Foundations of Physics

, Volume 46, Issue 8, pp 1006–1021 | Cite as

Does Bohm’s Quantum Force Have a Classical Origin?



In the de Broglie–Bohm formulation of quantum mechanics, the electron is stationary in the ground state of hydrogenic atoms, because the quantum force exactly cancels the Coulomb attraction of the electron to the nucleus. In this paper it is shown that classical electrodynamics similarly predicts the Coulomb force can be effectively canceled by part of the magnetic force that occurs between two similar particles each consisting of a point charge moving with circulatory motion at the speed of light. Supposition of such motion is the basis of the Zitterbewegung interpretation of quantum mechanics. The magnetic force between two luminally-circulating charges for separation large compared to their circulatory motions contains a radial inverse square law part with magnitude equal to the Coulomb force, sinusoidally modulated by the phase difference between the circulatory motions. When the particles have equal mass and their circulatory motions are aligned but out of phase, part of the magnetic force is equal but opposite the Coulomb force. This raises a possibility that the quantum force of Bohmian mechanics may be attributable to the magnetic force of classical electrodynamics. It is further shown that relative motion between the particles leads to modulation of the magnetic force with spatial period equal to the de Broglie wavelength.


Bohmian mechanics Quantum force Quantum potential Zitterbewegung interpretation of quantum mechanics  Classical spinning particle 


  1. 1.
    Bohm, D.: A suggested interpretation of the quantum theory in terms of “hidden” variables. I. Phys. Rev. 85(2), 166–179 (1952)ADSMathSciNetCrossRefMATHGoogle Scholar
  2. 2.
    Hestenes, D.: The zitterbewegung interpretation of quantum mechanics. Found. Phys. 20(10), 1213–1232 (1990)ADSMathSciNetCrossRefGoogle Scholar
  3. 3.
    Dirac, P.A.M.: The quantum theory of the electron. Proc. R. Soc. Lond. A 117, 610 (1928)ADSCrossRefMATHGoogle Scholar
  4. 4.
    Schrödinger, E.: Über die kräftefreie Bewegung in der relativistischen Quantenmechanik. Sitzungber. Preuss. Akad. Wiss. Phys.-Math. Kl.24, 418 (1930)Google Scholar
  5. 5.
    Huang, K.: On the zitterbewegung of the Dirac electron. Am. J. Phys. 20, 479 (1952)ADSCrossRefMATHGoogle Scholar
  6. 6.
    Hestenes, D.: op. citGoogle Scholar
  7. 7.
    Rivas, M.: Kinematical Theory of Spinning Particles. Kluwer Academic, Dordrecht (2001)MATHGoogle Scholar
  8. 8.
    Rivas, M.: An interaction Lagrangian for two spin 1/2 elementary Dirac particles. J. Phys. A 40, 2541–2552 (2007)ADSMathSciNetCrossRefMATHGoogle Scholar
  9. 9.
    Jackson, J.D.: Classical Electrodynamics. Wiley, New York (1998)MATHGoogle Scholar
  10. 10.
    Hestenes, D.: Zitterbewegung in quantum mechanics. Found. Phys. 40(1), 1–54 (2010)ADSMathSciNetCrossRefMATHGoogle Scholar
  11. 11.
    Rivas, op. cit., Section 6.1.1, 254–264Google Scholar
  12. 12.
    Holland, P.: Quantum potential energy as concealed motion. Found. Phys. 45(2), 134–141 (2015)ADSMathSciNetCrossRefMATHGoogle Scholar
  13. 13.
    Dennis, G., de Gosson, M., Hiley, B.: Bohm’s Quantum Potential as an Internal Energy, (preprint) arXiv:1412.5133 (2014)
  14. 14.
    Salesi, G.: Spin and the Madelung fluid. Mod. Phys. Lett. A 11, 1815 (1996)ADSMathSciNetCrossRefMATHGoogle Scholar
  15. 15.
    Salesi, G., Recami, E.: A velocity field and operator for spinning particles in (nonrelativistic) quantum mechanics. Found. Phys. 28, 763–773 (2010)ADSMathSciNetCrossRefGoogle Scholar
  16. 16.
    Dolce, D.: Compact time and determinism for bosons. Found. Phys. 41(178), 178–203 (2011)ADSMathSciNetCrossRefMATHGoogle Scholar
  17. 17.
    Dolce, D.: Gauge interaction as periodicity modulation. Ann. Phys. 327(6), 1562–1592 (2012)ADSMathSciNetCrossRefMATHGoogle Scholar
  18. 18.
    Dolce, D., Perali, A.: The role of quantum recurrence in superconductivity, carbon nanotubes, and related gauge symmetry breaking. Found. Phys. 44(9), 905–922 (2014)ADSMathSciNetCrossRefMATHGoogle Scholar
  19. 19.
    De Luca, J.: Minimizers with discontinuous velocities for the electromagnetic variational method. Phys. Rev. E 82, 026212 (2010)ADSMathSciNetCrossRefGoogle Scholar
  20. 20.
    Driver, R.D.: A “backwards” two-body problem of classical relativistic electrodynamics. Phys. Rev. 178(5), 2051–2057 (1969)ADSMathSciNetCrossRefGoogle Scholar
  21. 21.
    Raju, C.K.: The electrodynamic 2-body problem and the origin of quantum mechanics. Found. Phys. 34(6), 937–962 (2004)ADSMathSciNetCrossRefMATHGoogle Scholar
  22. 22.
    De Luca, J.: Stiff three-frequency orbit of the hydrogen atom. Phys. Rev. E 73, 026221 (2006)ADSCrossRefGoogle Scholar
  23. 23.
    Raju, C.K., Raju, S.: Radiative damping and functional differential equations. Mod. Phys. Lett. A 26(35), 2627–2638 (2011)ADSMathSciNetCrossRefMATHGoogle Scholar
  24. 24.
    Price, H.: Time’s Arrow and Archimedes Point: New Directions for the Physics of Time. Oxford University Press, Oxford (1996)Google Scholar
  25. 25.
    Price, H., Wharton, K.: A Live Alternative to Quantum Spooks, (preprint) arXiv:1510.06712 (2015)
  26. 26.
    Price, H., Wharton, K.: Disentangling the Quantum World, (preprint) arXiv:1508.01140 (2015)
  27. 27.
    Dirac, P.A.M.: Classical theory of radiating electrons. Proc. R. Soc. A 167, 148 (1938)ADSCrossRefMATHGoogle Scholar
  28. 28.
    Wheeler, J.A., Feynman, R.P.: Interaction with the absorber as the mechanism of radiation. Rev. Mod. Phys. 17, 157–181 (1945)ADSCrossRefGoogle Scholar
  29. 29.
    Schild, A.: Electromagnetic two-body problem. Phys. Rev. 131, 2762 (1963)ADSMathSciNetCrossRefMATHGoogle Scholar
  30. 30.
    De Luca, J.: Variational principle for the Wheeler-Feynman electrodynamics. J. Math. Phys. 50, 062701 (2009)ADSMathSciNetCrossRefMATHGoogle Scholar
  31. 31.
    De Luca, J.: Double-slit and electromagnetic models to complete quantum mechanics. J. Comput. Theor. Nanosci. 8(6), 1040–1051 (2010)CrossRefGoogle Scholar
  32. 32.
    De Luca, J.: Variational electrodynamics of atoms. Prog. Electromagn. Res. B 53, 147–186 (2013)CrossRefGoogle Scholar
  33. 33.
    Hiley, B.J.: Is the electron stationary in the ground state of the Dirac hydrogen atom in Bohms Theory?, (preprint) arXiv:1412.5887v1 (2014)
  34. 34.
    Hiley, B.J., Callaghan, R.E.: Clifford algebras and the Dirac-Bohm quantum Hamilton-Jacobi equation. Found. Phys. 42, 192–208 (2012)ADSMathSciNetCrossRefMATHGoogle Scholar
  35. 35.
    Bohm, D.: op. citGoogle Scholar
  36. 36.
    Griffiths, J.D.: Introduction to Elementary Particles, p. 162. Wiley-VCH, Weinheim (2008)Google Scholar

Copyright information

© Springer Science+Business Media New York 2016

Authors and Affiliations

  1. 1.Mercer IslandUSA

Personalised recommendations