Spherical harmonics

Question

Is it possible theta to become negative in spherical harmonics? If so, whats the qualitative idea of when this might happen, and perhaps an example where it does?

Answer 1

It depends on which convention for spherical coordinates you are using. If you're using $\theta$ for the angle from the north pole, that is by definition between $0$ and $\pi$ radians ($180$ degrees). If you're using $\theta$ for the longitude, there are some who allow it to be negative, but usually it's required to be from $0$ to $2\pi$ radians ($360$ degrees).

Answer 2

Supposing you use the convention that a vector ${\bf r}$ in polar coordinates is given by $$ {\bf r} = \left( \begin{array}{c} r \sin \theta \cos \phi \\ r \sin \theta \sin \phi \\ r \cos \theta \end{array} \right) $$ with $r \in [0,\infty), \theta \in [0,\pi],\phi \in [0,2 \pi]$ and the spherical harmonics $Y_l^m(\theta,\phi)$ then negative values of $\theta$ normally do not occur.

That does no mean however that negative values are forbidden, but that care should be taken. With the above restrictions each point in ${\mathbb R}^3$ has a unique $(r,\theta,\phi)$ description. Allowing negative values of $\theta$ would result in an additional different characterisation of points and is probably undesirable. This is usually solved by mapping the values back on the allowed domain, so if we consider a point $(r,\vartheta,\varphi)$ with $-\pi \leq \vartheta < 0$ we would obtain the point $$ {\bf r} = \left( \begin{array}{c} r \sin \vartheta \cos \varphi \\ r \sin \vartheta \sin \varphi \\ r \cos \vartheta \end{array} \right) = \left( \begin{array}{c} r \sin (-|\vartheta|) \cos \varphi \\ r \sin (-|\vartheta|) \sin \varphi \\ r \cos -|\vartheta| \end{array} \right) = \left( \begin{array}{c} - r \sin |\vartheta| \cos \varphi \\ - r \sin |\vartheta| \sin \varphi \\ r \cos |\vartheta| \end{array} \right) = \left( \begin{array}{c} r \sin |\vartheta| \cos (\varphi+\pi) \\ r \sin |\vartheta| \sin (\varphi+\pi) \\ r \cos |\vartheta| \end{array} \right) $$ So we map $(r, \vartheta,\varphi)$ for negative $\vartheta$ on $(r, |\vartheta|,\varphi+\pi)$. Strictly speaking we might have to use $\varphi-\pi$ to bring it back in to the proper domain.

The spherical harmonics themselves are not affected by this because $$Y_l^m(\theta,\phi)=Y_l^m(-\theta,\phi+\pi)$$ $$Y_l^m(\theta,\phi)=Y_l^m( \theta,\phi+2 \pi)$$ But other functions in polar coordinates might, for instance $\sin \theta$, which is not a spherical harmonic, would be an example where allowing negative angles would become an issue.