Chapter 10, Problem 10.64E

The average value of radius in a circular system, $\langle r\rangle$, is determined by the expression $\langle r\rangle ={\displaystyle {\int }_0^{\infty }{\Psi }^{\ast }\cdot r\cdot \Psi \cdot 4\pi r^{2}dr}$.

(a) Evaluate $\langle r\rangle$ for $\Psi ={\left(\frac{1}{\pi a^3}\right)}^{1/2}{e}^{-r/a}$, where $a$ is a constant.

See the integral table in Appendix $1$.

(b) What is the numerical value of $\langle r\rangle$ if $a=5.29\times {10}^{-11}\text{m}$?