Past exam of the options II course of the University of Oxford / 2012 / J / J2 / b / Solution

The Born probability is the integral of $|{\psi }_n{|}^2$ over the indicated interval. Changing variables to $\xi$ cancels the dimensional normalization factors and givesSince $\frac{d}{d\xi }(\xi e^{-{\xi }^2})=e^{-{\xi }^2}-2{\xi }^2{e}^{-{\xi }^2}$, integration givesAs $\ell \to \infty$, the correction term tends to zero and $P_0(\ell )\to 1$ by normalization. Hence $\boxed{P_1(\ell )\to 1}$. This is expected because the expanding interval exhausts the whole position space and the state is normalized: the particle is somewhere on the line with total probability one.