- #1
touqra
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In solving the 3D hydrogen atom, we obtain a spherical harmonic, Y such that,
[tex]Y_{lm}(\theta,\phi) = \epsilon\sqrt{\frac{(2l+1)}{(4\pi)}}\sqrt{\frac{(l-|m|)!}{(l+|m|)!}}e^{im\phi}P^m_l(cos \theta)[/tex]
where [tex]\epsilon = (-1)^m[/tex] for [tex]m \geq 0[/tex] and [tex]\epsilon = 1[/tex] for [tex]m \leq 0[/tex].
In quantum, m = -l, -l+1, ..., l-1, l.
But according to the formula above, when m = l, we should have zero and not a finite value, since [tex]l - |m| = 0[/tex]. Which means the wavefunction should be zero when m = l.
Where did I go wrong?
[tex]Y_{lm}(\theta,\phi) = \epsilon\sqrt{\frac{(2l+1)}{(4\pi)}}\sqrt{\frac{(l-|m|)!}{(l+|m|)!}}e^{im\phi}P^m_l(cos \theta)[/tex]
where [tex]\epsilon = (-1)^m[/tex] for [tex]m \geq 0[/tex] and [tex]\epsilon = 1[/tex] for [tex]m \leq 0[/tex].
In quantum, m = -l, -l+1, ..., l-1, l.
But according to the formula above, when m = l, we should have zero and not a finite value, since [tex]l - |m| = 0[/tex]. Which means the wavefunction should be zero when m = l.
Where did I go wrong?
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