Schrödinger's Equation and the Hydrogen Atom — Question 2

Question 2

The most probable radius differs from the location of maximum spatial density

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 ground-state wave function is $\psi(r)=Ne^{-r/a}$ with N positive. Use spherical volume element $d^3r=4\pi r^2dr$ for an isotropic density and $\int_0^\infty r^2e^{-br}dr=2/b^3$.

1. Normalize the wave function and obtain the radial probability density.

2. Find where the spatial density and the radial density are maximal.

Question 2 - Solution

The most probable radius differs from the location of maximum spatial density

(a) Method and calculation

$$1=4\pi N^2\frac{2}{(2/a)^3}=\pi a^3N^2\Rightarrow\boxed{N=1/\sqrt{\pi a^3}}.$$$$\boxed{P_r(r)=4\pi r^2|\psi|^2=\frac{4r^2}{a^3}e^{-2r/a}}.$$

(b) Interpretation and further calculation

$$\rho(r)=\frac{e^{-2r/a}}{\pi a^3}\quad\Rightarrow\quad\boxed{\rho\text{ maximal at }r=0}.$$$$\frac{d}{dr}\ln P_r=\frac2r-\frac2a=0\Rightarrow\boxed{r_{\mathrm{most\ probable}}=a}.$$

Physical check

The radial distribution includes the growing area of a spherical shell. Maximum density at the origin does not mean a finite probability of occupying exactly one point.