The low temperature exciton scattering rate in a structure consisting of a series of undoped single GaAs quantum wells of widths 5,10,20 and 80 nm separated by undoped 34 nm A10.3Ga0.7As barriers has been investigated in the frequency domain as a function of electric field. The low temperature n=1 heavy hole homogeneous exciton linewidth studied using a combination of resonant Rayleigh scattering and photoluminescence excitation spectroscopy is found to show dramatic increases, to values in excess of the inhomogeneous width, for certain values of electric field which depend on the particular well being studied. The change in exciton scattering rate is interpreted as being due to resonant tunnelling of carriers between neighbouring wells of different widths as bound states are brought into alignment by the external electric field.