# Symbol denoting parity eigenvalue

What is the symbol reserved for designating the parity of a parity eigenstate?

For example an eigenstate $$\phi$$ of the squared angular momentum operator $$\hat{\mathbf{L}}^2$$ is characterized by a symbol $$L$$ where $$\hat{\mathbf{L}}^2 \phi = \hbar^2 L (L + 1) \phi$$.

So if we have an operator $$\hat{P}$$ which performs the inversion $$\mathbf{x} \to -\mathbf{x}$$ of the spatial coordinates $$\mathbf{x}$$, we might use the symbol $$\sigma$$ to denote the eigenvalue of +1 or -1 of a parity eigenstate $$\psi$$, i.e. $$\hat{P}\psi = \sigma \psi$$. What is the conventional symbol we should use to replace $$\sigma$$ here? Is there any consistency across different branches of physics (particle physics, atomic physics, molecular physics, condensed matter, etc.)?

In nuclear physics the symbol is usually $$P$$. In that context the parity is rarely abbreviated without also using the total angular momentum. For example, the deuteron has $$J^P=1^+$$. I am trying to recall whether I have seen $$J^\pi$$ used by authors who really like non-Latin letters. I feel like that's unfashionable; however, the Wikipedia article on forbiddenness in beta decays uses $$\Delta\pi$$ for "change in parity," and that notation was so unsurprising that I nearly forgot about it as a contemporary and actively-maintained example.

In atomic physics one generally uses the entire term symbol $$^{2S+1}L_J$$, with a letter representing the orbital angular momentum $$L$$; for example, the noble gases have electronic ground state $$^1S_0.$$ New students must learn the ordered sequence $$L\in\{S,P,D,F,G,H,\cdots\}$$, and must also learn that the overall parity is $$(-1)^L$$, so it's unusual to abbreviate the parity on its own.

If the focus of a discussion is just on the parity, it's more common to use text like "positive parity" or "parity-even" than a symbolic phrase like "$$P=+1$$," even in contexts where one might write prefer "$$L=2$$" for the orbital angular momentum instead of "$$d$$-wave."

My experience is that nontrivial molecules are generally not eigenstates of parity, so parity eigenvalues generally don't arise in discussions of molecules or larger assemblies, at least with enough frequency that there would be a standard symbol.

• Excluding extremely weak parity-violation effects associated with the weak nuclear interaction, one can in fact assign definite parities to molecule eigenstates to describe their behavior under inversion of the spatial coordinates. That is of course not to say that every molecule has an inversion symmetric equilibrium structure. Ammonia (NH3) is a good example, which possesses large "inversion splittings" in its infrared spectra, with state pairs differentiated by their parity under inversion, even though its equilibrium structure is not inversion symmetric (though it is achiral). Commented Jul 19 at 16:07
• You have correctly detected my background as a weak nuclear interaction person who does not know much about molecules. By "nontrivial" I was thinking of things like lipid chains, proteins, or antelopes, all of which are equally mysterious from my subatomic perspective.
– rob
Commented Jul 19 at 16:20
• I think $P$ is the standard notation for parity eigenvalues in condensed matter too, at least in the context of fermion number parity. Commented Jul 19 at 17:50