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I have been asked to find the most probable position of electron in infinitesimal space $dV$ orbiting a Hydrogen atom. I know that probability $P$ of finding the electron in a volume $dV$ is given by $$ P = |\psi(\vec r,t)|^2 dV $$ where $\psi(\vec r,t) = A e^{-r/a_0}e^{-iE_1t/\bar{h}}$.

The time dependence is irrelevant and this gives me \begin{align} P &= |A|^2e^{-2r/a_0}dV \\ &= |A|^2e^{-2r/a_0}d^3r \\ &= |A|^2e^{-2r/a_0}[r^2 \sin(\theta)dr\, d\theta\, d\phi] \end{align} My question is what do I do next? When I differentiate it by $dr$ it will give me $a_0$, but that represents the spherical shell where the probability is highest. Not infinitesimal space with the highest probability.

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You're done already when you get to $$ P = |A|^2e^{-2r/a_0}dV. $$ Trying to break down $dV$ into its constituent differentials in spherical coordinates will only serve to confuse you.

On the other hand, if you want to give a correct probability, it is essential that you normalize correctly, i.e., that you get an explicit value of $|A|^2$ so that the total probability over all of space is unity.

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  • $\begingroup$ Ok thank you. As you sad I was a bit confused by it. Finding normalisation constant is not a problem that is a next part of the exercise I have no problem doing. The question follows with: "If you were able to measure the position only once where would you be measuring?" how do I answer? Do you have any suggestion? $\endgroup$ – majyno Oct 28 '20 at 14:29
  • $\begingroup$ @majyno This is not a homework-help site. $\endgroup$ – Emilio Pisanty Oct 28 '20 at 14:52

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