Case Study 2: Hydrogen-Like Atoms — From He$^+$ to Muonic Hydrogen and the Proton Radius Puzzle

The Power of Scaling

One of the most useful features of the hydrogen atom solution is that it generalizes effortlessly to any system consisting of a single particle orbiting a point charge. The substitution rules are simple:

• Replace $e^2$ (the electron charge squared) with $Ze^2$ (where $Z$ is the nuclear charge)
• Replace $m_e$ with the reduced mass $\mu = m_1 m_2/(m_1 + m_2)$

The energy levels become:

$$E_n = -\frac{\mu Z^2 e^4}{2(4\pi\epsilon_0)^2\hbar^2 n^2} = -Z^2\frac{\mu}{m_e}\frac{13.6\;\text{eV}}{n^2}$$

and the effective Bohr radius scales as:

$$a = \frac{4\pi\epsilon_0\hbar^2}{\mu Z e^2} = \frac{m_e}{\mu Z}a_0$$

These scaling laws apply to a surprising range of physical systems.

Hydrogen-Like Ions

The simplest generalizations are hydrogen-like ions: atoms stripped of all but one electron.

As $Z$ increases, the electron is pulled closer to the nucleus (smaller Bohr radius) and bound more tightly (larger binding energy). For heavy hydrogen-like ions like U$^{91+}$, the binding energy exceeds 100 keV and the electron orbits so close to the nucleus that relativistic effects become significant. The non-relativistic Schrodinger equation breaks down, and the Dirac equation (Chapter 29) is needed.

These highly charged ions are produced and studied at facilities like GSI (Germany) and RHIC (USA). They provide some of the most stringent tests of quantum electrodynamics (QED) in strong electric fields.

Exotic Hydrogen-Like Atoms

The scaling laws also apply to exotic systems where the orbiting particle is not an electron.

Muonic Hydrogen

A muon ($\mu^-$) is a particle identical to the electron in every respect except mass: $m_\mu = 207 m_e$. When a muon replaces the electron in hydrogen, the resulting system --- muonic hydrogen --- has energy levels and Bohr radius scaled by the reduced mass:

$$\mu_{\text{reduced}} = \frac{m_\mu m_p}{m_\mu + m_p} \approx 186 m_e$$

The muonic hydrogen Bohr radius is:

$$a_\mu = \frac{m_e}{\mu_{\text{reduced}}}a_0 \approx \frac{a_0}{186} \approx 2.84 \times 10^{-3}\;\text{\AA} = 284\;\text{fm}$$

This is only about 300 times the proton radius ($r_p \approx 0.88\;\text{fm}$). The muon orbits much closer to the proton than the electron does, making muonic hydrogen exquisitely sensitive to the finite size of the proton.

Positronium

Positronium is a bound state of an electron and a positron ($e^+$): a "hydrogen atom" where the proton is replaced by an antiparticle of the electron itself. Since $m_{e^+} = m_e$:

$$\mu = \frac{m_e \cdot m_e}{m_e + m_e} = \frac{m_e}{2}$$

The energy levels are exactly half those of hydrogen: $E_n = -6.8\;\text{eV}/n^2$. The Bohr radius is $2a_0$. Positronium annihilates (electron and positron destroy each other, producing photons), but the ground state lifetime is long enough ($\sim 10^{-10}$ s for para-positronium) for precision spectroscopy.

Muonium

Muonium ($\mu^+ e^-$) is a bound state of a positive muon and an electron. Since $m_\mu \gg m_e$, the reduced mass is nearly $m_e$, and the energy levels are nearly identical to hydrogen. Muonium is important because it is a purely leptonic atom --- no nuclear structure --- making it an ideal testing ground for QED without the complications of nuclear physics.

The Proton Radius Puzzle

In 2010, a team led by Randolf Pohl at the Paul Scherrer Institute in Switzerland made a measurement that shook the foundations of atomic physics. By measuring the Lamb shift (the energy difference between the $2S_{1/2}$ and $2P_{1/2}$ states) in muonic hydrogen using laser spectroscopy, they determined the proton's charge radius with extraordinary precision:

$$r_p(\text{muonic}) = 0.84184(67)\;\text{fm}$$

This value disagreed with the previously accepted value from ordinary hydrogen spectroscopy and electron-proton scattering:

$$r_p(\text{electronic}) = 0.8768(69)\;\text{fm}$$

The discrepancy was $7\sigma$ --- far too large to be a statistical fluctuation. This became known as the proton radius puzzle, and it set off a decade of intense experimental and theoretical investigation.

Why Muonic Hydrogen Is More Sensitive

The key physics is straightforward. The finite size of the proton shifts the energy levels slightly from their point-nucleus values. For s-states (which have nonzero probability density at $r = 0$), the energy shift is approximately:

$$\Delta E \approx \frac{2\alpha^4 m_r^3 c^2}{3n^3}\left(\frac{r_p}{\hbar/m_r c}\right)^2$$

where $m_r$ is the reduced mass and $\alpha$ is the fine-structure constant. The shift scales as $m_r^3$, so it is $(m_{\mu,r}/m_{e,r})^3 \approx 186^3 \approx 6.4 \times 10^6$ times larger in muonic hydrogen than in ordinary hydrogen. This enormous enhancement factor is why a single muonic hydrogen measurement could determine $r_p$ more precisely than decades of electronic hydrogen measurements.

The Resolution (Still Evolving)

Between 2010 and 2022, the situation evolved significantly:

1. 2010--2017: The puzzle deepened. New muonic hydrogen measurements confirmed the smaller radius. Theoretical efforts to explain the discrepancy through new physics (a new force between muons and protons, for example) were largely unsuccessful.

2. 2017--2019: New electronic hydrogen measurements at the Max Planck Institute, using different techniques (the $2S \to 4P$ transition), found a proton radius consistent with the muonic value: $r_p = 0.8335(95)\;\text{fm}$.

3. 2019--2022: The PRad experiment at Jefferson Lab measured the proton radius via electron-proton scattering and found $r_p = 0.831(7)(12)\;\text{fm}$, again consistent with the muonic value.

4. 2022--present: The consensus has shifted toward the smaller radius ($\sim 0.84\;\text{fm}$), with the discrepancy attributed to systematic errors in the older electronic measurements. However, not all experimental groups agree, and the situation remains an active area of research.

The proton radius puzzle illustrates several important points:

• Precision tests matter. The hydrogen atom is not "solved" in the sense that we can stop measuring it. Every improvement in precision is an opportunity to discover new physics.
• Simple systems probe deep physics. The hydrogen atom, the simplest atom, remains at the frontier of fundamental physics.
• Scaling laws reveal new physics. By replacing the electron with a muon, experimenters gained access to nuclear structure that would otherwise require billion-dollar accelerators.

Rydberg Atoms: The Other Extreme

While muonic hydrogen explores the small-$r$ regime (probing nuclear structure), Rydberg atoms explore the large-$r$ regime. A Rydberg atom is a hydrogen atom (or any atom) excited to a very high principal quantum number, $n \gg 1$.

For a hydrogen Rydberg atom with $n = 100$:

• $\langle r \rangle = \frac{3}{2}n^2 a_0 = 15{,}000\;a_0 \approx 0.79\;\mu\text{m}$ --- larger than a bacterium
• $E_{100} = -13.6/10{,}000\;\text{eV} = -1.36\;\text{meV}$ --- barely bound
• Orbital period: $T \propto n^3 \approx 10^{-10}\;\text{s}$
• Sensitivity to electric fields: polarizability $\propto n^7$

Rydberg atoms are so large and sensitive that they are used in precision measurements of fundamental constants, as single-photon detectors, and as qubits in quantum computing (the Rydberg blockade mechanism). In astrophysics, Rydberg transitions with $n \sim 100$ are observed in radio recombination lines from interstellar gas clouds.

The correspondence principle is beautifully illustrated by Rydberg atoms: at $n = 100$, the electron's motion is nearly classical, and the quantum-mechanical transition frequencies approach the classical orbital frequency. As Bohr himself emphasized, the quantum world merges smoothly into the classical world as quantum numbers become large.

Discussion Questions

1. The proton radius puzzle initially suggested the possibility of a new force between muons and protons that does not affect electrons. What constraints would such a force need to satisfy? How would it affect the muon's anomalous magnetic moment ($g - 2$)?

2. Positronium has energy levels $E_n = -6.8\;\text{eV}/n^2$ and eventually annihilates into photons. How does the annihilation process depend on the quantum numbers of the state? (Hint: consider conservation of charge conjugation parity.)

3. For a hydrogen atom in the state $n = 100$, compute $\langle r \rangle$, $\Delta E$ (energy spacing to the next level), and the classical orbital period $T = 2\pi r^{3/2}/\sqrt{GM}$ (using the Coulomb analog). Does the correspondence principle hold?

4. The muonic hydrogen experiment required producing muons, slowing them down to thermal energies, and letting them replace electrons in hydrogen atoms --- all within the muon's 2.2-microsecond lifetime. Discuss the experimental challenges this imposes.

Quantitative Exercises

CS2.1. Compute the ground state energy and Bohr radius for muonic hydrogen. If the $2S - 2P$ splitting in regular hydrogen is about 1058 MHz (the Lamb shift), estimate the $2S - 2P$ splitting in muonic hydrogen.

CS2.2. For positronium, compute (a) the ground state energy, (b) the Bohr radius, (c) the binding energy, and (d) the wavelength of the Lyman-$\alpha$ transition.

CS2.3. A Rydberg atom of hydrogen is in the $n = 50$ state. Compute: (a) the binding energy in meV, (b) the expectation value $\langle r \rangle$ in micrometers, (c) the wavelength of the photon emitted in the $n = 50 \to n = 49$ transition. In what part of the electromagnetic spectrum is this transition?

CS2.4. The proton radius enters the hydrogen energy levels through the finite-size correction $\Delta E_{ns} = \frac{2}{3}\frac{Z\alpha^4 m_r^3 c^4}{\hbar^2 n^3}r_p^2$. Compute this energy shift for the $1s$ state of ordinary hydrogen using $r_p = 0.84\;\text{fm}$. Express your answer in MHz (using $\Delta\nu = \Delta E/h$). Compare with the total $1s$ binding energy. Why is this correction so small?

CS2.5. At what value of $Z$ does the ground state energy of a hydrogen-like ion reach $m_e c^2 = 511\;\text{keV}$ (where the non-relativistic approximation breaks down completely)? Compare with the highest-$Z$ elements in the periodic table.