Parity Operator | Quantum Mechanics

Parity operator corresponds to space reflection about the origin i.e.

$\hat{P}\psi(x)=\psi(-x)$

In general,

$\hat{P}\psi(\overrightarrow{r})=\psi(-\overrightarrow{r});$ 

where

$\overrightarrow{r}=x\hat{i}+y\hat{i}+z\hat{k}$ 

and

 $\overrightarrow{-r}=-x\hat{i}-y\hat{j}-z\hat{k}$

When parity operator acts on a wavefunction, the following changes take place in various coordinate systems:

(i) Cartesian co-ordinate system:

$x\rightarrow-x,y\rightarrow-y,z\rightarrow-z$

(ii) Spherical polar co-ordinate system:

$r\rightarrow r,\,\theta\rightarrow\pi-\theta,\,\phi\rightarrow\pi+\phi$

(iii) Cyllindrical co-ordinate system:

$p\rightarrow p,\phi\rightarrow\pi+\phi,z\rightarrow-z$

Properties:

(1) Eigenvalues of the Parity Operator is 1, -1. $^{\Longleftrightarrow}$

    $\hat{P}\psi(x)=\psi(-x)=\psi(x)\rightarrow$ even parity.

   $\hat{P}\psi(x)=\psi(-x)=-\psi(x)\rightarrow$ odd parity.

(2) Parity operator is hermitian in nature.

(3) Parity operator commutes with hamiltonian operator if the potential under which particle is moving i.e. $V\left(x\right)$ is symmetric in nature.