Hydrogen-like ions and isotopes · Quantum Mechanics

Intuition

Strip an atom of all but one electron and it is hydrogen again, with a nucleus of charge $Ze$ . Nothing needs solving afresh: replace $e^{2}$ by $Ze^{2}$ and every result scales. The energies grow as $Z^{2}$ , the sizes shrink as $1/Z$ . Helium with one electron is bound four times as strongly as hydrogen, uranium with one electron some eight thousand times. Changing the orbiting particle instead — a muon, or a positron’s partner — changes the reduced mass and scales everything again.

A stronger magnet pulls a steel ball closer and holds it harder. A nucleus of higher charge does both to its electron: closer by a factor $Z$ , harder by a factor $Z^{2}$ .

The levels $n=1$ to 4 of hydrogen and of the helium ion He $^{+}$ on one energy scale, in units of 13.6 eV. Every He $^{+}$ level is four times deeper: eV for its ground state against . Its level coincides with hydrogen’s ground state.

Scaling with charge and mass

For a nucleus of charge $Ze$ and an orbiting particle of reduced mass $\mu$ , the hydrogen results scale with $Z$ and with $\mu/m_{e}$ .

\begin{aligned}E_{n}&=-Z^{2}\,\frac{\mu}{m_{e}}\,\frac{13.6\ \text{eV}}{n^{2}}\\ a&=\frac{a_{0}}{Z}\,\frac{m_{e}}{\mu}\end{aligned}

Examples

He $^{+}$ ( $Z=2$ ): ground state $-54.4$ eV, size $a_{0}/2$ . Li (): eV.

The scaling with Z

V=-\frac{Zk}{r}\ \Longrightarrow\ a=\frac{a_{0}}{Z},\qquad E_{n}=-\frac{Z^{2}mk^{2}}{2\hbar^{2}n^{2}}

Every hydrogen formula depends on the strength $k=e^{2}/4\pi\varepsilon_{0}$ of the attraction. A charge $Ze$ multiplies it by . Lengths go as , so they shrink by ; energies go as , so they grow by .

Proof steps

k\ \to\ Zk

The nucleus has charge $Ze$ instead of $e$ .

a_{0}=\frac{\hbar^{2}}{mk}\ \to\ \frac{\hbar^{2}}{mZk}=\frac{a_{0}}{Z}

The Bohr radius depends on $k$ as $1/k$ .

E_{n}=-\frac{mk^{2}}{2\hbar^{2}n^{2}}\ \to\ -\frac{mZ^{2}k^{2}}{2\hbar^{2}n^{2}}

The energies depend on $k^{2}$ .

m\ \to\ \mu:\qquad a\propto\frac{1}{\mu},\quad E\propto\mu

The same formulas give the dependence on the reduced mass.

Applications

Practice

Energies as Z^{2}

A one-electron ion with nuclear charge $Z$ has energies $Z^{2}$ times hydrogen’s.

E_{n}=-\frac{13.6\ \text{eV}\,Z^{2}}{n^{2}}

Try it

What is the ground-state energy of Li $^{2+}$ ( $Z=3$ ), in eV? Give one decimal place.