Chapter 5, Problem 48AP

The wave function of an electron in the lowest (that is, ground) state of the hydrogen atom is

$\begin{array}{l}\psi \left(r\right)={\left(\frac{1}{\pi a_0^3}\right)}^{1/2}\exp \left(\frac{r}{a_0}\right)\\ a_o=0.529\times {10}^{-10}\text{m}\end{array}$

(a) What is the probability of finding the electron inside a sphere of volume $1.0{\text{pm}}^2$ , centered at the nucleus $\left(\text{1 pm}=\text{1}{0}^{-\text{12}}\text{m}\right)$ ?

(b) What is the probability of finding the electron in a volume of $1.0{\text{pm}}^2$ at a distance of 52.9 pm from the nucleus, in a fixed but arbitrary direction?

(c) What is the probability of finding the electron in a spherical shell of 1.0 pm in thickness, at a distance of 52.9 pm from the nucleus?