- #1
touqra
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In solving the 3D hydrogen atom, we obtain a spherical harmonic, Y such that,
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)
where \epsilon = (-1)^m for m \geq 0 and \epsilon = 1 for m \leq 0.
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 l - |m| = 0. Which means the wavefunction should be zero when m = l.
Where did I go wrong?
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)
where \epsilon = (-1)^m for m \geq 0 and \epsilon = 1 for m \leq 0.
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 l - |m| = 0. Which means the wavefunction should be zero when m = l.
Where did I go wrong?
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