# Can the Parity Operator in polar coordinates be defined as $\hat\Pi\psi(r,\theta,\phi) = \psi(r,\theta+\pi,\phi).$?

I was reading about Symmetries & Conservation Laws from Introduction to Quantum Mechanics, David J. Griffiths when I came across this question about the parity operator in three dimensions:

Problem 6.1 Consider the parity operator in three dimensions.

1. Show that $$\hat\Pi\psi(\mathbf{r}) = \psi'(\mathbf{r}) = \psi(-\mathbf{r})$$ is equivalent to a mirror reflection followed by a rotation.
2. Show that, for $$\psi$$ expressed in polar coordinates, the action of the parity operator is $$\hat\Pi\psi(r,\theta,\phi) = \psi(r,\pi-\theta,\phi+\pi).$$
3. Show that for the hydrogenic orbitals, $$\hat\Pi\psi_{nlm}(r,\theta,\phi)=(-1)^l\psi_{nlm}(r,\theta,\phi).$$ That is, $$\psi_{nlm}$$ is an eigenstate of the parity operator, with eigenvalue $$(-1)^l$$. Note: This result applies to the stationary states of any central potential $$V(\mathbf{r}) = V(r)$$. For a central potential, the eigenstates may be written in the separable form $$R_{nl}(r)Y_l^m(\theta,\phi)$$ where only the radial function $$R_{nl}$$ —which plays no role in determining the parity of the state— depends on the specific functional form of $$V(r)$$.

I have solved this problem and checked it with the solution manual, the way it was supposed to be done.

But I think I can get "another answer" for part 3, if in part 2, instead of writing the parity operator the way it has been, we write: $$\hat\Pi\psi(r,\theta,\phi) = \psi(r,\theta+\pi,\phi).$$ This seems correct to me as the cartesian coordinates change from $$(r\sin\theta\cos\phi,r\sin\theta\sin\phi,r\cos\theta)$$ to $$(-r\sin\theta\cos\phi,-r\sin\theta\sin\phi,-r\cos\theta)$$.

But when I apply the parity operator this way on a hydrogenic orbital, I get $$\hat\Pi\psi_{nlm}(r,\theta,\phi)=(-1)^{l+m}\psi_{nlm}(r,\theta,\phi).$$

Does this mean that I cannot "define" Parity Operator for polar coordinates that way? Does it have something to do with the domain of $$\theta$$?

• $\theta$ varies in $[0,\pi]$ only… Commented Jul 3 at 19:13

Take a look at the derivation from which $$Y_\ell^m(\theta,\phi)$$ is obtained, in Section 4.1.2 of the book (I have the third edition). You should notice that "$$m$$ is an integer" arises from the requirement that $$\phi$$ is periodic over $$2\pi$$.
In your new definition of the coordinate system, which requires that $$\theta$$ be the one periodic over $$2\pi$$, $$\phi$$ becomes no longer periodic so $$m$$ need not be an integer. The consequence is that the equation for $$\Theta(\theta)$$ becomes no longer an associated Legendre equation whose solution is $$P_\ell^m(\cos \theta)$$, breaking down the statement that "$$Y$$ is a spherical harmonic".
As such, it is pretty much nonsensical for you to write $$Y_\ell^m(\theta+\pi,\phi)=(-1)^{\ell-m}Y_\ell^m(\theta,\phi)$$, since $$\theta$$ here must satisfy $$\theta\in[0,\pi]$$.
• I'm afraid that I do not understand why only one of $\theta$ or $\phi$ can be periodic over $2\pi$. Why could they both not be periodic? Commented Jul 4 at 18:47
• @Kapil This has to do with how the spherical coordinate itself is defined. A coordinate system must be unambiguous, which means that one set of coordinates---in this case $(r,\theta,\phi)$---must correspond to exactly one point. For this to be true, you need to choose either $\theta$ or $\phi$ to be periodic over $2\pi$, and bound the other over $[0,\pi]$. This is just a convention, upon which the entirety of that Section in the book is built. Your new definition is justified, but that means you have to derive everything again as your definition is not compatible. Commented Jul 5 at 2:36