Question

how to find n,l from radical part of schrodingers eqn

Answer 1

The quantum numbers $n$ and $l$ are determined by the mathematical properties and boundary conditions of the radial wave function's solution to the Schrödinger equation. $l$ is determined by the behavior of $R(r)$ near $r=0$, where $R(r) \propto r^l$. $n$ is determined by the quantization of energy levels ($E_n$) and the number of radial nodes ($n-l-1$).

Answer 2

The radial part of the Schrödinger equation is: $\frac{1}{r^2} \frac{d}{dr} \left( r^2 \frac{dR}{dr} \right) + \frac{2\mu}{\hbar^2} [E - V(r)] R - \frac{l(l+1)}{r^2} R = 0$ For $R(r)$ to be finite at $r \to 0$, $R(r) \propto r^l$, which determines $l$. Normalizability and bound state conditions lead to quantized energy levels $E_n$, determining $n$. The number of radial nodes is $n-l-1$.