= Solution
In SI base dimensions,
$$
[e]=IT,\quad[\epsilon_0]=I^2T^4M^{-1}L^{-3},\quad[m]=M,\quad[h]=ML^2T^{-1}.
$$
Equating current, time, <mass> and <length> powers respectively yields
$$
\alpha+2\beta=0,\quad\alpha+4\beta-\delta=0,\quad-\beta+\gamma+\delta=0,\quad-3\beta+2\delta=1.
$$
Thus
$$
\boxed{\alpha=-2,\quad\beta=1,\quad\gamma=-1,\quad\delta=2,\qquad a_0\propto\frac{\epsilon_0h^2}{me^2}.}
$$
<Dimensional analysis> cannot determine the dimensionless proportionality constant.
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