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Parity Operator | Quantum Mechanics

Parity Operator

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


In general,






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

(i) Cartesian co-ordinate system:


(ii) Spherical polar co-ordinate system:

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

(iii) Cyllindrical co-ordinate system:

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


(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.

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