Past exam of the quantum theory course of the University of Oxford / 2023 / 1 / b / ii / Solution

The orthonormal basis makes the squared norm equal to the sum of squared expansion coefficients:Choose a positive real wavefunction normalization constant, fixing an irrelevant global phase:As a direct check, ${\int }_0^{2\pi R}{\sin }^4(x/R)dx=3\pi R/4$.