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In quantum mechanics, we consider the squared modulo of a wave function has meaning of probability, so does it mean a wave packet should be unitless? I am reading some materials online about the Gaussian wave packet http://www.colorado.edu/physics/phys2170/phys2170_sp07/downloads/Gaussian.pdf, there at eq. 7.10, we have

##|\psi|^2 = \frac{1}{\sqrt{\pi}\sigma}\exp(-x^2/\sigma)##

I think ##\sigma## has the unit of meter so whatever inside the ##\exp## has no unit. But what about the coefficient? It has unit of 1/meter. When I am reading it, it looks like that ##|\psi|^2## is density of probability but not probability.