What are the differences between the two graphs of electron probability density?

[jd12345]

Ok I am a noob in quantum mechanics so please keep the level down for me to understand.
My text gives a graph of probability density(ψ^2) of 1s orbital agaisnt distance r -
The graph is maximum near the nucleus and then decreases - i always thought electron has max probability at the bohr's radius but the graph seemsto show max near the nucleus. IS it correct.

Searching at many places i have actually found two graphs - one which shows maximum at nucleus and then it decreases and
other which shows zero at nucleus and then increases upto a point and then decreases

Are both different ? Which is correct?

 

[Jano L.]

Hello jd12345,
in fact both graphs are correct:

1) The graph of the function with the maximum at the proton gives the probability that the electron is inside small ball-like region of volume $\Delta V$ of physical space located in a distance $r$ divided by that volume: $f_1(r) = \frac{\Delta p}{\Delta V}$.

2) The graph with the maximum around the Bohr radius gives the probability that the electron is inside a shell of radius $r$ divided by the thickness of the shell: $f_1(r) =\frac{\Delta p}{\Delta r}.$

It turns out that for the first psi - function of the hydrogen atom,

$$f_1(r) = Ce^{-2r/a_{\mathrm{B}}}$$

$$f_2(r) = 4\pi r^2 f_1(r) = C4\pi r^2 e^{-2r/a_{\mathrm{B}}}.$$

where $C$ is a normalization constant and $a_{\mathrm{B}}$ is the Bohr radius.