Question

The value of parity in s and t space in quantum mechanics

Answer 1

For parity of space, $(x_1, x_2, ...)$, we can introduce parity operator $P: \bar{x} \rightarrow -\bar{x}$, where $\bar{x}$ is from Hilbert space. Using the fact, that $P^2 = 1$, we have

$P^2 \psi(x) = p^2 \psi(x) \; \Rightarrow p = \pm 1$

Here eigenvalues of parity is a quantum number, that is different for different systems. Moreover, the value of parity can change in weak processes. So, the value of space parity is $\pm 1$, but the choice of +1 or -1 depends on particular systems and processes.

For time we can introduce time reversal operator $T: t \to -t$. In can be shown that in general case $T \psi(t)=\psi^*(−t) \neq \eta \psi (t)$. There are no observable eigenvalues of T. Contrary to P we cannot search for T-allowed or forbidden transitions.

However, it is important to notice, that combination of CPT (where C is charge conjugate operator) will not change any quantum system in any process. From theoretical side, CPT symmetry is fundamental property of nature. From experimental side, no violations of CPT symmetry were observed so far.