JWKB wavefunction

The JWKB wavefunction defined by Eq. (5.1.3) is appropriate for either bound (two turning points) or unbound states (one turning point), provided that R is restricted to the region where E > V (R). The normalization factor for the unbound yJWKB at energy E is... 

FIGURE 6.3 wavefunctions for the last I -wave i = — 1 bound state (solid line) with E i ofh = —10.6 MHz and for the I -wave scattering state (dashed line) for E/h = 0.02 MHz E/k-Q = 1 xK) for two atoms. Both wavefunctions are given a common JWKB... 

However, the more peaked is the scaling function, the more singular becomes the scaling transform s behavior at a = T = 0. We prefer to work with scaling transforms that are regular in a. This means that the scaling function cannot decay faster than the JWKB zeroth order estimate for the physical wavefunction. 

The scalet equation is analytic at e = 0. This is immediately apparent from the M matrix s regular e dependence in Eq.(1.92). This is true in general, for any rational fraction potential, as long as the underlying scaling function dies off more slowly than the lowest order JWKB estimate for the physical wavefunction. 

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