\hat{H} (R Y_{l m}) = Y_{l m} [- \frac{1}{2 r ^{2}} \frac{d}{d r} (r^{2} R^{'}) + \frac{l ( l + 1 )}{2 r ^{2}} R - \frac{R}{r}] .

Consequently

R Y_{l m}

is an energy eigenfunction provided the radial factor satisfies

- \frac{1}{2 r ^{2}} \frac{d}{d r} (r^{2} R^{'}) + [\frac{l ( l + 1 )}{2 r ^{2}} - \frac{1}{r}] R = ER .

R

does not give an eigenfunction; regularity, decay and wavefunction normalization select admissible bound quantum states.

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