Referring to the answer by DSVA (Most probable point for finding an electron in the 1s orbital of a Hydrogen atom)
There's a maximum of finding the electron at a certain distance away from the core (but not a single point at that distance)
I face a problem in solving the maximum probability of finding electron in a 2p orbital. $$\psi=R_{2,1}Y_{1,0}=\sqrt\frac{1}{32\pi a_o^3} \left(\frac{r}{a_o}\right) \exp\left(\frac{-r}{2a_o}\right) \cos\theta $$
Using probability density function, differentiate and equate to zero$$ \frac{d\psi^2}{dr}=constants\left(2r-\frac{r^2}{a_o}\right)\exp\left(\frac{-r}{a_o}\right)=0$$ I obtain$$r=2a_o$$ Using radial probability distribution, differentiate and equate to zero$$P(r)=r^2|R(r)|^2 $$ Referring Atkins' Physical Chemistry (pg. 312), it is stated that spherical harmonics is normalised to 1.$$\frac{dP}{dr}=constants\left(4r^3-\frac{r^4}{a_o}\right)\exp\left(\frac{-r}{a_o}\right)=0 $$ I obtain$$r=4a_o$$
My question: When should we use radial probability or probability density to find maximum probability of finding electron and its most probable distance? What does the difference of values mean?