# Inside radius of atomic electron cloud vs Z

by NLB
 Homework Sci Advisor HW Helper Thanks P: 12,870 ##R_{nl}(r)## is the radial component of the single-electron wavefunction. The probability density function is the square modulus. i.e. The probability of finding an electron in state |n,l> between r and r+dr is ##p(r)dr=|R_{nl}(r)|^2dr## so ##\int_0^\infty p(r)dr = 1## right? The mean radius is $$\langle r \rangle=\int_0^\infty R_{nl}^\star r R_{nl} dr$$ ... since ##R_{nl}## is real, that is where they get the ##r|R_{nl}(r)|^2## from. It's just the expectation value of r: E[r]. Similarly $$\langle r^2 \rangle=\int_0^\infty r^2 R_{nl}^2 dr$$ ... would be E[r^2].