I googled for the above question, and I got the answer to be $$[\Psi]~=~L^{-\frac{3}{2}}.$$
Can anyone give an easy explanation for this?
Answer
The physical interpretation of the wavefunction is that $|\psi(\vec r)|^2dV$ gives the probability of finding the electron in a region of volume $dV$ around the position $\vec r$. Probability is a dimensionless quantity. Hence $|\psi(\vec r)|^2$ must have dimension of inverse volume and $\psi$ has dimension $\mathrm{m}^{-3/2}$.
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