Measurements of planar channelling radiation generated in a variety of crystallographic planes of a quartz single crystal have been performed at the radiation source ELBE at electron energies of 17, 25 and 32 MeV. The quantum theory of channelling radiation has been applied for the interpretation of the observed spectra. Planar continuum potentials, transverse electron states, transition energies and line widths have been calculated. Taking the population dynamics of states during channelling into account, spectral distributions of the channelling radiation could be simulated, which reveal good quantitative agreement with the measured spectra.