Past exam of the physics course of the year 1 of the natural sciences course of the University of Cambridge / 2019 / Section B / 8 / e / Solution

Use adjacent segments $[-\ell ,0]$ and $[0,\ell ]$, with $\ell =2\pi /k_1$. Integrate the threshold probability density from part (d). The left segment contains two complete density periods, so its average ${\cos }^2$ is $1/2$:The right-hand density is constant:The common stationary-state normalization cancels, yieldingThe larger local probability weight on the right is compatible with zero transmitted probability current at threshold: density and flux measure different quantities.