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[SOLVED] Spherical Harmonics easy question

dwsmith

Well-known member
Feb 1, 2012
1,673
$$
Y_{\ell}^m = \sqrt{\frac{(2\ell + 1)(\ell - m)!}{4\pi(\ell + m)!}}P^m_{\ell}(\cos\theta)e^{im\varphi}
$$

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:

topsquark

Well-known member
MHB Math Helper
Aug 30, 2012
1,123
$$
Y_{\ell}^m = \sqrt{\frac{(2\ell + 1)(\ell - m)!}{4\pi(\ell + m)!}}P^m_{\ell}(\cos\theta)e^{im\varphi}
$$

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
[tex]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}[/tex]

-Dan