WKBJ approximation

Expression (13.80) is called WKBJ (ray theoretic) Green s function, because it is associated with the names of several physicists, G. Wentzel, H. A. Kramers, L. Brillouin, and H. Jeffreys, who independently introduced this approximation in connection with the solution of different physical problems (Morse and Feshbach, 1953). I would also recommend an excellent book by Bleistein et al. (2001), where the interested reader can find a more thorough mathematical analysis of the WKBJ approximation. 

In the special case of a 1-D model, the WKBJ approximation of Green s function takes the form (Bleistein et al., 2001, p. 69)... 

Note that the 1-D formula (13.83) contains one additional negative power of u in comparison with a 3-D WKBJ approximation (13.80). This term was introduced by analogy with the expression (13.79). Certainly, in the model with a constant wavespeed, c z) = cq, formula (13.83) naturally reduces to the expression for a 1-D Green s function in a homogeneous medium ... 

We would like to develop a similar approach to 1-D inversion in a medium with variable background wavespeed. Unfortunately, in a general case there is no simple analytical expression for the Green s function. However, when investigating a high frequency acoustic field, one can use the WKBJ approximation (13.83) for the calculation of the Green s function ... 

Thus, we arrive at integral equation (15.106) for the anomalous square slowness. The key to the solution of this equation for a general variable-background wavespeed is in using the WKBJ approximation (13.80) for the Green s function. By substituting (13.80) into (15.106), we finally find... 

III.G., the WKBJ approximation is applied to retrieve the analytic expression derived by Brown using the Kramers transition state theory for the longest relaxation time for uniaxial anisotropy in the limit of high potential barriers. This is also extended to yield the formula for the relaxation time for high uniaxial anisotropy in the presence of a longitudinal field. 

This approach is named after its founders—Wentzel (1926), Kramers (1926), Brillouin (1926a,b), and Jeffreys (1925). The WKBJ method is one of the powerful approximate approaches of quantum mechanics. Although in the present discussion we are concerned only with its application to obtaining approximate eigenvalues for bound states of the onedimensional Schrodinger equation, such application does not cover all of its range and its force. 

The WKBJ method enables us to obtain the approximate expression for the total number of vibrational states for a given diatomic potential function. 

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