MHB Spherical Harmonics easy question

AI Thread Summary
The discussion centers on the normalization of spherical harmonics, specifically for the case where both $\ell$ and $m$ equal 1. The user calculates the spherical harmonic $Y_1^1$ and arrives at a result of $\frac{1}{2}\sqrt{\frac{3}{2\pi}}e^{i\varphi}\sin \theta$, while Mathematica provides a solution of $-\frac{1}{2} e^{i\varphi} \sqrt{\frac{3}{2\pi}} \sin\theta$. The discrepancy arises from differences in normalization conventions used in various texts. One participant suggests that their book incorporates a factor of $(-1)^m$ in the definition, which could explain the negative sign in Mathematica's output. Understanding these normalization differences is crucial for consistent results in spherical harmonics calculations.
Dustinsfl
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$$
Y_{\ell}^m = \sqrt{\frac{(2\ell + 1)(\ell - m)!}{4\pi(\ell + m)!}}P^m_{\ell}(\cos\theta)e^{im\varphi}
$$

For $\ell = m = 1$, we have
$$
\sqrt{\frac{(2 + 1)(0)!}{4\pi(2)!}}P^1_{1}(\cos\theta)e^{i\varphi} = \frac{1}{2}\sqrt{\frac{3}{2\pi}}e^{i\varphi}\sin \theta
$$

But Mathematica is telling me the solution is
$$
-\frac{1}{2} e^{i\varphi} \sqrt{\frac{3}{2\pi}} \sin\theta
$$

What is going wrong?
 
Last edited:
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dwsmith said:
$$
Y_{\ell}^m = \sqrt{\frac{(2\ell + 1)(\ell - m)!}{4\pi(\ell + m)!}}P^m_{\ell}(\cos\theta)e^{im\varphi}
$$

For $\ell = m = 1$, we have
$$
\sqrt{\frac{(2 + 1)(0)!}{4\pi(2)!}}P^1_{1}(\cos\theta)e^{i\varphi} = \frac{1}{2}\sqrt{\frac{3}{2\pi}}e^{i\varphi}\sin \theta
$$

But Mathematica is telling me the solution is
$$
-\frac{1}{2} e^{i\varphi} \sqrt{\frac{3}{2\pi}} \sin\theta
$$

What is going wrong?
I'm not sure about how your book normalizes spherical harmonics, but mine has
Y_l^m (\theta, \phi) = (-1)^m \sqrt{\frac{(2l+1)(l-m)!}{4 \pi (l+ m)!}} P_l^m(cos(\theta)) e^{im \phi}

-Dan
 
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