  We report results from shell model studies of muon capture on $^{40}$Ca to
low-lying levels of $^{40}$K. We discuss the comparison between calculated
capture rates, measured capture rates and analogous transitions in
($e$,$e^{\prime}$) scattering in terms of the particle-hole structure of the
$^{40}$Ca-$^{40}$K nuclei. We highlight the $^{40}$Ca$(0^+,
0)$$\to$$^{40}$K$(0^-, 2626)$ axial charge transition and its sensitivity to
the induced pseudoscalar coupling $g_p$ of the proton's weak interaction. In
addition, we address the hindrance of unique first-forbidden transitions due to
particle-hole interactions and the emergence of allowed Gamow-Teller
transitions due to ground state correlations. Lastly, we examine the
longitudinal alignment of $^{40}$K recoils following muon capture, and discuss
this possibility for independently determining the induced coupling $g_p$.
