Parity dependence on the orbital quantum number (l)

[soccer_dude13]

Hi,
I know that the angular momentum eigenfunctions in spherical coordinates are spherical harmonics, Y_lm ( \theta, \phi ) \propto (-1)^mP_lm(cos\theta)e^im\phi.
Applying the parity operator to Y_lm ( \theta,\phi ) means that \theta -> \pi - \theta and \phi -> \phi +\pi.
This implies that e^im\phi will pick up a (-1)^m factor. However, from the definition of the P_lm(cos\theta)'s I don't see how I can pick up a factor of (-1)^l-2m in order to give parity the final correct dependence of (-1)^l. In fact I don't see how we can creep up a dependence on l, at all.

 

[DrDu]

Consider first the special case of the P_ll, i.e. the case with m=l. The P_ll are just the ordinary Legendre functions which are even polynomials in x=cos theta for even l and odd polynomials for odd l. The associated Legendre Polynomials P_lm contain m further derivatives with respect to x so that the odd/evenness changes with m.