JWKB method

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]. 

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

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] 

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 

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