Bring the chapter together: symmetries as probability-keeping maps, generators and their exponentials, translations and momentum, rotations and angular momentum, conservation laws, parity and its selection rule, time reversal and the degeneracies symmetry forces. No worked example sits above the answers.

A detective who knows what cannot have changed finds the culprit faster than one who checks everything. Symmetry arguments are that knowledge: they settle what must vanish, what must be conserved and what must be degenerate before any calculation.

A symmetry is unitary or antiunitary; a continuous one is $e^{-is\hat{G}/\hbar}$ with a Hermitian generator; commuting with $\hat{H}$ conserves the generator; clashing symmetries force degeneracies.

• Parity: $\hat{\Pi}\hat{x}\hat{\Pi}=-\hat{x}$, eigenvalues $\pm1$; $\hat{x}$ connects only states of opposite parity.
• Time reversal: $\psi\to\psi^{*}$, antiunitary, $\hat{p}\to-\hat{p}$, .
• $\hat{L}_{z}=\hat{x}\hat{p}_{y}-\hat{y}\hat{p}_{x}=-i\hbar\,\partial_{\varphi}$ and .
• Energy, momentum, angular momentum and charge are conserved by invariance under time translations, space translations, rotations and phase changes.