I was trying to find the s-wave scattering cross section due to a potential barrier $V_0>0$. For the solution near $r=0$. I have the DE,
$$\left[\frac{d^2}{dr^2}+\frac{2}{r}\frac{d}{dr}-\frac{l(l+1)}{{\hbar}^2} -U_0+k^2\right]y=0$$ where $U_0=\frac{2mV_0}{{\hbar}^2}$ and $k^2=\frac{2mE}{{\hbar}^2}$, $E$ and $V_0$ are constants. How do I solve this equation near $r=0$?
If $V_0<0$ then the solution is a spherical Bessel function.
Thanks for the help in advance.
On
You forgot to write the unknown in your equation. Let $y(r)$ be the unknown function.
$$\frac{d^2 y}{dr^2}+\frac{2}{r}\frac{dy}{dr}+\left(-l(l+1)/{\hbar}^2 -U_0+k^2\right)y=0$$ Why keeping so many constant parameters, which hides a very simple equation.
Let $\quad a=-l(l+1)/{\hbar}^2 -U_0+k^2$ $$\frac{d^2 y}{dr^2}+\frac{2}{r}\frac{dy}{dr}+ay=0$$ Change of function $\quad y(r)=\frac{u(r)}{r}\quad\to\quad y'=\frac{u'}{r}-\frac{u}{r^2}\quad\to\quad y''=\frac{u''}{r}-2\frac{u'}{r^2}+\frac{2u}{r^3}$ $$\left(\frac{u''}{r}-2\frac{u'}{r^2}\right)+\frac{2u}{r^3}+\frac{2}{r}\left(\frac{u'}{r}-\frac{u}{r^2}\right)+a\frac{u}{r}=0$$ $$\frac{d^2 u}{dr^2}+au=0$$ $$u(r)=c_1e^{\sqrt{-a}\:r}+c_2e^{-\sqrt{-a}\:r}$$ If $a>0$ replace the exponentials by sinusoidal functions. $$y(r)=\frac{1}{r}\left(c_1e^{\sqrt{-a}\:r}+c_2e^{-\sqrt{-a}\:r} \right)$$ To have approximates for $r$ close to $0$, expend the exponential (or sinusoidal) functions into series.
You didn't name the dependent variable, so I'll call it $\Phi$. Rewrite $\alpha^2 = k^2 - U_0 - l(l+1)/\hbar^2$ to obtain
$$ r^2 \frac{d^2\Phi}{dr^2} + 2r \frac{d\Phi}{dr} + \alpha^2r^2\Phi = 0 $$
Substitute $\rho = \alpha r$ we get
$$ \rho^2 \frac{d^2\Phi}{d\rho^2} + 2\rho \frac{d\Phi}{d\rho} + \rho^2\Phi = 0 $$
The solutions are spherical Bessel functions of order $0$
$$ j_0(\rho) = \frac{\sin \rho}{\rho} $$ $$ y_0(\rho) = -\frac{\cos \rho}{\rho} $$
If $\alpha^2 < 0$ then the solutions are modified spherical Bessel functions $$ i_0(\rho) = \frac{\sinh \rho}{\rho} $$ $$ k_0(\rho) = \frac{e^{-\rho}}{\rho} $$