I found that the derivation of most “google result” is not clear enough. So here is my derivation. Before process, people may need to review the pervious post.

Most people start from,

u_l = A( \hat{h}_l^- - S_l \hat{h}_l^+ )

that annoying me because it is somehow not “natural”. Why there is a “minus” sign? Why the \hat{h}_l^- is the first term? For my self, a more natural way is,

u_l = a \hat{h}_l^+ + b \hat{h}_l^-

where a, b are complex numbers, but that is still not so natural, because in numerical calculation, for simplicity, there is no complex number, we only have,

u_l = \alpha \hat{j}_l + \beta \hat{n}_l

The first term is alway there as it is the free solution and bounded at r = 0. the second term is caused by the potential.

The goal is to find a solution take the form

\displaystyle \psi = A \left( e^{i \vec{k} \cdot \vec{r}} + f(\theta) \frac{e^{ikr}}{r} \right)

where the first term is free wave and the second term is scattered wave. The solution for elastics scattering is

\displaystyle \psi = \sum C_l P_l (\cos\theta) \frac{u_l}{kr} \rightarrow \sum C_l P_l(\cos\theta) (\alpha \hat{j}_l + \beta \hat{n}_l)

we used the substitution,

\displaystyle R_l(r) = \frac{u_l(\rho)}{\rho},  \rho = kr.

The radial function can be solved using Rungu-Kutta method on the equation,

\displaystyle \frac{d^2}{d\rho^2} u_l = \frac{2 m_\mu}{\hbar^2} (V-E) u_l + \frac{l(l+1)}{\rho^2}

and the solution of u_l at far away is,

u_l \rightarrow  \alpha \hat{j}_l + \beta \hat{n}_l.

the arrow means r \rightarrow \infty. So, the problem is how to rewrite the solution. In the way, we will see how the phase shift or the S-matrix was found.

The free solution is the spherical wave,

\displaystyle e^{i \vec{k} \cdot \vec{r}} = \sum_l (2l+1) i^l P_l(\cos\theta) j_l(kr)

The spherical Bessel function j_l(kr) cna be express as Heankel function

h_l^{\pm} = n_l \pm i j_l \rightarrow e^{\pm i (kr - l \frac{\pi}{2})}

The + sign is outgoing wave.

\displaystyle u_l \rightarrow (\alpha \hat{j}_l + \beta \hat{n}_l)

\displaystyle = \frac{\alpha}{2i} (\hat{h}_l^{+} - \hat{h}_l^{-}) + \frac{\beta}{2}(\hat{h}_l^{+} + \hat{h}_l^{-})

\displaystyle = \frac{\alpha + i \beta}{2i} (\hat{h}_l^{+} - \hat{h}_l^{-}) + \beta \hat{h}_l^{+}

\displaystyle = (\alpha - i \beta ) \left( \frac{\hat{h}_l^+ - \hat{h}_l^-}{2i} + \frac{\beta}{\alpha - i \beta} \hat{h}_l^+\right)

\displaystyle = (\alpha - i \beta ) \left( \hat{j}_l + \frac{\beta}{\alpha - i \beta} \hat{h}_l^+\right)

Since the u_l should be normalized, we can se \alpha = \cos \delta and \beta = \sin\delta.

\displaystyle \frac{\beta}{\alpha - i \beta } = \sin(\delta) e^{i\delta}

We put u_l back

\displaystyle \psi \rightarrow \sum_l C_l P_l (cos\theta)(\alpha - i \beta ) \left( \hat{j}_l + \sin(\delta) e^{i\delta} \hat{h}_l^+\right)

By setting

\displaystyle C_l = A i^l \frac{2l+1}{\alpha - i \beta} ,

we have the first term is the free wave function. In the second term, \hat{h}_l^+ \rightarrow e^{i(kr - l \frac{\pi}{2}}) / kr . Notice that

e^{i l \frac{\pi}{2}} = i^{-l}

That cancel the i^l term in C_l . And we have

 \displaystyle f(\theta) = \sum (2l+1) P_l (\cos\theta) \frac{\sin(\delta) e^{i\delta}}{k}

some people will write the u_l as \hat{h}_l^{\pm} and the S-matrix,

\displaystyle u_l = \frac{\alpha + i \beta} {2i} \hat{h}_l^+ - \frac{\alpha - i \beta}{2i} \hat{h}_l^-

\displaystyle = -\frac{\alpha - i \beta}{2i} \left( \hat{h}_l^- - \frac{\alpha + i \beta}{\alpha - i \beta} \hat{h}_l^+ \right)

\displaystyle = A' (\hat{h}_l^- - S_l \hat{h}_l^+)


\displaystyle S_l =\frac{\alpha + i \beta}{\alpha - i \beta} = e^{2i\delta} .

Remember that this is the S-matrix for elastics scattering.