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Based on the description in wikipedia and the book: Modern Quantum Mechanics (Sakurai & Napolitano), any product of two complex spherical harmonics follows the contraction rule: $$Y_{\ell_1}^{m_1}Y_{\ell_2}^{m_2}=\sum_{\ell=|\ell_1-\ell_2|}^{\ell_1+\ell_2}\sum_{|m|\le\ell}C_{\ell_1,\ell_2,\ell}^{m_1,m_2,m}Y_{\ell}^m,$$ where the constant $C_{\ell_1,\ell_2,\ell}^{m_1,m_2,m}$ can be determined via the Wigner's 3-j symbols. I am wondering if there exists a similar property for the real spherical harmonics, which is defined as the real/image part for the complex spherical harmonics: $$Y_{\ell,m}=\sqrt2(-1)^m\Re Y_\ell^m,Y_{\ell,-m}=\sqrt2(-1)^m\Im Y_\ell^m,m>0.$$ In other words, I would like to show something like $$Y_{\ell_1,m_1}Y_{\ell_2,m_2}=\sum_{\ell=|\ell_1-\ell_2|}^{\ell_1+\ell_2}\sum_{|m|\le\ell}C_{\ell_1,\ell_2,\ell}^{m_1,m_2,m}Y_{\ell,m}.$$

Here is what I have explored so far:

The 3-j symbols vanish unless $m=m_1+m_2$, so by the contraction rule of complex spherical harmonics as well as the definition $$Y_{\ell}^m=\sqrt{\frac{2\ell+1}{4\pi}\frac{(\ell-m)!}{(\ell+m)!}}P_\ell^me^{im\varphi}$$ we have a similar product formula for the associated Legendre polynomial $$P_{\ell_1}^{m_1}P_{\ell_2}^{m_2}=\sum_{\ell=|\ell_1-\ell_2|}^{\ell_1+\ell_2}A_{\ell_1,\ell_2,\ell}^{m_1,m_2}P_{\ell}^{m_1+m_2}$$ with $A_{\ell_1,\ell_2,\ell}^{m_1,m_2}$ being constants. Since the real spherical harmonics can be written as $$Y_{\ell,m}\simeq P_\ell^m\sin m\varphi,\textrm{ or }P_\ell^m\cos m\varphi$$ up to a mutiplicative constant depending on the sign of $m$, it suffices to show that $P_{\ell_1}^{m_1}P_{\ell_2}^{m_2}\cos m_1\varphi\cos m_2\varphi$ is a linear combination of real spherical harmonics (assuming $m_1$ and $m_2$ are both positive for now). But it seems that the term $\cos(m_1-m_2)\varphi$ produced by $\cos m_1\varphi\cos m_2\varphi$ cannot be expressed as a summation of real spherical harmonics if multiplied with $P_{\ell}^{m_1+m_2}$.

Any help would be appreciated.

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  • $\begingroup$ The more elegant proof is go back to the space of polynomials on $\mathbb{R}^3$. $\endgroup$ Sep 5, 2023 at 11:31

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Thanks for the hint from @user10354138. In fact, the unnormalized $P_\ell^m\cos m\varphi$ and $P_\ell^m\sin m\varphi$ form a standard basis for spherical harmonics of degree $\ell$, and in particular, they are also polynomials of degree $\ell$. Hence the multiplication of $P_{\ell_1}^{m_1}$ and $P_{\ell_2}^{m_2}$ is a polynomial of degree $(\ell_1+\ell_2)$, which can be expressed by a finite sum of spherical harmonics of degree less than $(\ell_1+\ell_2)$. One may refer to Section 4.1 of the book Spherical Harmonics and Approximations on the Unit Sphere: An Introduction (Kendall Atkinson and Weimin Han) for details on the polynomial spaces. As a consequence, we have $$Y_{\ell_1,m_1}Y_{\ell_2,m_2}=\sum_{\ell\le\ell_1+\ell_2}\sum_{|m|\le\ell}C_{\ell_1,\ell_2,\ell}^{m_1,m_2,m}Y_{\ell,m}.$$

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