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The relativistic energy spectrum of hydrogen
Author(s) -
Marsch E.
Publication year - 2005
Publication title -
annalen der physik
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.009
H-Index - 68
eISSN - 1521-3889
pISSN - 0003-3804
DOI - 10.1002/andp.200410137
Subject(s) - physics , dirac equation , hydrogen atom , hamiltonian (control theory) , two body dirac equations , hydrogen like atom , electron , pauli exclusion principle , quantum electrodynamics , principal quantum number , quantum mechanics , atomic physics , total angular momentum quantum number , relativistic wave equations , exotic atom , bound state , quantum number , relativistic quantum mechanics , position and momentum space , angular momentum , hydrogen , angular momentum coupling , quantum , quantum dynamics , mathematical optimization , mathematics , group (periodic table) , quantum dissipation
The method originally used by Dirac to derive his equation for a single electron is here applied to the two‐particle system electron plus proton, considered as an elementary fermion, to obtain a relativistic two‐particle Dirac‐Breit‐type Hamiltonian for the hydrogen atom. This problem can be solved exactly. Thus the relativistic energy levels of the hydrogen atom as a system bound by the static Coulomb force are obtained. The radial part of the resulting Hamiltonian operates in a four‐dimensional space describing particles, respectively antiparticles, i. e. electron and positron as well as proton and antiproton. The spin of a particle is described by the normal two‐component Pauli spinor, and therefore standard theoretical tools for dealing with angular‐momentum coupling can be exploited. The classical energy states of the hydrogen atom are retained in the appropriate non‐relativistic limit, in particular the energy levels resulting from Schrödinger's equation. The exact energy spectrum shows the expected dependence on the reduced mass of the two‐particle system, and thus describes the recoil of the core properly. The fine structure of the hydrogen spectrum arises from a dependence of the energy levels upon the quantum number of the total angular momentum.