Radial wavefunctions · Quantum Mechanics

Intuition

The radial functions of hydrogen are polynomials times decaying exponentials. The most telling picture is not $R$ itself but the radial probability density $r^{2}R^{2}$ , the chance of finding the electron at distance $r$ in any direction. For 1s it peaks at exactly the Bohr radius; 2s has an inner bump and a node before its main peak; 2p, with no node, peaks at four Bohr radii. The higher $n$ , the further out the electron lives, roughly as $n^{2}$ .

A population map shows people per square kilometre; a map of how many live at each distance from a city centre must multiply by the ring’s area. The radial density does that, multiplying by $4\pi r^{2}$ for the shell at each radius.

Radial probability densities $P(r)=r^{2}R^{2}$ of the 1s, 2s and 2p states against $r$ in Bohr radii. 1s peaks at $r =$ ; 2s has a small inner bump, a node at and its main peak further out; 2p peaks at .

The first radial functions

In units of the Bohr radius, $R_{n\ell}$ is a polynomial of degree $n-\ell-1$ times $r^{\ell}e^{-r/n}$ , normalised with .

\begin{aligned}R_{10}&=2e^{-r},\qquad R_{20}=\tfrac{1}{\sqrt2}\bigl(1-\tfrac{r}{2}\bigr)e^{-r/2}\\ R_{21}&=\tfrac{1}{2\sqrt6}\,r\,e^{-r/2}\end{aligned}

Properties

The radial density $P(r)=r^{2}R^{2}$ is the probability per unit distance of finding the electron at distance $r$ .
has nodes; 2s has one, at .

The most probable radius of 1s

P_{1s}(r)=\frac{4r^{2}}{a_{0}^{3}}e^{-2r/a_{0}}\ \Longrightarrow\ r_{\text{max}}=a_{0}

The radial density is $r^{2}$ , which grows, times a decaying exponential. Its derivative is zero where the growth of $r^{2}$ exactly balances the decay, which is at one Bohr radius.

Proof steps

P(r)=r^{2}R_{10}^{2}=\frac{4r^{2}}{a_{0}^{3}}e^{-2r/a_{0}}

Square the radial function and multiply by $r^{2}$ .

P'(r)=\frac{4}{a_{0}^{3}}\Bigl(2r-\frac{2r^{2}}{a_{0}}\Bigr)e^{-2r/a_{0}}

Product rule.

2r-\frac{2r^{2}}{a_{0}}=0\ \Rightarrow\ r=a_{0}

The nonzero root.

P(a_{0})=\frac{4}{a_{0}}e^{-2}\approx\frac{0.54}{a_{0}}

A maximum, since $P$ vanishes at $0$ and at infinity.

Applications

Practice

Radial Density

The probability of finding the electron between $r$ and $r+dr$ in any direction is $r^{2}R^{2}dr$ . For 1s it peaks at one Bohr radius.

P(r)=r^{2}R_{n\ell}(r)^{2}

Try it

At what $r$ , in Bohr radii, does the 2p radial density $P\propto r^{4}e^{-r}$ peak?