The normalized angular wave functions are called spherical harmonics: $$Y^m_l(\theta,\phi)=\epsilon\sqrt{\frac{(2l+1)}{4\pi}\frac{(l-|m|)!}{(l+|m|)!}}e^{im\phi}*P^m_l(cos\theta)$$(adsbygoogle = window.adsbygoogle || []).push({});

How do I obtain this from this(http://www.physics.udel.edu/~msafrono/424-2011/Lecture 17.pdf) (Page 8)?

The normalisation condition for Y is $$\int^{2\pi}_0\int^{\pi}_0|Y|^2\sin\theta d\theta d\phi=1$$ Note that 0<∅<2π and 0<θ<π.

$$Y=\Theta(\theta)*\Phi(\phi)$$.

$$\Theta(\theta)=A*P^m_l(cos\theta)$$ where A is the coefficient, x=cosθ, P is the associated Legendre Polynomial.

$$\Phi(\phi)=e^{im\phi}$$

$$Y^m_l(\theta,\phi)=A*e^{im\phi}P^m_l(cos\theta)$$

where $$A=\epsilon\sqrt{\frac{(2l+1)}{4\pi}\frac{(l-|m|)!}{(l+|m|)!}}$$.

I know how to obtain Φ but not A from Θ(θ), which is why I don't know how to obtain the spherical harmonics equation above.

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# I Spherical Harmonics

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