Chapter 39 - Wave Functions and Uncertainty - Exercises and Problems - Page 1175: 10

$\rm m^{1/2}$

We know that the probability is $|\psi|^2\delta x$ dimensionless, but the unit of $\delta x$ is $\rm m$. So, $$|\psi|^2\delta x\Rightarrow\rm (Units\; of \;|\psi|^2)\cdot (m)=1$$ Hence, $$\rm Units\; of \;|\psi|^2=\dfrac{1}{m}$$ $$\rm Units\; of \;\psi=\sqrt{\dfrac{1}{m}}$$ $$\boxed{\rm Units\; of \;\psi=\color{red}{\bf m^{1/2}}}$$