quantum mechanics - What is the unit (dimension) of the 3-dimensional position space wavefunction $Psi$ of an electron?

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 $\mathrm{m}^{-3/2}$.