Why are the radial wavefunction and radial distribution function different? I'm having trouble understanding the following two graphs, which are the radial wavefunction of the hydrogen 1s orbital and the corresponding radial distribution function:



Specifically, why is the radial distribution function zero at $r = 0$, but not the radial wavefunction?

Also, why does the radial distribution function have a maximum at the Bohr radius $a_0$?

(Mathematica input for the graphs: Plot[{2*Exp[-x], 4*Exp[-2 x]*x^2}, {x, 0, 6}, AxesLabel -&gt; {r/Subscript[a, 0], Subscript[f, "1s"][r]}, PlotRange -&gt; {{0, 6}, {0, 2}}, PlotLegends -&gt; {"Radial wavefunction", "Radial distribution function"}, BaseStyle -&gt; {FontSize -&gt; 14}, ImageSize -&gt; Medium])
The radial wave function $R(r)$ is simply the value of the wave function at some radius $r$, and its square is the probability of the finding an electron in some infinitesimal volume element around a point at distance $r$ from the nucleus.

But, the infinitesimal volume of space at radius $r$ is $4\pi r^{2} dr$ (it's a spherical shell with thickness $dr$ at radius $r$). That means that probability of finding an electron at radius $r$ is proportional to $R^{2}(r)4\pi r^{2}$. But the behavior of this function is such that the probability of finding the electron at radius 0 is also 0.