The WKB Propagator

An approximate solution of Eq. (4.46) can be obtained for a general potential energy V (r) by means of the WKB method and expressions derived for a number of quantities. Using Kramers modification A = (i/ + ) in the differential equation (4.46), one defines the local wave number as 

Comment The WKB approximation uses a power series in ft and since the units chosen here have ft = 1 it may be useful to briefly outline the essence of the WKB approximation without those units. The solution is expressed as exp[iS(r)/fi] yielding 

The Wronskian of the two independent solutions is 2i and the Green s function, which should vanish when r or r equals zero or infinity, is 

The Green s function (4.60) has, for real v, a branch cut along the real E-axis, thus exhibiting a completely continuous spectrum. The spectral density function can be determined as discussed before. The complex energy variable is written d E = t + ip and the real quantity 

Several distinct cases can be recognized depending on whether r and r are in forbidden or allowed regions and the number of turning points, i.e., zeros of A (r), lie in the interval (r,r ). Assuming for simplicity that r is less than r, one can distinguish the following cases  

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