Sunday 13 September 2015

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


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