Sommerfeld Model

The Sommerfeld model extended Bohr’s old quantum atom by allowing quantized elliptical orbits and by adding relativistic corrections. It made old quantum theory look unexpectedly powerful: hydrogen fine structure, additional quantum numbers, and a more systematic action-quantization rule all entered the story before modern wave mechanics existed.

Its success is also its warning label. Sommerfeld’s theory improved the Bohr model while keeping classical electron orbits. It sharpened the need for a deeper mechanics because its rules worked in selected cases without becoming a general theory of atoms.

From Bohr to Sommerfeld

The Bohr Model treated hydrogen using circular orbits, angular-momentum quantization, and transition frequencies. Sommerfeld generalized the orbit picture by treating the electron as a classical Kepler particle whose periodic motions satisfy quantum conditions.

The guiding old-quantum-theory rule was action quantization:

$$J_i = \oint p_i\,dq_i = n_i h.$$

The integral is taken over one period of the corresponding classical coordinate. In the Coulomb problem, the radial and angular motions can be assigned separate action variables. In one common old-quantum notation,

$$J_r=n_rh, \qquad J_\phi=n_\phi h, \qquad n=n_r+n_\phi.$$

This was more flexible than Bohr’s original circular-orbit rule. It connected atomic spectra to the geometry of classical phase-space orbits, not just to a single angular-momentum postulate.

For the modern semiclassical method descended from this idea, see Bohr–Sommerfeld Quantization. That treatment explains the additional phase corrections required at turning points.

Elliptical Orbits

In classical Coulomb motion, bound orbits are ellipses with the nucleus at one focus. The energy fixes the semimajor axis, while angular momentum controls the eccentricity. Circular Bohr orbits are only the special case of zero eccentricity.

Sommerfeld’s model enlarged Bohr’s circular-orbit picture to quantized Kepler ellipses. The historical quantum numbers came from radial and angular action integrals, not from modern orbital wavefunctions.

Allowing ellipses introduced a radial quantum number. A highly eccentric orbit has radial motion in and out as well as angular motion around the nucleus. Sommerfeld’s action conditions quantized both pieces.

The nonrelativistic Coulomb energy still depends only on the principal combination $n$:

$$E_n = - \frac{\mu Z^2e^4} {2(4\pi\epsilon_0)^2\hbar^2n^2},$$

where $\mu$ is the reduced mass. The old theory therefore reproduced the gross hydrogen spectrum while adding more labels to the allowed motions. This resembles, but is not the same as, the later wave-mechanical degeneracy among angular-momentum states.

Additional Quantum Numbers

Sommerfeld’s extension introduced more quantum labels than the Bohr model. The old labels varied across authors and conventions, but the basic roles were:

Old-quantum label	Rough role in the model	Modern caution
$n$	principal energy label	survives as the principal quantum number
$n_r$	radial action label	not identical to a direct observable trajectory count
$n_\phi$ or $k$	angular action label	related historically to angular momentum, but not the modern $\ell$ without corrections
$m$	orientation label in a field	later reorganized by angular-momentum operators

The similarity to modern quantum numbers is real but should not be overstated. Modern hydrogen states are wavefunctions labeled by eigenvalues of commuting operators such as $H$, $L^2$, and $L_z$. They are not classical ellipses with definite electron positions.

The mismatch is especially visible in angular momentum. Old quantum theory did not include the correct zero-angular-momentum $s$ states in the same way wave mechanics does, and later semiclassical treatments require phase corrections such as the Langer modification.

Relativistic Corrections

Sommerfeld also treated the electron’s Kepler motion relativistically. Near the nucleus, especially in eccentric orbits, the electron speed can be a significant fraction of $c$. Relativity changes the orbital motion and produces a small splitting of levels that were degenerate in the simplest Bohr formula.

The relevant dimensionless strength is the fine-structure constant,

$$\alpha = \frac{e^2}{4\pi\epsilon_0\hbar c} \simeq \frac{1}{137}.$$

The size of fine-structure corrections is roughly smaller than the gross Bohr energy by a factor of $(Z\alpha)^2$:

$$\Delta E_{\mathrm{fs}} \sim \lvert E_n\rvert (Z\alpha)^2.$$

In one common old-quantum notation, the Sommerfeld fine-structure expansion for a hydrogenic atom can be written schematically as

$$E_{n,k} \simeq - \frac{\mu c^2(Z\alpha)^2}{2n^2} \left[ 1+ \frac{(Z\alpha)^2}{n} \left( \frac{1}{k} - \frac{3}{4n} \right) \right],$$

where $k=1,\ldots,n$ is an old angular quantum number. For a fixed point nucleus use $\mu=m_e$; substituting the reduced mass incorporates the leading mass correction but not all relativistic recoil effects. The important historical point is not the notation; it is that the energy began to depend on a second label once relativistic motion was included.

Modern fine structure is explained using relativistic quantum mechanics and spin-dependent corrections. The Dirac equation later produced a formula with a related structure, but with quantum numbers and physical interpretation reorganized. That later agreement does not make Sommerfeld’s ellipses literal.

What the Model Could Not Predict

The hydrogenic agreement was striking, but the construction depended on the separability and symmetry of the Coulomb problem. Relativistic orbits also precess: the ellipses in the figure represent the nonrelativistic model, not the exact relativistic trajectories.

The rules did not supply a general method for line intensities, many-electron atoms, or spin. Modern fine structure combines relativistic dynamics with the electron’s spin. The Dirac energy formula has the same dependence after identifying $k=j+1/2$, but the states, degeneracies, and allowed transitions require the modern theory. See limits of old quantum theory for the wider breakdown.

A modern orbital is therefore not a blurred Sommerfeld ellipse. It is a wavefunction with radial and angular dependence, determined by an eigenvalue equation and boundary conditions.

Common Mistakes

• Treating Sommerfeld ellipses as literal pictures of modern atomic orbitals.
• Assuming the old angular quantum number is identical to the modern orbital quantum number $\ell$.
• Presenting the fine-structure agreement as a proof that classical electron orbits were almost right.
• Forgetting that spin is absent from the Sommerfeld model.
• Applying action quantization without checking separability, turning-point phases, or coordinate dependence.
• Concluding that old quantum theory failed because it produced no correct results. Its problem was that it had no general, coherent mechanics.

Exercises

1. In the old Coulomb model, $n=n_r+n_\phi$. Why does the nonrelativistic energy depending only on $n$ imply a degeneracy among different old orbit shapes?

Solution

Different pairs $(n_r,n_\phi)$ can have the same sum $n$. If the nonrelativistic energy depends only on that sum, then distinct elliptical orbit shapes have the same energy. Relativistic corrections partly split this degeneracy by making the energy depend on an additional angular label.

2. Estimate the fine-structure scale for the hydrogen ground-state energy using $\lvert E_1\rvert\simeq 13.6\,\mathrm{eV}$ and $\alpha\simeq 1/137$.

Solution

The rough scale is

$$\Delta E_{\mathrm{fs}} \sim \lvert E_1\rvert\alpha^2 \simeq 13.6\,\mathrm{eV} \left( \frac{1}{137} \right)^2 \simeq 7.2\times 10^{-4}\,\mathrm{eV}.$$

This is a scale estimate for a correction, not a splitting of the ground state into different orbital levels. For $n=k=1$, the displayed expansion includes the extra factor $1/4$, giving a shift about $-1.8\times10^{-4}\,\mathrm{eV}$. The fine-structure correction is much smaller than the gross binding energy.

3. For a classical Coulomb ellipse, the eccentricity satisfies $\varepsilon^2=1+2EL^2/(\mu Z^2 k_e^2e^4)$, where $k_e=1/(4\pi\epsilon_0)$. Use the old rules to find the eccentricity at $n=2$, $k=1$ and at $n=k=2$.

Solution

Insert $E=-\mu Z^2k_e^2e^4/(2\hbar^2n^2)$ and $L=k\hbar$. Then $\varepsilon^2=1-k^2/n^2$. The first case gives $\varepsilon=\sqrt3/2$; the second gives zero, a circle. Both have the same nonrelativistic energy because $n$ is the same.

4. Explain why a modern hydrogen orbital should not be interpreted as a Sommerfeld ellipse.

Solution

A Sommerfeld ellipse is a classical trajectory with a definite path. A modern hydrogen orbital is a stationary wavefunction, labeled by operator eigenvalues and interpreted through probability density. It does not assign the electron a definite Kepler ellipse. Similar quantum numbers record a historical relationship, not an identity of physical pictures.

References

• B. H. Bransden and C. J. Joachain, Physics of Atoms and Molecules, 2nd ed., Pearson, 2003.
• H. Haken and H. C. Wolf, The Physics of Atoms and Quanta, 7th ed., Springer, 2005.
• M. Jammer, The Conceptual Development of Quantum Mechanics, 2nd ed., American Institute of Physics, 1989.
• J. Mehra and H. Rechenberg, The Historical Development of Quantum Theory, Springer, 1982-2001.
• A. Sommerfeld, “Zur Quantentheorie der Spektrallinien,” Annalen der Physik 51, 1-94 and 125-167, 1916.
• A. Sommerfeld, Atombau und Spektrallinien, Vieweg, 1919; English translation Atomic Structure and Spectral Lines, Methuen, 1923.
• W. Wilson, “The Quantum-Theory of Radiation and Line Spectra,” Philosophical Magazine 29, 795-802, 1915.