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Asymptotic Forms

Small x:

$\displaystyle \lim_{ x \rightarrow 0} j_\ell(x)$ $\displaystyle =$ $\displaystyle \frac{2^\ell \ell!}{(2 \ell + 1)!}
x^\ell$ (13.24)
$\displaystyle \lim_{ x \rightarrow 0} n_\ell(x)$ $\displaystyle =$ $\displaystyle - \frac{(2 \ell)!}{2^\ell \ell!}
\frac{1}{x^{\ell +1}}.$ (13.25)

Note that for small $ x$ ($ r << k$ ) $ j_\ell(kr)$ is proportional to $ r^\ell$ and $ n_\ell(kr)$ is proportional to $ 1/r^{\ell+1}$ , which are the regular and irregular solutions to the separated Laplace equation. This is the correct way to obtain the static limit.

Large x:

$\displaystyle \lim_{ x \rightarrow \infty} j_\ell(x)$ $\displaystyle =$ $\displaystyle \frac{1}{x} \cos(x - (\ell+1)
\frac{\pi}{2})$ (13.26)
$\displaystyle \lim_{ x \rightarrow \infty} n_\ell(x)$ $\displaystyle =$ $\displaystyle \frac{1}{x} \sin(x - (\ell+1)
\frac{\pi}{2}).$ (13.27)

Note that both solutions are regular (go to zero smoothly) at infinity and are the same (trig) function shifted by $ \pi/2$ over $ x$ there. Note that they are not square integrable on $ \RE^3$ (for your quantum course) but are still better than plane waves in that regard. Something to think about ...


next up previous contents
Next: Hankel Functions Up: Properties of Spherical Bessel Previous: The Lowest Few Functions   Contents
Robert G. Brown 2017-07-11