For the free particle with quantum number $l=0$, the regular solution to the radial Schrodinger equation is $R_0 (\rho)=\frac{\sin{\rho}}{\rho}$ while the irregular solution is $R_0 (\rho)=\frac{\cos{\rho}}{\rho}$. Is there a reason for this nomenclature -- (ir)regular? Thanks.
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The reason for this nomenclature is the behavior at $r=0$: $$\lim_{r\to0^{+}}\frac{\sin r}{r} = 1,$$ $$\lim_{r\to0^{+}}\frac{\cos r}{r} = \infty .$$ $\frac{\sin r}{r}$ is regular at $r=0$ while $\frac{\cos r}{r}$ is irregular. |
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