# What measurable quantity is associated with parity?

In quantum mechanics, we learn that for any Hamiltonian with a symmetry, there exists a unitary operator associated with that symmetry. Consider the parity operator which is defined by its operation on the position basis, $$\Pi\left|x\right>=\left|-x\right>.$$ One can show that $$\Pi$$ is Hermitian with eigenvalues of $$\pm1.$$ If Hermitian operators are associated with measurable quantities, what is the measurable quantity associated with parity? How would you go out and measure parity in the lab?

More generally, one of the postulates of quantum mechanics states that every measurable quantity $$\mathcal{A}$$ is associated with a Hermitian operator $$A.$$ We could write $$\mathcal{A}\implies A.$$ Another way to frame this question is to ask whether the converse is also true: that is, for every Hermitian operator $$A,$$ can we associate a measurable quantity $$\mathcal{A}$$? Does $$A \implies \mathcal{A}?$$

• Parity has eigenvectors and eigenvalues, so expectation values between suitable states... Parity eigenvalues of particles can be measured that way. Expectation values of suitable operators is normally what's involved in predicting the outcome of every measurement. Sep 17, 2021 at 20:09