Normalisation constant of the spherical harmonics

The normalisation constant ensures that the spherical harmonics $Y_l^{\vert m_l\vert}(\theta,\phi)$ are properly scaled, thereby maintaining the probabilistic interpretation of quantum states.

It is defined mathematically as:

$N^2\int_0^{2\pi}\int_0^{\pi}Y_l^{\vert m_l\vert*}(\theta,\phi)Y_l^{\vert m_l\vert}(\theta,\phi)sin\theta d\theta d\phi=1\;\;\;\;\;\;\;\;414$

where $Y_l^{\vert m_l\vert}(\theta,\phi)=P_l^{\vert m_l\vert}(cos\theta)e^{im_l\phi}$ .

Substituting $x=cos\theta$ in eq390 gives

$\int_0^{\pi}P_l^{\vert m_l\vert}(cos\theta)P_l^{\vert m_l\vert}(cos\theta)sin\theta d\theta=\frac{(l+\vert m_l\vert)!}{(l-\vert m_l\vert)!}\frac{2}{2l+1}\;\;\;\;\;\;\;\;415$

Therefore, eq414 becomes

$N^2=\frac{(l+\vert m_l\vert)!}{(l-\vert m_l\vert)!}\frac{2}{2l+1}\int_0^{2\pi}d\phi=1$

$N=\frac{1}{\sqrt{2\pi}}\biggr\[\biggr$\frac{2l+1}{2}\biggr$\frac{(l-\vert m_l\vert)!}{(l+\vert m_l\vert)!}\biggr\]^{\frac{1}{2}}\;\;\;\;\;\;\;\;416$

Normalisation constant of the spherical harmonics

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