# Parity of spin states

Since orbital angular momentum commutes with the parity operator and since both are hermitian it is possible to build a common basis. These are the spherical harmonics, whose parity is known.

Now, we also know that since the spin operator commutes with the parity operator we can build common eigenkets, but I have never been told which is the parity of a spin state.

Moreover, when calculating the parity of, for example, a particle I always multiply the contribution of the orbital angular momentum by the intrinsic parity of the particle, and the spin part is never mentioned.

I guess that the spin contribution is in that intrinsic parity. So i would like someone to tell me about the parity of the spin states and how they are related to the intrinsic parity of particles(if in fact my guess is right)

## 1 Answer

First off, if the spin state did have a parity attached to it, then the parity would have to be independent of the magnetic quantum number, or else rotational invariance would be violated. This means that, e.g., every spin-1/2 state would have the same parity. So, e.g., for an electron, that there can't be any way to split its intrinsic parity into a contribution from its intrinsic spin state and a contribution from the fact that it's an electron. I think the fact that this kind of factoring is uninteresting is fundamentally because the parity operator is defined in terms of the spatial coordinates, whereas intrinsic spin can't be generated by motion in space.

So in general, all we can do is assign a parity to the entire internal state of a particle. This parity is arbitrarily defined to be + for the proton, neutron, and electron, which then causes it to be - for, e.g., the pion. WP has a description of the fact that this involves an initial arbitrary choice: https://en.wikipedia.org/wiki/Parity_%28physics%29#Fixing_the_global_symmetries