Question

The average value of the radius r for a radial function Rn,l(r) of a hydrogen-like atom:

<r>= | Rn.l(rjrRn,l(r)dT rRna(r)dr=- [enfinity Rm(r)rRml(r)r2 dr

The most probable value of the radius r_mp is located where:

$\frac{\mathrm{d} }{\mathrm{d} r}(r^2R_{n,l}(r)^2)=0$

Calculate < r > and r_mp for a hydrogen-like atom with charge Z in the 1s and 2s states. You will find the necessary integral and R_n,l(r) formulas on the equation sheet. You may use numerical software or your graphing calculator to find the roots of the cubic polynomial that you should get when finding r_mp for the 2s state. Take the largest root as r_mp.

Answer 1

"Radical fxns of hydrogen-like atoms: 1s n = 1, l = 0 Rn, l = 2 (z/a_0)^3/2 exp (-zr/a_0) 2s n = 2, l = 0 Rn, l = (z/2a_0)^3/2 (2 - zr/a_0) exp (-zr/2 a_0) z = atomic no. for H-like atom 1s < r > = < R_n, l (r) R_n, l > = integral [{2 (z/a_0)^3/2 exp (- zr/a_0)}