- #1
soccer_dude13
- 1
- 0
Hi,
I know that the angular momentum eigenfunctions in spherical coordinates are spherical harmonics, Ylm ( [tex]\theta[/tex], [tex]\phi[/tex] ) [tex]\propto[/tex] (-1)mPlm(cos[tex]\theta[/tex])eim[tex]\phi[/tex].
Applying the parity operator to Ylm ( [tex]\theta,\phi[/tex] ) means that [tex]\theta[/tex] -> [tex]\pi[/tex] - [tex]\theta[/tex] and [tex]\phi[/tex] -> [tex]\phi[/tex] +[tex]\pi[/tex].
This implies that eim[tex]\phi[/tex] will pick up a (-1)m factor. However, from the definition of the Plm(cos[tex]\theta[/tex])'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.
I know that the angular momentum eigenfunctions in spherical coordinates are spherical harmonics, Ylm ( [tex]\theta[/tex], [tex]\phi[/tex] ) [tex]\propto[/tex] (-1)mPlm(cos[tex]\theta[/tex])eim[tex]\phi[/tex].
Applying the parity operator to Ylm ( [tex]\theta,\phi[/tex] ) means that [tex]\theta[/tex] -> [tex]\pi[/tex] - [tex]\theta[/tex] and [tex]\phi[/tex] -> [tex]\phi[/tex] +[tex]\pi[/tex].
This implies that eim[tex]\phi[/tex] will pick up a (-1)m factor. However, from the definition of the Plm(cos[tex]\theta[/tex])'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.