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Relativistic Hydrogen: Dirac vs Schrodinger, Fine Structure, Zitterbewegung

What you are seeing: hydrogen energy levels from the nonrelativistic Schrodinger equation set against the exact Dirac-Coulomb (Sommerfeld) solution. The Dirac levels depend only on n and j, so 2s(1/2) and 2p(1/2) stay degenerate while 2p(3/2) lies above, the fine structure of order (Z alpha)^4. Sweep the nuclear charge Z and watch the relativistic deepening and the splitting grow. One panel draws both level ladders at true scale with an auto-zoomed n=2 inset, a second shows Zitterbewegung (the Dirac position trembling at 2 m c^2 / hbar over a sub-luminal drift), and a third plots the n=2 splitting against Z on log-log axes of slope 4.

Figure 1. Hydrogen-like levels from the nonrelativistic Schrodinger equation against the exact Dirac-Coulomb spectrum: the relativistic levels are deeper and split by total angular momentum j (the fine structure, of order (Z alpha)^4), and a free Dirac packet trembles (Zitterbewegung) at angular frequency 2 m c^2 / hbar. Method: closed-form Schrodinger and exact Dirac (Sommerfeld) levels; deterministic.
nuclear charge Z50
packet momentum p (mc)0.60

WHAT TO TRY

  • Sweep the nuclear charge Z up: the Dirac levels deepen below the Schrodinger ones and the fine-structure splitting grows as (Z alpha)^4. Relativity matters most for heavy nuclei.
  • Watch 2s(1/2) and 2p(1/2) stay degenerate while 2p(3/2) sits above: the Dirac energy depends only on n and j, which is the exact fine structure before the Lamb shift.
  • Read Z alpha approach 1: as Z climbs toward 137 the 1s level dives toward -mc^2 and the Sommerfeld formula signals where a point-nucleus Dirac treatment breaks down.