The Sommerfeld-Dirac Paradox Reconsidered: Too Clever by Half

In 1916 Arnold Sommerfeld derived a relativistic version of the Bohr hydrogen atom that correctly predicted the observed ”fine splitting” in hydrogen spectral lines. A decade later, Dirac derived the first relativistic wavefunction model for a spin-1/2 electron that gave the same prediction. This created a paradox: how could Sommerfeld possibly have obtained the correct splitting without including (the then inconceivable notion of) electron spin? We present a new perspective. There is no paradox.

Transcript

1. The Sommerfeld-Dirac Paradox Reconsidered Too Clever by 1/2 Dr. Neil J. Gunther Performance Dynamics Research APS Global Physics Summit Anaheim, California March 20, 2025 © 2025 Performance Dynamics Research The Sommerfeld-Dirac Paradox Reconsidered April 8, 2025 1 / 21
2. Abstract In 1916 Arnold Sommerfeld derived a relativistic version of the Bohr hydrogen atom: both being Newtonian particle models. It correctly accounted for the observed ”fine splitting” in the hydrogen spectrum. A decade later, Dirac derived the first rela- tivistic wavefunction model of the hydrogen atom by incorporating Pauli spin matrices. This created a paradox: How could Sommerfeld possibly have obtained the correct spectrum without including (the then inconceivable notion of) electron spin? Surprisingly, opinions about resolving this paradox have been varied and con- tentious during the intervening decades, with most physicists who are aware of the paradox assuming Sommerfeld just made a lucky mistake. Other luminaries, like Heisenberg and Weinberg, have suggested that the resolution might reside in the un- derlying dynamical symmetries. We offer a new perspective by introducing a discrim- inator, comprised of the appropriate quantum numbers, that leads to what we call the ¨ Uber-Relativistic Approximation (URA). Viewed in this setting, it becomes clear that Sommerfeld did not make a mistake. Rather, the URA reveals that what appears to be the same energy eigenvalue formula mathematically for both Sommerfeld and Dirac models, is actually two different approximations that share some numerical values in common. This framework also enables us to explain why prior opinions have been so varied and incomplete. © 2025 Performance Dynamics Research The Sommerfeld-Dirac Paradox Reconsidered April 8, 2025 2 / 21
3. My paradoxical heritage My M.Sc. advisor Prof. Christie Eliezer Eliezer (and me) with Dirac in Australia My M.Sc. was on Lie groups and dynamical symmetries [1, 2]. Eliezer was one of Dirac’s few research students. He told me about the Sommerfeld-Dirac paradox1 as a possible thesis topic. I assumed it was already resolved using QFT, e.g., QED vacuum effects, Bethe-Salpeter ladder diagrams, axiomatic C∗ algebras, etc. (none of which either Eliezer or I understood in detail) so, I forgot about it. 1 The Abstract contains a statement of the Sommerfeld-Dirac paradox. © 2025 Performance Dynamics Research The Sommerfeld-Dirac Paradox Reconsidered April 8, 2025 3 / 21
4. I was wrong More recently, I became aware of this litany claiming to have resolved the paradox2: 1 Bergmann (1942): “cancellation of two omissions: wavefunction + spin” [3] 2 Bethe & Saltpeter (1957): “wrong interpretation of quantum numbers” [4] 3 Heisenberg (1968): “a miracle of (latent) dynamical symmetries” [5] 4 Biedenharn (1983): “rotating frame makes it consistent” [6] 5 Weinberg (1995): “an accident” [7] 6 Granovskii (2004): “if he’d done good science then k = + 1/2” [8] 7 Possibly others I’m not aware of This many diverse opinions suggests something has not been properly understood. They can’t all be right. Separately, each might contain a grain of truth. Has the paradox really been resolved? 2 See slide 16 for a more analytic summary. © 2025 Performance Dynamics Research The Sommerfeld-Dirac Paradox Reconsidered April 8, 2025 4 / 21
5. Hydrogen gas spectrum Sunlight (rainbow) is a continuous spectrum (He & plasma excitations) But the Sun (most stars) is a continuous hydrogen bomb H emission spectrum (lower temp) exhibits discrete lines (Why?) Balmer (1885): Energy (wavelength) spacing is the inverse square of line numbers, ns = 1, 2, 3, . . . and ni = 2, 3, . . . (Why?) 1 λ = 13.6eV hc 1 n2 s − 1 n2 i (1) © 2025 Performance Dynamics Research The Sommerfeld-Dirac Paradox Reconsidered April 8, 2025 5 / 21
6. Arnold Sommerfeld and Niels Bohr © 2025 Performance Dynamics Research The Sommerfeld-Dirac Paradox Reconsidered April 8, 2025 6 / 21
7. Bohr model of hydrogen atom (1913) Planetary model with a proton “sun” and an electron “planet” Gravity replaced by the Coulomb electrostatic force (inverse square law) Orbital paths assumed circular Radiative collapse prevented by quantizing the orbital energy (via angular momentum) In the spirit of Planck (1900) Ground state: E1 = −13.6 eV En = − 13.6 eV n2 , n = 1, 2, 3, . . . (2) cf. Balmer empirical formula in (1) © 2025 Performance Dynamics Research The Sommerfeld-Dirac Paradox Reconsidered April 8, 2025 7 / 21
8. Spectral line structure Hydrogen single α line (red) is actually a doublet at 1000x magnification Bohr (circular) model does not account for this detail (no shame) Gap proportional to fine structure constant (introduced by Sommerfeld) α = e2 c ≈ 1 137 (or 0.00729927) (3) Today it’s the QED coupling constant But what causes this splitting? © 2025 Performance Dynamics Research The Sommerfeld-Dirac Paradox Reconsidered April 8, 2025 8 / 21
9. Sommerfeld’s corrections to Bohr’s model Bohr classical particle model is not relativistically invariant 1. Sommerfeld imposed elliptical orbits (SRT makes them precess [9]) 2. Discrete energy levels via phase-space quantization: pr dr = n and pθ dθ = k 3. Ellipse semi axes aspect ratio a/b = n/k [11] En = mc2 1 + α n − k + √ k2 − α2 2 −1/2 (4) 4. n = 1, 2, 3, . . . (Bohr integers) [10] 5. k = 1, 2, . . . , n (Sommerfeld integers) [9, 11] 6. n = k (circular Bohr orbits) © 2025 Performance Dynamics Research The Sommerfeld-Dirac Paradox Reconsidered April 8, 2025 9 / 21
10. Dirac’s spin-1/2 eigenvalues En = mc2 1 + α n − |κ| + √ κ2 − α2 2 −1/2 (5) 1 n = 1, 2, 3, . . . (Bohr orbit levels) 2 = 0, 1, 2, . . . , (n − 1) (orbital angular momentum) 3 s = ±1 2 (electron spin angular momentum) 4 j = + |s| = + 1/2 (total angular momentum) 5 κ = ±(j + 1/2) (spin-orbit alignment) Dirac spin operator (S = 1 2 σ) couples to orbital angular momentum (L) L and S opertors are not dynamic invariants but J and κ are The 1/2 in items 4 and 5 comes from fermionic spin © 2025 Performance Dynamics Research The Sommerfeld-Dirac Paradox Reconsidered April 8, 2025 10 / 21
11. But how could this be? The Paradox 1 Relativistic Dirac energy levels En(N2 ) are for a spin-1/2 electron (1928). 2 En(N2 ) intrinsically involves extended (4 × 4) Pauli spin matrices (c.1927). 3 Relativistic Sommerfeld energy levels En(N1 ) are for a spinless electron (1916). 4 En(N1 ) is an ad hoc semi-classical particle electron model. 5 En(N2 ) is a quantum mechanical electron wavefunction model. 6 How could such different models possibly produce identical energy levels? © 2025 Performance Dynamics Research The Sommerfeld-Dirac Paradox Reconsidered April 8, 2025 11 / 21
12. Discriminating between energy expressions Energy levels in (4) and (5) have a common algebraic form En(Ni ) = mc2 1 + (Zα)2 N2 i , i = 1, 2, 3 (6) where Ni = n − Ki + K2 i − α2 and Definition 1 (Discriminator) Ki =          k if i = 1, Bohr-Sommerfeld (BS) for s=0 electron (7a) |κ| if i = 2, Dirac Equation (DE) for s=1/2 electron (7b) + 1 2 if i = 3, Klein-Gordon (KG) for s=0 electron (7c) K3 corresponds to the Klein-Gordon spinless electron wavefunction (c.1924). Will be used for comparison with spinless electron particle K1 . Ki uniquely identifies the corresponding H energy-levels En(Ni ) in (6). © 2025 Performance Dynamics Research The Sommerfeld-Dirac Paradox Reconsidered April 8, 2025 12 / 21
13. Comparing quantum numbers 1 The BS integer quantum number, k, is defined by ellipse aspect ratio a/b = n/k where n = 1, 2, 3, . . . (8) = 0, 1, 2, 3, . . . , (n − 1) (9) k ≡ + 1 = 1, 2, 3, . . . , n (10) 2 The DE composite quantum number, |κ|, is given by |κ| = j + 1 2 = + 1 2 + 1 2 |κ| ≡ + 1 = 1, 2, 3, . . . , n (11) From (10) and (11), BS K1 and DE K2 are also identical integer sequences. But from (7c), KG discriminator K3 = + 1 2 generates a half-integer sequence. © 2025 Performance Dynamics Research The Sommerfeld-Dirac Paradox Reconsidered April 8, 2025 13 / 21
14. Using Ki to compute H energy levels Table 1: 2P → 2P3/2 , 2P1/2 splitting Model n k j |κ| ∆E (eV) BS 2 1, 2 - - - 0.00004529 DE 2 - 0, 1 1/2, 3/2 1, 2 0.00004529 KG 2 - 0, 1 - - 0.00012072 Splitting energy difference: 4.53 × 10−5 eV (Table 1) KG predicted splitting is 3x too big !!! Principal level n2 = −10.2 eV Lyman doublet (121.500, 121.501) nm (Table 2) Table 2: Lyman-α spectral doublet Model n k j |κ| ∆Eα (eV) λα (nm) 1 1 - - - - - BS 2 1 - - - 10.2044 121.500 2 2 - - - 10.2044 121.501 1 - 0 1/2 1 - - DE 2 - 0 1/2 1 10.2044 121.500 2 - 1 3/2 2 10.2044 121.501 © 2025 Performance Dynamics Research The Sommerfeld-Dirac Paradox Reconsidered April 8, 2025 14 / 21
15. Klein-Gordon En (N3 ) wasn’t accessible (luckily) 1 KG levels En(N3 ) are different from En(N1 ) and En(N2 ). 2 A 1/2-integer quantum number would have looked bizarre in 1916. No experimental or theoretical justification. For both Bohr and Sommerfeld, discrete energy spectra had to be indexed by integers (cardinals), because of the Balmer formula. Still is for n and today. 3 K3 was not accessible to Sommerfeld because the 1/2 arises from boundary conditions on the KG wavefunction—an unknown concept in 1916. 4 Even if K3 had somehow been explored ab initio by Sommerfeld, it yields the wrong spectrum and would have been rejected—just as Schr¨ odinger did a decade later. 5 Sommerfeld had no choice but to make K1 = k an integer sequence. But elliptical orbits alone do not produce En(N1) energy levels. Full SRT Hamiltonian is needed to produce algebraic form En(N1) in (6). And En(N1) clearly worked numerically in 1916. © 2025 Performance Dynamics Research The Sommerfeld-Dirac Paradox Reconsidered April 8, 2025 15 / 21
16. Paradoxical propositions Several conflicting aspects have fueled the debate over the S-D paradox. 1 En(N1 ) should’ve been En(N3 ). Both s = 0 particles. 2 En(N1 ) and En(N2 ) have identical algebraic forms. 3 K1 in (7a) and K2 in (7b) have identical integer sequences. 4 En(N1 ) and En(N2 ) are solutions to very different differential equations, viz., ∂2/∂t2 and ∂/∂t, respectively. 5 K1 is a function of k — an ad hoc semi-classical quantum number (integer) — while K2 is a function of κ, a quantum mechanical pseudo-scalar eigenvalue. Seen in this way, it is astonishing indeed that Sommerfeld could arrive at the correct fine-structure energy levels a decade before (i) formalized quantum theory and (ii) Dirac’s relativistic spinor eigenvalues (5). Seemingly irreconcilable (miracle) but some have claimed a resolution. [3, 4, 6] But, is this the right way to look at it? © 2025 Performance Dynamics Research The Sommerfeld-Dirac Paradox Reconsidered April 8, 2025 16 / 21
17. ¨ Uber relativistic approximation K1 ≡ K2 because the k values over-estimate SRT contribution. They’re chosen, ad hoc, as inputs into the analysis via the angular phase-space quantization condition pθ dθ = k for Coulomb bound states. Conversely, K2 values are outputs defined by the eigenvalues obtained from solving the relativistic quantum wave equation for Coulomb bound states. En(N1 ) does not correctly predict spin-1/2 effects ... The 3 Zs: 1) Zitterbewegung 2) Zeeman splitting 3) Z > 1 spectra Similarly, En(N2 ) does not correctly predict 1) Hyperfine splitting 2) Lamb shift 3) Anomalous (g − 2)/2 Physics is an empirical science. In that vein, En(N1 ), might be more properly called the ¨ uber-relativistic3 approximation to En(N2 ). 3 In the sense of being overly zealous. © 2025 Performance Dynamics Research The Sommerfeld-Dirac Paradox Reconsidered April 8, 2025 17 / 21
18. Resolved: There is no paradox Theorem 2 The effect of Sommerfeld’s corrections to the Bohr model, viz., SRT and K1, exaggerates the relativistic contribution, that enforces k = + 1, by virtue of constraining the aspect ratio of the classical particle orbit, a/b (semi axes), to be a rational number, n/k, in the phase integral quantization. This constraint conceals a multitude of sins (“spins of omission”) unknown in 1916. Proof. K1 with reduced semiminor axis (b) corresponds to K 1 = K1 − 1/2 which gives the correct SRT contribution but the wrong H spectrum, viz., En(N3), not En(N2). Conversely, replacing K3 with K 1 , yields the Sommerfeld particle spectrum, K1. There is no analytic route to K2. Corollary 3 (All about approximations) 1) There was no miracle [5] or cancellation of errors [3] or accident [7] or bad science [8]. 2) And SRT orbital precession [9] breaks dynamical SO(4) symmetry [5]. 3) BS is a spinless semi-classical (¨ uber relativistic) approximation to DE. 4) DE is only an approximation to QED [12]. © 2025 Performance Dynamics Research The Sommerfeld-Dirac Paradox Reconsidered April 8, 2025 18 / 21
19. Conclusion “Sometimes an idea which looks completely paradoxical at first, if analyzed to completion in all detail and experimental situations, may not be paradoxical.” —R. P. Feynman4 4 Nobel Lecture (1965). © 2025 Performance Dynamics Research The Sommerfeld-Dirac Paradox Reconsidered April 8, 2025 19 / 21
20. References I [1] N.J. Gunther, Dynamical symmetry Groups: The Study and Interpretation of Certain Invariants as Group Generators in Quantum Mechanics, M.Sc., La Trobe University, Submitted to the Dept. of Applied Mathematics (1976) [2] N.J. Gunther, “Dynamical Symmetry Breaking in The Lewis-Riesenfeld Oscillator”, J. Math.Phys. (21)7, 286 (1980) [3] P.G. Bergmann, Introduction to the Theory of Relativity, Dover 1976 (Orig. pub. 1942) [4] H.A. Bethe and E.E. Salpeter, Quantum Mechanics of One- and Two-Electron Atoms, Springer-Verlag (1957) [5] W. Heisenberg, “Ausstrahlung von Sommerfelds Werk in der Gegenwart,” Physikalische Bl¨ atter, 24 530-537 (1968) [6] L.C. Biedenharn, “The ‘Sommerfeld Puzzle’ Revisited and Resolved,” Foundations of Physics, vol. 13, no. 1 (1983) [7] S. Weinberg, The Quantum Theory of Fields, Vol. 1: Foundations, Cambridge Univ. Press (1995) [8] Y. Granovskii, “Sommerfeld Formula and Dirac’s Theory,” Uspekhi Fiz. Nauk, 47(5), 523-524 (2004) [9] M. Born, Mechanics of the Atom, Ungar, 1960 (Orig. German 1924), [10] N. Bohr, “On the constitution of atoms and molecules,” The London, Edinburgh and Dublin Philosophical Magazine and Journal of Science, 26 (151): 1-25 (July 1913) [11] A. Sommerfeld, Atombau und Spektrallinien, Friedrich Vieweg und Sohn, Braunschweig (1919) [12] R.P. Feynman, “Quantum Electrodynamics,” W.A. Benjamin (1973) © 2025 Performance Dynamics Research The Sommerfeld-Dirac Paradox Reconsidered April 8, 2025 20 / 21
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