Past exam of the mathematics course of the University of Cambridge / 2014 / ib / Paper 1 / 15A / ii / Solution

Choose the prepared wavefunction real and positive on its support. Its normalization isIn the ramp-state expansion in a symmetric infinite well, write $\psi ={\sum }_{n\ge 1}{c}_n{\psi }_n$. Orthogonality givesFor $\kappa =n\pi /(2a)$, a primitive is $-x\cos (\kappa (x+a))/\kappa +\sin (\kappa (x+a))/{\kappa }^2$. Evaluating both endpoints givesAn equivalent parity-separated form isThese formulas with ${\sum }_n{c}_n{\psi }_n$ are the requested explicit expansion. The convergence is in the square-integrable norm; at the idealized discontinuity at the right wall, a pointwise boundary value does not change the quantum state or its probability amplitudes.