I stumbled upon the following question in a textbook. The direct translation would be, for a potential: $$ V(r)=\frac{\alpha}{r^2} $$ perform partial wave analysis. What I assume this means is, use partial wave expansion to find $\delta_l$, the scattering phases and $\sigma_l$. Both are directly connected via: $$ \sigma_l=\frac{4\pi}{k^2}(2l+1)\sin^2(\delta_l) $$ As question b) is "use the Born Approximation", I would assume that section a) explicitly does not want us to use the Born Approximation. So for better or worse, I presume the only thing that can be done is write down the partial wave expansion of the incoming plain wave: $$ \exp(-ikz)=\sum_l (2l+1)P_l(\cos(\theta))i^l\left(\exp(i(kr-l\frac{\pi}{2}))-\exp(-i(kr-l\frac{\pi}{2}))\right)\frac{1}{ikr} $$ write down a plain wave example for the wave function in side the potential and somehow derive $\delta_l$ with term by term comparison. This, I hoped, could be done via assuming that $\psi_{in}(a)=\psi_{ext}(a)$, and if need be take the limit $a \rightarrow \infty$. However, the interior solution is a bit weird. The potential is spherically symmetric, so we can assume $\psi(r,\theta,\phi)=R(r)Y_{lm}(\theta,\phi)$ And then write down the usual radial equation, where everything has been renormalized $(\beta=\alpha*(2m/\hbar^2))$: $$ \left[\frac{d^2}{dr^2}-\frac{l(l+1)}{r^2}+\frac{\beta}{r^2}+E\right]u(r)=0 $$ The Bessel functions are the solutions too: $$ \left[\frac{d^2}{dr^2}-\frac{l(l+1)}{r^2}+1\right]u(r)=0 $$ Motivated by this, we can rewrite: $$ \beta=C_ll(l+1) $$ Leaving us with: $$ \left[\frac{d^2}{dr^2}-\frac{(1-C_l)(l(l+1))}{r^2}+E\right]u(r)=0 $$ Checking with a table gives us the following, amazing solution: $$ u(r)=c_1\sqrt{r}J_{0.5\sqrt{4(1-C_l)l(l+1)+1}}(\sqrt{E}r)+c_2\sqrt{r}Y_{0.5\sqrt{4(1-C_l)l(l+1)+1}}(\sqrt{E}r) $$ where $j_l(x)$ is the Bessel function of the first kind and $Y_l$ is the Bessel function of the second kind. Because the second kind diverges at $r=0$, we know $c_2=0$. Up to this point the question has been reasonably difficult, but now comes the really weird part. While all mentioned Bessel function do exist for non integer $l$, the incoming wave consists only of integer $l$ in the $j_l$. It is now entirely possible that $C_l$ is such, that the index simply is no integer. What does this mean? Does this mean the Ansatz is bad, and no stitching to gether of the solution can be done? Also for some $C$, the index is never $1$, or $0$, what does this imply? Naively, this would mean, that even if a wave with $l=0$ comes in, it doesnt leave with $l=0$. This would mean, that angular momentum is not preserved, which clashes with the obvious rotationally symmetry of $V(r)$.

So the deeper questions to me are:

A) Can Partial wave analysis be meaningfully applied here?

B) If yes, which mistake did I make

C) If no, how else to tackle this, if Born Approximation is explicitly disallowed?



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