Skip to main content

Interfering Quantum Trajectories Without Which-Way Information


Quantum trajectory-based descriptions of interference between two coherent stationary waves in a double-slit experiment are presented, as given by the de Broglie–Bohm (dBB) and modified de Broglie–Bohm (MdBB) formulations of quantum mechanics. In the dBB trajectory representation, interference between two spreading wave packets can be shown also as resulting from motion of particles. But a trajectory explanation for interference between stationary states is so far not available in this scheme. We show that both the dBB and MdBB trajectories are capable of producing the interference pattern for stationary as well as wave packet states. However, the dBB representation is found to provide the ‘which-way’ information that helps to identify the hole through which the particle emanates. On the other hand, the MdBB representation does not provide any which-way information while giving a satisfactory explanation of interference phenomenon in tune with the de Broglie’s wave particle duality. By counting the trajectories reaching the screen, we have numerically evaluated the intensity distribution of the fringes and found very good agreement with the standard results.

This is a preview of subscription content, access via your institution.

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5


  1. Bohr, N.: The quantum postulate and the recent development of atomic theory. Nature (London) 121, 580 (1928)

    ADS  Article  MATH  Google Scholar 

  2. Einstein, A.: On the development of our views concerning the nature and constitution of radiation. Phys. Z. 10, 817 (1909)

    Google Scholar 

  3. Einstein, A., Infeld, L.: The Evolution of Physics: From Early Concepts to Relativity and Quanta. The Cambridge University Press, Cambridge (1937)

    MATH  Google Scholar 

  4. Holland, P.R.: The Quantum Theory of Motion: An Account of the de Broglie–Bohm Causal Interpretation of Quantum Mechanics. Cambridge University Press, Cambridge (1995)

    MATH  Google Scholar 

  5. De-Broglie, L.: Recherches sur la théorie des quanta(Research on the Theory of the Quanta). Ph.D. thesis, Migration-université en cours d’affectation (1924)

  6. De-Broglie, L.: La mécanique ondulatoire et la structure atomique de la matière et du rayonnement (Wave mechanics and the atomic structure of matter and radiation). J. Phys. Radium 8, 225 (1927)

    Article  MATH  Google Scholar 

  7. Bacciagaluppi, G., Valentini, A.: Quantum Theory at the Crossroads: Reconsidering the 1927 Solvay Conference. Cambridge University Press, Cambridge (2009)

    Book  MATH  Google Scholar 

  8. Bohm, D.: A suggested interpretation of the quantum theory in terms of hidden ‘variables. 1’. Phys. Rev. 85, 166 (1952)

    ADS  MathSciNet  Article  MATH  Google Scholar 

  9. Kocsis, S., Braverman, B., Ravets, S., Stevens, M.J., Mirin, R.P., Shalm, L.K., Steinberg, A.M.: Observing the average trajectories of single photons in a two-slit interferometer. Science 332, 1170 (2011)

    ADS  Article  MATH  Google Scholar 

  10. Floyd, E.R.: Welcher weg? A trajectory representation of a quantum youngs diffraction experiment. Found. Phys. 37, 1403 (2007)

    ADS  Article  MATH  Google Scholar 

  11. John, M.V.: Modified de Broglie-Bohm approach to quantum mechanics. Found. Phys. Lett. (1988–2006), 15, 329 (2002)

  12. John, M.V.: Probability and complex quantum trajectories. Ann. Phys. 324, 220 (2009)

    ADS  MathSciNet  Article  MATH  Google Scholar 

  13. John, M.V.: Probability and complex quantum trajectories: Finding the missing links. Ann. Phys. 325, 2132 (2010)

    ADS  MathSciNet  Article  MATH  Google Scholar 

  14. John, M.V., Mathew, K.: Coherent states and modified de Broglie-Bohm complex quantum trajectories. Found. Phys. 43, 859 (2013)

    ADS  MathSciNet  Article  MATH  Google Scholar 

  15. Mathew, K., John, M.V.: Tunneling in energy eigenstates and complex quantum trajectories. Quantum Stud. 2, 403 (2015)

    MathSciNet  Article  MATH  Google Scholar 

  16. Dürr, D., Goldstein, S., Zanghi, N.J.: Quantum equilibrium and the origin of absolute uncertainty. J. Stat. Phys. 67, 843 (1992)

    ADS  MathSciNet  Article  MATH  Google Scholar 

  17. Carroll, R.: Quantum Theory, Deformation and Integrability. Elsevier, Amsterdam (2000)

    MATH  Google Scholar 

  18. Chou, C.C., Wyatt, R.E.: Quantum trajectories in complex space. Phys. Rev. A 76, 012115 (2007)

    ADS  Article  Google Scholar 

  19. Chou, C.C., Wyatt, R.E.: Quantum trajectories in complex space: One-dimensional stationary scattering problems. J. Chem. Phys. 128, 154106 (2008)

    ADS  Article  Google Scholar 

  20. Kolasiński, K., Szafran, B.: Electron paths and double-slit interference in the scanning gate microscopy. N. J. Phys. 17, 063003 (2015)

    Article  Google Scholar 

  21. Jönsson, C.: Electron diffraction at multiple slits. Am. J. Phys. 42, 4 (1974)

    ADS  Article  Google Scholar 

  22. Gondran, M., Gondran, A.: Measurement in the de Broglie-Bohm interpretation: double-slit, stern-gerlach, and EPR-B. Phys. Res. Int. 2014, 1 (2014)

    Article  Google Scholar 

  23. Mahler, D.H., Rozema, L., Fisher, K., Vermeyden, L., Resch, K.J., Wiseman, H.M., Steinberg, A.: Experimental nonlocal and surreal Bohmian trajectories. Sci. Adv. 2, e1501466 (2016)

    ADS  Article  Google Scholar 

  24. Feynman, R.P., Leighton, R.B., Sands, M.: The Feynman Lectures on Physics, vol. 1. Addison-Wesley, Boston (1964)

    MATH  Google Scholar 

  25. Philippidis, C., Dewdney, C., Hiley, B.: Quantum interference and the quantum potential. Il Nuov. Cim. B (1971-1996), 52, 15 (1979)

  26. Luis, A., Sanz, Á.S.: What dynamics can be expected for mixed states in two-slit experiments? Ann. Phys. 357, 95 (2015)

    ADS  Article  MATH  Google Scholar 

  27. Sanz, Á.S.: Investigating puzzling aspects of the quantum theory by means of its hydrodynamic formulation. Found. Phys. 45, 1153 (2015)

    ADS  MathSciNet  Article  MATH  Google Scholar 

  28. Davidović, M., Sanz, Á.S., Božić, M.: Description of classical and quantum interference in view of the concept of flow line. J. Russ. Laser Res. 36, 329 (2015)

    Article  Google Scholar 

  29. Sanz, Á.S., Miret-Artés, S.: A trajectory-based understanding of quantum interference. J. Phys. A 41, 435303 (2008)

    ADS  MathSciNet  Article  MATH  Google Scholar 

  30. Ghose, P.: On the incompatibility of standard quantum mechanics and conventional de Broglie-Bohm theory. Pramana-J. Phys. 59, 417 (2002)

    ADS  Article  Google Scholar 

  31. Floyd, E.R.: Modified potential and Bohm’s quantum-mechanical potential. Phys. Rev. D 26, 1339 (1982)

    ADS  Article  Google Scholar 

Download references


We wish to thank an anonymous Reviewer for helping to point out the important feature of same-color crossings in the MdBB trajectory plots. We also thank Professor K. Babu Joseph for useful discussions.

Author information

Authors and Affiliations


Corresponding author

Correspondence to Moncy V. John.

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Mathew, K., John, M.V. Interfering Quantum Trajectories Without Which-Way Information. Found Phys 47, 873–886 (2017).

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI:


  • Quantum trajectory
  • Double-slit experiment
  • de Broglie–Bohm theory
  • Complex trajectories
  • Which-way information