# [SOLVED]Spherical Harmonics easy question

#### dwsmith

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

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
$$Y_{\ell}^m = \sqrt{\frac{(2\ell + 1)(\ell - m)!}{4\pi(\ell + m)!}}P^m_{\ell}(\cos\theta)e^{im\varphi}$$

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