= Repeated projective-measurement survival probability
{title2=$\mathbb P_N(T)=\left|\langle\psi,e^{-iHT/(N\hbar)}\psi\rangle\right|^{2N}$}
For a normalized <quantum state> $\psi$ and a time-independent <Hamiltonian operator> $H$, consider ideal <projective measurements> of the <orthogonal projection> $P=|\psi\rangle\langle\psi|$ at intervals $\Delta t=T/N$. Write $U=e^{-iH\Delta t/\hbar}$. A successful check leaves the <post-measurement state> $\psi$ up to <global phase>. Thus each successive success, conditional on earlier successes, has <conditional probability> $|\langle\psi,U\psi\rangle|^2$, and multiplication gives the displayed formula. Equivalently,
$$
(PU)^N\psi=\langle\psi,U\psi\rangle^N\psi,\qquad
\mathbb P_N(T)=\|(PU)^N\psi\|^2.
$$
This is the <probability> of the entire sequence of successful checks. It underlies the <quantum Zeno effect>.
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