Parity Operator

Definition

The operator $P$, defined as follows, is called the parity operator.

$$ P\psi (x) = \psi (-x) $$

Description

It is an operator that translates the position variable of a wave function symmetrically.

The parity operator $P$ is used in quantum mechanics to distinguish between two degenerate eigenfunctions. Let’s suppose there are two degenerate wave functions as follows.

$$ \psi_{1}(x)=e^{ikx},\quad \psi_{2}(x)=e^{-ikx} $$

Then, solving the eigenvalue equation for the energy operator $H$ does not allow for differentiation between the two wave functions.

$$ H\psi_{1} = \frac{\hbar^2 k^2}{2m}\psi_{1} \\[1em] H\psi_2 = \frac{\hbar^2 k^2}{2m}\psi_2 $$

Now, let’s do the following.

$$ u_{+}(x)=\psi_{1}(x) +\psi_{2}(x) \\[1em] u_{-}(x)=\psi_{1}(x)-\psi_{2}(x) $$

Then, the eigenvalues for the parity operator are different for each wave function as $+1$ and $-1$, enabling us to distinguish between the two functions.

$$ \begin{align*} Pu_{+} &= e^{-ikx}+e^{ikx} =u_{+} \\ Pu_{-} &= e^{-ikx}-e^{ikx} =-u_{-} \end{align*} $$

On the other hand, by the definition of the parity operator, a wave function can never be an eigenfunction of the parity operator. However, it can be shown that the wave function is an eigenfunction of $P^{2}$, with the eigenvalue being $1$.

Properties

$$ P^{2} \psi (x) = \psi (x) $$

Proof

$$ P^{2}\psi (x)=P(P\psi (x))=P\psi (-x)=\psi (x) $$

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