  Although highly successful on cosmological scales, Cold Dark Matter (CDM)
models predict unobserved over-dense `cusps' in dwarf galaxies and overestimate
their formation rate. We consider an ultra-light axion-like scalar boson which
promises to reduce these observational discrepancies at galactic scales. The
model, known as Fuzzy Dark Matter (FDM), avoids cusps, suppresses small-scale
power, and delays galaxy formation via macroscopic quantum pressure. We compare
the substructure and density fluctuations of galactic dark matter haloes
comprised of ultra-light axions to conventional CDM results. Besides
self-gravitating subhaloes, FDM includes non-virialized over-dense wavelets
formed by quantum interference patterns which are an efficient source of
heating to galactic discs. We find that, in the solar neighborhood, wavelet
heating is sufficient to give the oldest disc stars a velocity dispersion of
$\sim 30 \: \mathrm{km} \: \mathrm{s}^{-1}$ within a Hubble time if energy is
not lost from the disc, the velocity dispersion increasing with stellar age as
$\sigma_D \propto t^{0.4}$ in agreement with observations. Furthermore, we
calculate the radius-dependent velocity dispersion and corresponding scale
height caused by the heating of this dynamical substructure in both CDM and FDM
with the determination that these effects will produce a flaring that
terminates the Milky Way disc at $15 - 20 \: \mathrm{kpc}$. Although the source
of thickened discs is not known, the heating due to perturbations caused by
dark substructure cannot exceed the total disc velocity dispersion. Therefore,
this work provides a lower bound on the FDM particle mass of $m_a > 0.6 \times
10^{-22} \mathrm{eV}$. Furthermore, FDM wavelets with this particle mass should
be considered a viable mechanism for producing the observed disc thickening
with time.
