Past exam of the quantum theory course of the University of Oxford / 2023 / 1 / a / i / Solution

The free particle has Hamiltonian operator $H=P^2/(2m)$, with momentum operator $P=-i\hslash {\partial }_x$. Its time-dependent Schrödinger equation isHere $x$ measures arc length for the particle on a circle, so the wavefunction normalization uses $dx$, rather than $d(x/R)$.