= Nonselective two-level Zeno dynamics
{title2=$P_b(T)=\tfrac12[1-\cos^n(\Omega T/n)]$}
Start in $a$, drive resonantly with <Hamiltonian operator> $\hbar\Omega X/2$, and perform $n$ ideal basis measurements separated by $T/n$. An interval changes either basis state with probability $p=\sin^2(\Omega T/(2n))$. Ignoring measurement records leaves a two-state <Markov chain> whose population difference is multiplied by $1-2p=\cos(\Omega T/n)$ each step. This proves the displayed final-population formula.
For a pi pulse, $\Omega T=\pi$, it is $\pi^2/(4n)+O(n^{-2})$. The <repeated projective-measurement survival probability> instead counts records in which every result is $a$, giving $P_{\rm all\,a}=\cos^{2n}(\pi/(2n))$. Final return after an earlier departure separates the two finite-$n$ events.
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