integral approximation of arbitrary order generated from an appropriately chosen base function is a still more efficient and often highly accurate method for the treatment of several problems, not only in quantum mechanics, but in various fields of theoretical physics. With... 

The phase shifts rji are calculated by the method of JWKB approximation with Langer s modification - ... 

The JWKB semiclassical method The Schrodinger equation can be written in the form... 

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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