Schrödinger's Equation and the Hydrogen Atom — Question 4

Question 4

The mean and rms radii of hydrogen are distinct from its most probable radius

Treat hydrogen with a fixed nucleus, a nonrelativistic Coulomb potential and no spin, fine structure or external field. Write the Bohr length as a. For the 1s state use $P_r(r)=4r^2e^{-2r/a}/a^3$ and $\int_0^\infty r^ne^{-br}dr=n!/b^{n+1}$.

1. Calculate $\langle r\rangle$ and $\langle r^2\rangle$.

2. Find the rms radius and standard deviation, and compare the three radius measures with the radial mode a.

Question 4 - Solution

The mean and rms radii of hydrogen are distinct from its most probable radius

(a) Method and calculation

$$\boxed{\langle r\rangle=\frac4{a^3}\frac{3!}{(2/a)^4}=\frac32a}.$$$$\boxed{\langle r^2\rangle=\frac4{a^3}\frac{4!}{(2/a)^5}=3a^2}.$$

(b) Interpretation and further calculation

$$\boxed{r_{\mathrm{rms}}=\sqrt3\,a},\qquad\boxed{\sigma_r=\sqrt{3-9/4}\,a=\frac{\sqrt3}{2}a}.$$$$\boxed{r_{\mathrm{mode}}=a<\langle r\rangle=1.5a<r_{\mathrm{rms}}=1.73205a}.$$

Physical check

The radial variable is nonnegative and its distribution is skewed. These unequal measures describe different features of the same state.