Expectation values in hydrogen · Quantum Mechanics

Intuition

Averages tell what a measurement on many atoms would find. The mean distance of a 1s electron is one and a half Bohr radii, more than the most probable distance because the tail reaches far. The mean potential energy is exactly twice the total energy, and the mean kinetic energy exactly minus it: the virial theorem of the Coulomb force, the same balance as for planets. So a hydrogen atom in its ground state has 13.6 eV of kinetic energy and $-27.2$ eV of potential energy.

The most common income in a country is less than the average income, because a long tail of large incomes pulls the average up. The 1s density has such a tail, and its mean radius lies beyond its peak.

The 1s radial density in Bohr radii, with its peak at $r=a_{0}$ and its mean $\langle r\rangle=\tfrac32a_{0}$ . The long tail pulls the mean beyond the most likely distance.

Averages in hydrogen

Radial averages are integrals of powers of $r$ against the radial density. Closed forms exist for every state.

\begin{aligned}\langle r\rangle_{n\ell}&=\frac{a_{0}}{2}\bigl(3n^{2}-\ell(\ell+1)\bigr)\\ \Bigl\langle\frac{1}{r}\Bigr\rangle_{n\ell}&=\frac{1}{n^{2}a_{0}}\end{aligned}

Properties

The virial theorem for $1/r$ : $\langle T\rangle=-E_{n}$ and $\langle V\rangle=2E_{n}$ .

The mean radius of 1s

\langle r\rangle_{1s}=\int_{0}^{\infty}r\,\frac{4r^{2}}{a_{0}^{3}}e^{-2r/a_{0}}\,dr=\frac32a_{0}

Weight each radius by the radial density and integrate. With $x=2r/a_{0}$ the integral becomes a factorial integral, $\int x^{3}e^{-x}dx=3!$ .

Proof steps

\langle r\rangle=\int_{0}^{\infty}r\,P(r)\,dr=\frac{4}{a_{0}^{3}}\int_{0}^{\infty}r^{3}e^{-2r/a_{0}}\,dr

The average of $r$ against the radial density.

\begin{aligned}&x=\frac{2r}{a_{0}}\ \Rightarrow\\ &\int_{0}^{\infty}r^{3}e^{-2r/a_{0}}dr=\Bigl(\frac{a_{0}}{2}\Bigr)^{4}\int_{0}^{\infty}x^{3}e^{-x}dx\end{aligned}

Change the variable.

\int_{0}^{\infty}x^{3}e^{-x}dx=3!=6

The factorial integral of the Analysis course.

\langle r\rangle=\frac{4}{a_{0}^{3}}\cdot\frac{a_{0}^{4}}{16}\cdot6=\frac32a_{0}

Multiply out.

Applications

Practice

The Mean Radius

The mean distance of the electron is the integral of $r$ times the radial density. For hydrogen’s states it has a closed form.

\langle r\rangle_{n\ell}=\frac{a_{0}}{2}\bigl(3n^{2}-\ell(\ell+1)\bigr)

Try it

What is $\langle r\rangle$ for the 2p state, in Bohr radii?