Relativistic Quantum Mechanics

  • Alexander Komech


In 1928, Dirac introduced the new wave equation, which was a relativistic covariant generalization of the Schrödinger equation. In this case, the wave function has 4 complex components and the coefficients of the equation are 4×4 Dirac matrices.

Dirac calculated the first two approximations of this equation in the limit as c→∞. The approximation up to order 1/c coincides with the Pauli equation, while the second approximation displays, up to order 1/c 2, the Russell–Sounders spin-orbital coupling, as well as some other effects.

The resulting equation admits the Lagrangian and Hamiltonian formulations, which provide the corresponding conserved observables and the continuity equation for charge and current. The angular momentum automatically includes the spin component with the factor 1/2, as predicted by Goudsmit and Uhlenbeck.

A few months later, Gordon and Darwin independently solved the Dirac spectral problem for Hydrogen. Now the energies depend on the angular momentum, in contrast to the nonrelativistic case. This dependence perfectly explains the ‘fine structure’ of the spectrum.

The energy for the Dirac equation is not bounded from above and from below, suggesting instability of solutions. This problem was solved in quantum electrodynamics by imposing anticommutation relations for the electron field.


Dirac Equation Dirac Operator Spectral Problem Lagrangian Density Gordon Equation 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


  1. 12.
    H. Bethe, E. Salpeter, Quantum Mechanics of One- and Two-Electron Atoms (Springer, Berlin, 1957) zbMATHGoogle Scholar
  2. 13.
    J.D. Bjorken, S.D. Drell, Relativistic Quantum Mechanics (McGraw-Hill, New York, 1964) Google Scholar
  3. 39.
    C.G. Darwin, The wave equation of the electron. Proc. R. Soc. Lond. A 118, 654–680 (1928) ADSzbMATHCrossRefGoogle Scholar
  4. 45.
    P.A.M. Dirac, The quantum theory of the electron, I. Proc. R. Soc. Lond. A 117, 610–624 (1928) ADSzbMATHCrossRefGoogle Scholar
  5. 71.
    I.M. Gelfand, R.A. Minlos, Z.Ya. Shapiro, Representations of the Rotation and Lorentz Groups and Their Applications (Pergamon, Oxford, 1963) Google Scholar
  6. 74.
    W. Gordon, Die Energieniveaus der Wassenstoffatoms nach der Diracschen Quantentheorie des Elektrons. Z. Phys. 48, 11–14 (1928) ADSzbMATHCrossRefGoogle Scholar
  7. 81.
    K. Hannabus, An Introduction to Quantum Theory (Clarendon, Oxford, 1997) Google Scholar
  8. 92.
    C. Itzykson, J.B. Zuber, Quantum Field Theory (McGraw-Hill, New York, 1980) Google Scholar
  9. 158.
    J.J. Sakurai, Advanced Quantum Mechanics (Addison-Wesley, Reading, 1967) Google Scholar
  10. 159.
    G. Scharf, Finite Quantum Electrodynamics. The Causal Approach (Springer, Berlin, 1995) zbMATHCrossRefGoogle Scholar
  11. 160.
    L. Schiff, Quantum Mechanics (McGraw-Hill, New York, 1955) zbMATHGoogle Scholar

Copyright information

© Springer Science+Business Media Dordrecht 2013

Authors and Affiliations

  • Alexander Komech
    • 1
  1. 1.Faculty of MathematicsUniversity of ViennaViennaAustria

Personalised recommendations