Past exam of the MPhil in micro- and nanotechnology enterprise course of the University of Cambridge / 2018 / Paper 2 / Section A(ii) / 14 / ii / a / Solution

Only $n=1,2$ have energies below $6ϵ$. Squaring the corresponding expansion coefficients gives $0.2^2+0.3^2=0.13$.

The printed rounded coefficients have normThus the expected rounded answer is $P(E<6ϵ)\simeq 0.13$. If the decimal coefficients are treated as literal exact inputs and the state normalized, the exact probability for that input is $0.13/S\simeq 0.129916$. This distinction avoids treating rounding error as a physical departure from wavefunction normalization.