  Motivated by recent breakthrough in molecular beam epitaxy of GaAs/AlGaAs
quantum wells [Y. J. Chung \textit{et al.}, Nature Materials \textbf{20}, 632
(2021)], we examine contributions to mobility and quantum mobility from various
scattering mechanisms and their dependencies on the electron density. We find
that at lower electron densities, $n_e \lesssim 1 \times 10^{11}$ cm$^{-2}$,
both transport and quantum mobility are limited by unintentional background
impurities and follow a power law dependence, $\propto n_e^{\alpha}$, with
$\alpha \approx 0.85$. Our predictions for quantum mobility are in reasonable
agreement with an estimate obtained from the resistivity at filling factor
$\nu= 1/2$ in a sample of Y. J. Chung \textit{et al.} with $n_e = 1 \times
10^{11}$ cm$^{-2}$. Consideration of other scattering mechanisms indicates that
interface roughness (remote donors) is a likely limiting factor of transport
(quantum) mobility at higher electron densities. Future measurements of quantum
mobility should yield information on the distribution of background impurities
in GaAs and AlGaAs.
