Exploring Quantum Physics - Week 3 Bonus Question 2

Question:

Prove that the only eigenvalues of the parity operator, $\hat{I}$, are 1 and −1.

Solution:

Recall that $\hat{I}(f(x)) = f(-x)$, and let $\lambda$ be its eigenvalue, so we have:

$\begin{eqnarray*} f(x) &=& \hat{I}(f(-x)) \\ &=& \lambda f(-x) \\ &=& \lambda (\hat{I}(f(x))) \\ &=& \lambda (\lambda f(x)) \\ 1 &=& \lambda^2 \end{eqnarray*}$

So the only eigenvalues are 1 and -1.