  The local electronic structure of muons (Mu) as dilute pseudo-hydrogen in
single-crystalline $\beta$-Ga$_2$O$_3$ has been studied by the muon spin
rotation/relaxation ($\mu$SR). High-precision measurements over a long time
range of $\sim$25 $\mu$s have clearly identified two distinct Mu states: a
quasi-static Mu (Mu$_1$) and fast-moving Mu (Mu$_2$). By comparing this result
with predictions from the recently established ambipolarity model, these two
states are respectively attributed to the relaxed-excited states associated
with the donor ($E^{+/0}$) and acceptor ($E^{-/0}$) levels predicted by density
functional theory (DFT) calculations for the interstitial H. Furthermore, the
local electronic structure of Mu$_1$ is found to be an OMu-bonded state with
three-coordinated oxygen. The structure is almost identical with the thermal
equilibrium state of H, and it is found to function as an electron donor. The
other Mu$_2$ is considered to be in the hydride state (Mu$^-$) from the
ambipolarity model, suggesting that it is in fast diffusion motion through the
short-lived neutral state due to the charge exchange reaction with conduction
electrons (Mu$^0+e^-\rightleftarrows$Mu$^-$).
