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\begin{document}

High energy scattering off of an attractive potential can be handled
by the WKB approximation.

\( \psi _{r>0}=e^{i\delta _{0}}cos(\delta _{0})j_{0}(kr) \) goes to \( sin(k_{0}r+\delta _{0})/r \) in the limit of large r. \( k_{0}=\sqrt{\frac{2mE}{\hbar ^{2}}} \).

apply WKB to U(r)=\( r*\psi (r) \) for the S-wave

Then \( U=\frac{C}{\sqrt{k(r)}}sin(\int _{0}^{r}k(r)dr) \)\( =\frac{C}{\sqrt{k(r)}}sin(k_{0}r-\int _{0}^{r}(k(r)-k_{0})dr) \) This means \( \delta _{0}=\int _{0}^{\infty }(k(r)-k_{0})=\sqrt{\frac{2mE}{\hbar ^{2}}}\int _{0}^{\infty }dr(\sqrt{1-\frac{V(r)}{E}}-1) \) In the limit of high energy, \( \delta _{0}=\frac{-k_{0}}{2E}\int _{0}^{\infty }V(r)dr \) for V\(<\)0,
you get \( \delta _{0} \)\(>\)0 = a phase advance.

time delay is \( 2\hbar \frac{\partial \delta _{0}}{\partial E}=2\int _{0}^{\infty }(\frac{1}{v(x)}-\frac{1}{v_{0}})dr \) where v is the velocity.

Also works for repulsive potentials as long as the energy is sufficiently
large.

\end{document}
