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

The Parseval identity for the orthonormal stationary-state expansion is ${\sum }_{n\ge 1}|c_n{|}^2=\Vert \psi {\Vert }^2=1$. Splitting the amplitudes into even and odd indices givesMultiply by ${\pi }^2/3$ to obtain the requested identity. Its integer constants areThe series converges absolutely, since both parity contributions are $O(p^{-2})$. This derivation uses the normalization of the prepared state and completeness of the well stationary states, rather than a separate summation trick.