I was reading the following article regarding solution of wavefunction of hydrogen :(adsbygoogle = window.adsbygoogle || []).push({});

http://skisickness.com/2009/11/22/

To solve the angular part they gave the substitution of [itex] y = \sin \theta [/itex] and then assumed that Y is a polynomial i.e. [itex] Y(y) = \sum b_n x^n [/itex] and then arrived at the recursion formula :

[itex] b_{n-2} = - \frac{n^2 - m^2}{l(l+1) - (n-1)(n-2) }b_{n} [/itex] , where [itex]l[/itex] is maximum value of [itex]n[/itex] for which [itex]b_n[/itex] is non-zero.

Then they say that :

" There must be a minimum value of n; otherwise, the series will diverge at y=0. Given l, for the series to converge, it is necessary that |m|=l-2k, with k greater than or equal to zero and less than or equal to l/2. For even or odd l, this series gives l+1 solutions. This solution gives the eigenfunctions with both odd or both even m and l. "

I did not understand why the series converges only for this particular condition. Is it something to do with starting with [itex]b_l[/itex] and then finding [itex]b_{l-2}[/itex] and then [itex]b_{l-4}[/itex] and so on in terms of [itex]b_l[/itex].

Thanks for help!

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# Why should the quantum number m be less than equal to l.

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