What is the probability of finding an electron in a hydrogen atom in an infinitesimal space $dV$?

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.

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.

• 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? – majyno Oct 28 '20 at 14:29
• @majyno This is not a homework-help site. – Emilio Pisanty Oct 28 '20 at 14:52