Schrödinger's Equation and the Hydrogen Atom — Question 8

Question 8

The 2p radial distribution has one positive maximum and a different mean 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. The normalized radial function is $R_{21}(r)=[1/(2\sqrt6\,a^{3/2})](r/a)e^{-r/(2a)}$. Its radial probability density is $P_r=r^2|R_{21}|^2$, because the angular factor integrates to one. Use $\int_0^\infty r^ne^{-r/a}dr=n!a^{n+1}$.

1. Find $P_r$ and its most probable radius.

2. Check radial normalization and calculate the mean radius.

Question 8 - Solution

The 2p radial distribution has one positive maximum and a different mean radius

(a) Method and calculation

$$\boxed{P_r(r)=\frac{r^4}{24a^5}e^{-r/a}}.$$$$\frac d{dr}\ln P_r=\frac4r-\frac1a=0\Rightarrow\boxed{r_{\mathrm{mode}}=4a}.$$

(b) Interpretation and further calculation

$$\int_0^\infty P_rdr=\frac{4!a^5}{24a^5}=\boxed{1}.$$$$\boxed{\langle r\rangle=\frac{5!a^6}{24a^5}=5a}.$$

Physical check

The 2p radial mode equals the n=2 Bohr radius, but the distribution has a finite width and a different mean. This agreement does not justify a definite orbit.