Repeated projective-measurement survival probability (${\mathbb{P}}_N(T)={\left|⟨\psi ,e^{-iHT/(N\hslash )}\psi ⟩\right|}^{2N}$)

For a normalized quantum state $\psi$ and a time-independent Hamiltonian operator $H$, consider ideal projective measurements of the orthogonal projection $P=|\psi ⟩⟨\psi |$ at intervals $\Delta t=T/N$. Write $U=e^{-iH\Delta t/\hslash }$. A successful check leaves the post-measurement state $\psi$ up to global phase. Thus each successive success, conditional on earlier successes, has conditional probability $|⟨\psi ,U\psi ⟩{|}^2$, and multiplication gives the displayed formula. Equivalently,This is the probability of the entire sequence of successful checks. It underlies the quantum Zeno effect.