Definition 1: The expectation value of the observable related to the parity operator ##\hat{P}## is constant over time. That is,(adsbygoogle = window.adsbygoogle || []).push({});

[tex]\frac{d}{dt}\langle P\rangle=0[/tex]

[tex]\int\Psi^*(r)\ \hat{P}\ \Psi(r)\ dr=constant[/tex]

[tex]\begin{align}\int\Psi^*(r)\ \Psi(-r)\ dr=constant\end{align}[/tex]

Definition 2: If the physical process proceeds in exactly the same way when referred to an inverted coordinate system, then parity is said to be conserved. If, on the contrary, the process has a definite handedness, then parity is not conserved in that physical process.

In particular, the expectation values of all observables ##A##'s are invariant under the parity transformation. That is,

[tex]\begin{align}\int\Psi^*(r)\ \hat{A}\ \Psi(r)\ dr=\int\Psi^*(-r)\ \hat{A}\ \Psi(-r)\ dr\end{align}[/tex]

I suppose both definitions are equivalent. How, then, do we prove (1) implies (2) and vice versa?

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# Definitions of parity conservation

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