  We construct a quantum mechanical matrix model that approximates
$\mathcal{N}=1$ super-Yang-Mills on $S^3\times\mathbb{R}$. We do so by pulling
back the set of left-invariant connections of the gauge bundle onto the real
superspace, with the spatial $\mathbb{R}^3$ compactified to $S^3$. We quantize
the $\mathcal{N}=1$ $SU(2)$ matrix model in the weak-coupling limit using the
Born-Oppenheimer approximation and find that different superselection sectors
emerge for the effective gluon dynamics in this regime, reminiscent of
different phases of the full quantum theory. We demonstrate that the
Born-Oppenheimer quantization is indeed compatible with supersymmetry, albeit
in a subtle manner. In fact, we can define effective supercharges that relate
the different sectors of the matrix model's Hilbert space. These effective
supercharges have a different definition in each phase of the theory.
