Solution of Radial Function (Hydrogen Atom)
Derive ![\Big[\frac{d^{2}}{d\rho^{2}} - 2\lambda_{k,l} \frac{d}{d\rho} + \Big(\frac{2}{\rho} - \frac{l(l+1)}{\rho^{2}}\Big)\Big]Y_{k,l}(\rho)=0](https://s0.wp.com/latex.php?latex=%5CBig%5B%5Cfrac%7Bd%5E%7B2%7D%7D%7Bd%5Crho%5E%7B2%7D%7D+-+2%5Clambda_%7Bk%2Cl%7D+%5Cfrac%7Bd%7D%7Bd%5Crho%7D+%2B+%5CBig%28%5Cfrac%7B2%7D%7B%5Crho%7D+-+%5Cfrac%7Bl%28l%2B1%29%7D%7B%5Crho%5E%7B2%7D%7D%5CBig%29%5CBig%5DY_%7Bk%2Cl%7D%28%5Crho%29%3D0&bg=ffffff&fg=000000&s=0 "\Big[\frac{d^{2}}{d\rho^{2}} - 2\lambda_{k,l} \frac{d}{d\rho} + \Big(\frac{2}{\rho} - \frac{l(l+1)}{\rho^{2}}\Big)\Big]Y_{k,l}(\rho)=0")
Solution:
To solve the equation derived from problem (1), we look at the asymptotic behavior. First, let us assume a series of solution. But we must take into account as our 
A solution of the form:
But 
Substituting gives
Taking the first term:
![\frac{d^{2}}{d\rho^{2}}\Big(e^-\rho{\lambda_{k,l}}Y_{k,l}(\rho)\Big) = \lambda_{k,l}e^-\rho{\lambda_{k,l}}Y_{k,l}(\rho)+e^-\rho{\lambda_{k,l}}\frac{d}{d\rho}\big[Y_{k,l}(\rho)\big]](https://s0.wp.com/latex.php?latex=%5Cfrac%7Bd%5E%7B2%7D%7D%7Bd%5Crho%5E%7B2%7D%7D%5CBig%28e%5E-%5Crho%7B%5Clambda_%7Bk%2Cl%7D%7DY_%7Bk%2Cl%7D%28%5Crho%29%5CBig%29+%3D+%5Clambda_%7Bk%2Cl%7De%5E-%5Crho%7B%5Clambda_%7Bk%2Cl%7D%7DY_%7Bk%2Cl%7D%28%5Crho%29%2Be%5E-%5Crho%7B%5Clambda_%7Bk%2Cl%7D%7D%5Cfrac%7Bd%7D%7Bd%5Crho%7D%5Cbig%5BY_%7Bk%2Cl%7D%28%5Crho%29%5Cbig%5D&bg=ffffff&fg=000000&s=0 "\frac{d^{2}}{d\rho^{2}}\Big(e^-\rho{\lambda_{k,l}}Y_{k,l}(\rho)\Big) = \lambda_{k,l}e^-\rho{\lambda_{k,l}}Y_{k,l}(\rho)+e^-\rho{\lambda_{k,l}}\frac{d}{d\rho}\big[Y_{k,l}(\rho)\big]")
Therefore,
![\Big[\frac{d^{2}}{d\rho^{2}} - 2\lambda_{k,l}\frac{d}{d\rho} + \Big(\frac{2}{\rho}-\frac{l(l+1)}{\rho^{2}}\Big)\Big] Y_{k,l}(\rho)=0](https://s0.wp.com/latex.php?latex=%5CBig%5B%5Cfrac%7Bd%5E%7B2%7D%7D%7Bd%5Crho%5E%7B2%7D%7D+-+2%5Clambda_%7Bk%2Cl%7D%5Cfrac%7Bd%7D%7Bd%5Crho%7D+%2B+%5CBig%28%5Cfrac%7B2%7D%7B%5Crho%7D-%5Cfrac%7Bl%28l%2B1%29%7D%7B%5Crho%5E%7B2%7D%7D%5CBig%29%5CBig%5D+Y_%7Bk%2Cl%7D%28%5Crho%29%3D0&bg=ffffff&fg=000000&s=0 "\Big[\frac{d^{2}}{d\rho^{2}} - 2\lambda_{k,l}\frac{d}{d\rho} + \Big(\frac{2}{\rho}-\frac{l(l+1)}{\rho^{2}}\Big)\Big] Y_{k,l}(\rho)=0")
