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

Question

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 $\theta$?

Answer 1

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 $\theta\in[0,\pi]$.