Thread: Multiple Integrals Shrödinger Problem

I've been set the following problem and have managed to complete parts i) and ii). However, I'm struggling to grasp the concept for part iii). What I would like to be able to do is come up with an expression for the probability of the electron being at radius r and then use differentiation to maximise it. But all I have is the probability of it being in a certain volume dV. Could someone please give me a hint?

Many Thanks.

Q9. According to the Schr¨odinger equation of quantum mechanics, an electron in the
ground state of a hydrogen atom has a spherically-symmetric spatial probability density
distribution P(r) = Ke−2r/a, i.e. the probability of locating the electron in a volume
dV at radius r is P(r)dV . Here, r is the distance of the electron from the proton, a is
a fixed constant with dimension of length and K is a (dimensional) constant chosen to
make the probability of finding the electron anywhere equal to unity. Calculate:
i. the constant K;
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ii. the average value of r; and
iii. the most probable value of r.

The most probable value of r is when



Or