Orthogonality of the spherical harmonics

The orthogonality of the spherical harmonics $Y_l^{\vert m_l\vert}(\theta,\phi)$ states that the integral of the product of two distinct spherical harmonics over a specified interval is zero.

It is defined mathematically as:

$\int_0^{2\pi}\int_0^{\pi}Y_l^{\vert m_l\vert*}(\theta,\phi)Y_k^{\vert m_k\vert}(\theta,\phi)sin\theta d\theta d\phi=0\;\;\;\;\;l\neq k\;\;\;\;\;412$

Substituting eq400 in eq412 gives

$N_l^{\vert m_l\vert*}N_k^{\vert m_k\vert}\int_0^{2\pi}\int_0^{\pi}P_l^{\vert m_l\vert}(cos\theta)P_k^{\vert m_k\vert}(cos\theta)e^{i(\vert m_k\vert-\vert m_l\vert)\phi}sin\theta d\theta d\phi=0\;\;\;\;\;l\neq k$

In general, the integral over $\phi$ is

$\int_0^{2\pi}e^{i(\vert m_k\vert-\vert m_l\vert)\phi} d\phi=2\pi\delta_{ik}$

Since the associated Legendre polynomials $P_l^{\vert m_l\vert}(cos\theta)$ are orthogonal to one another, we have

Conditions	$\boldsymbol{\int_0^{2\pi}\int_0^{\pi}Y_l^{\vert m_l\vert*}Y_k^{\vert m_k\vert}sin\theta d\theta d\phi=0}$
$l\neq k\;and\;m_l\neq m_k$	zero
$l\neq k\;and\;m_l=m_k$	zero
$l=k\;and\;m_l\neq m_k$	zero
$l= k\;and\;m_l= m_k$	non-zero

Therefore,

$N_l^{\vert m_l\vert*}N_k^{\vert m_k\vert}\int_0^{2\pi}\int_0^{\pi}Y_l^{\vert m_l\vert*}(\theta,\phi)Y_k^{\vert m_k\vert}(\theta,\phi)sin\theta d\theta d\phi=\delta_{ik}\delta_{m_l}\delta_{m_k}\;\;\;\;\;413$

which is the expression showing the orthonormal property of the spherical harmonics.

Orthogonality of the spherical harmonics

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