# Math Help - Multiple Integrals Shrödinger Problem

1. ## 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;
4
ii. the average value of r; and
iii. the most probable value of r.

2. $
\displaystyle {
Pr(r) \; dr=P(r) \; dV=K e^{-2r/a} \; 4 \pi r^2 dr
}
$

The most probable value of r is when

$
Pr(r) \; is \; max.
$

Or

$
\displaystyle {
\frac{d \; Pr(r)}{dr}=0.
}
$