We are given the relation $dP=r^2|R(r)|^2dr$. Also, we are given $R(r)$ for the 1s orbital, and we are required to find the most probable radius of the electron in the 1s orbital of a hydrogen atom.

It seems that the most probable radius $r$ should be that value that maximizes $P(r)$, the probability of finding the electron. From elementary mathematics, this should mean we solve the equations $dP(r)/dr=0$ and $d^2P/d^2r <0$. However, the correct answer is obtained by solving $d/dr (r^2|R(r)|^2)=0$, i.e maximizing the probability density and not the probability.

Why is this the case? Why are we not maximizing the probability? And if what we are doing is maximizing the probability, then why is it contradicting the elementary mathematical fact that $dP(r)/dr=0$ $d^2P/d^2r <0$ , for a maxima?

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    $\begingroup$ I think this is grounded in basic misunderstanding of what is probability and probability density. $\endgroup$ May 25 at 6:53

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