Quantum Zeno effect

Remarks

The formula for $p_N(t)$ comes from the Born rule and the collapse rule: after each "yes" the state is $\psi$ again, and the $N$ factors are independent. For the bound, let $\mu$ be the spectral measure of $H$ in the state $\psi$, so that $\langle \psi, e^{-i\tau H}\psi \rangle = \int e^{-i\tau\lambda}\, d\mu(\lambda)$. Then $1 - \abs{\langle \psi, e^{-i\tau H}\psi \rangle}^2 = \iint \big(1 - \cos(\tau(\lambda - \lambda'))\big)\, d\mu(\lambda)\, d\mu(\lambda')$, and $1 - \cos x \le x^2/2$ bounds this by $\tau^2 (\Delta H)^2$. Taking $\tau = t/N$ and the $N$-th power gives the bound, and the right side tends to $1$. Example: on $\C^2$ let $H = \omega X$, where $X$ is the Pauli matrix that swaps the two basis vectors, and let $\psi = (1, 0)$. The survival amplitude is $\cos(\omega t/N)$, so $p_N(t) = \cos^{2N}(\omega t/N) \to 1$. Without measurements, the state is fully flipped at $t = \pi/(2\omega)$. The hypothesis that $\psi$ lies in the domain of $H$ matters. Without a finite energy variance, the short-time decay need not be quadratic, and the limit can fail to be $1$.

Going deeper

For a student

The effect combines unitary evolution with the projection postulate, and it shows that frequent measurement is itself a strong dynamical influence. The key estimate is the quadratic short-time behaviour $1 - \abs{\langle \psi, e^{-i\tau H}\psi \rangle}^2 \le \tau^2 (\Delta H)^2$, which is a form of the energy-time uncertainty relation. For a subspace instead of a single vector, Misra and Sudarshan studied the limit of the evolution under frequent projections, which leads to a dynamics confined to that subspace. Exponential decay laws, as for an unstable particle, hold only at intermediate times, which is why the effect does not contradict them.