Skip to main content
SpringerLink
Log in

Bohmian Chaos in Multinodal Bound States

  • Published:
Foundations of Physics Aims and scope Submit manuscript

Abstract

We consider the Bohmian trajectories in a 2-d quantum harmonic oscillator with non commensurable frequencies whose wavefunction is of the form \(\Psi =a\Psi _{m_1,n_1}(x,y)+b\Psi _{m_2,n_2}(x,y)+c\Psi _{m_3,n_3}(x,y)\). We first find the trajectories of the nodal points for different combinations of the quantum numbers m, n. Then we study, in detail, a case with relatively large quantum numbers and two equal \(m's\). We find (1) fixed nodes independent of time and (2) moving nodes which from time to time collide with the fixed nodes and at particular times they go to infinity. Finally, we study the trajectories of quantum particles close to the nodal points and observe, for the first time, how chaos is generated in a complex system with multiple nodes scattered on the configuration space.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8
Fig. 9
Fig. 10
Fig. 11
Fig. 12
Fig. 13

Similar content being viewed by others

Data Availability

The datasets generated during the current study are available from the corresponding author on reasonable request.

Notes

  1. In fact we have

    $$\begin{aligned} \Psi _{m,n}=\Psi _m(x)\cdot \Psi _n(y), \end{aligned}$$
    (8)

    where

    $$\begin{aligned} \Psi _l(x)=\frac{1}{\sqrt{2^ll!}}\left( \frac{m_x\omega _x}{\pi \hbar }\right) ^{\frac{1}{4}}e^{-\frac{m_x\omega _x x^2}{2\hbar }}H_l\left( \sqrt{\frac{m_x\omega _ x}{\hbar }}x\right) ,l=0, 1, 2,\dots , \end{aligned}$$
    (9)

    and similarly for y.

  2. This is an example of a non stationary state with stationary nodes and chaos. This answers a question raised by Cesa, Martin and Struyve [26], who could not find a system of this type and wondered if such a system would generate chaos.

References

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

    Article  ADS  MathSciNet  MATH  Google Scholar 

  2. Bohm, D.: A suggested interpretation of the quantum theory in terms of “hidden" variables. II. Phys. Rev. 85, 180 (1952)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  3. 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 

  4. Parmenter, R.H., Valentine, R.: Deterministic chaos and the causal interpretation of quantum mechanics. Phys. Lett. A 201, 1 (1995)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  5. Iacomelli, G., Pettini, M.: Regular and chaotic quantum motions. Phys. Lett. A 212, 29 (1996)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  6. Sengupta, S., Chattaraj, P.: The quantum theory of motion and signatures of chaos in the quantum behaviour of a classically chaotic system. Phys. Lett. A 215, 119 (1996)

    Article  ADS  Google Scholar 

  7. de Sales, J., Florencio, J.: Quantum chaotic trajectories in integrable right triangular billiards. Phys. Rev. E 67, 016216 (2003)

    Article  ADS  MathSciNet  Google Scholar 

  8. Efthymiopoulos, C., Contopoulos, G.: Chaos in Bohmian quantum mechanics. J. Phys. A 39, 1819 (2006)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  9. Contopoulos, G., Tzemos, A.C.: Chaos in Bohmian quantum mechanics: a short review. Regul. Chaotic Dyn. 25, 476 (2020)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  10. Efthymiopoulos, C., Kalapotharakos, C., Contopoulos, G.: Nodal points and the transition from ordered to chaotic Bohmian trajectories. J. Phys. A 40, 12945 (2007)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  11. Efthymiopoulos, C., Kalapotharakos, C., Contopoulos, G.: Origin of chaos near critical points of quantum flow. Phys. Rev. E 79, 036203 (2009)

    Article  ADS  MathSciNet  Google Scholar 

  12. Benseny, A., Albareda, G., Sanz, Á.S., Mompart, J., Oriols, X.: Applied Bohmian mechanics. Eur. Phys. J. D 68, 1 (2014)

    Article  Google Scholar 

  13. Frisk, H.: Properties of the trajectories in Bohmian mechanics. Phys. Lett. A 227, 139 (1997)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  14. Konkel, S., Makowski, A.: Regular and chaotic causal trajectories for the Bohm potential in a restricted space. Phys. Lett. A 238, 95 (1998)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  15. Wu, H., Sprung, D.: Quantum chaos in terms of Bohm trajectories. Phys. Lett. A 261, 150 (1999)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  16. Barker, J., Akis, R., Ferry, D.: On the use of Bohm trajectories for interpreting quantum flows in quantum dot structures. Superlattices Microstruct. 27, 319 (2000)

    Article  ADS  Google Scholar 

  17. Falsaperla, P., Fonte, G.: On the motion of a single particle near a nodal line in the de Broglie–Bohm interpretation of quantum mechanics. Phys. Lett. A 316, 382 (2003)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  18. Wisniacki, D., Borondo, F., Benito, R.: Dynamics of quantum trajectories in chaotic systems. Europhys. Lett. 64, 441 (2003)

    Article  ADS  Google Scholar 

  19. Sanz, A., Borondo, F., Miret-Artés, S.: Quantum trajectories in atom-surface scattering with single adsorbates: the role of quantum vortices. J. Chem. Phys. 120, 8794 (2004)

    Article  ADS  Google Scholar 

  20. Sanz, A., Borondo, F., Miret-Artés, S.: Role of quantum vortices in atomic scattering from single adsorbates. Phys. Rev. B 69, 115413 (2004)

    Article  ADS  Google Scholar 

  21. Wisniacki, D.A., Pujals, E.R.: Motion of vortices implies chaos in Bohmian mechanics. Europhys. Lett. 71, 159 (2005)

    Article  ADS  MathSciNet  Google Scholar 

  22. Wisniacki, D., Pujals, E., Borondo, F.: Vortex dynamics and their interactions in quantum trajectories. J. Phys. A 40, 14353 (2007)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  23. Chou, C.-C., Wyatt, R.E.: Quantum vortices within the complex quantum Hamilton–Jacobi formalism. J. Chem. Phys. 128, 234106 (2008)

    Article  ADS  Google Scholar 

  24. Sanz, A., Miret-Artés, S.: Interplay of causticity and vorticality within the complex quantum Hamilton–Jacobi formalism. Chem. Phys. Lett. 458, 239 (2008)

    Article  ADS  Google Scholar 

  25. Borondo, F., Luque, A., Villanueva, J., Wisniacki, D.A.: A dynamical systems approach to Bohmian trajectories in a 2d harmonic oscillator. J. Phys. A 42, 495103 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  26. Cesa, A., Martin, J., Struyve, W.: Chaotic Bohmian trajectories for stationary states. J. Phys. A 49, 395301 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  27. Sanz, Á.S.: Atom-diffraction from surfaces with defects: a Fermatian, Newtonian and Bohmian joint view. Entropy 20, 451 (2018)

    Article  ADS  Google Scholar 

  28. Tzemos, A.C., Contopoulos, G., Efthymiopoulos, C.: Origin of chaos in 3-d Bohmian trajectories. Phys. Lett. A 380, 3796 (2016)

    Article  ADS  MATH  Google Scholar 

  29. Contopoulos, G., Tzemos, A.C., Efthymiopoulos, C.: Partial integrability of 3d Bohmian trajectories. J. Phys. A 50, 195101 (2017)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  30. Tzemos, A.C., Contopoulos, G.: Integrals of motion in 3d Bohmian trajectories. J. Phys. A 51, 075101 (2018)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  31. Tzemos, A.C., Efthymiopoulos, C., Contopoulos, G.: Origin of chaos near three-dimensional quantum vortices: a general Bohmian theory. Phys. Rev. E 97, 042201 (2018)

    Article  ADS  Google Scholar 

  32. Tzemos, A.C., Contopoulos, G., Efthymiopoulos, C.: Bohmian trajectories in an entangled two-qubit system. Phys. Scr. 94, 105218 (2019)

    Article  ADS  Google Scholar 

  33. Tzemos, A.C., Contopoulos, G.: Chaos and ergodicity in an entangled two-qubit Bohmian system. Phys. Scr. 95, 065225 (2020)

    Article  ADS  Google Scholar 

  34. Tzemos, A.C., Contopoulos, G.: Ergodicity and Born’s rule in an entangled two-qubit Bohmian system. Phys. Rev. E 102, 042205 (2020)

    Article  ADS  MathSciNet  Google Scholar 

  35. Tzemos, A., Contopoulos, G.: The role of chaotic and ordered trajectories in establishing Born’s rule. Phys. Scr. 96, 065209 (2021)

    Article  ADS  Google Scholar 

  36. Valentini, A.: Signal-locality, uncertainty, and the subquantum h-theorem. I. Phys. Lett. A 156, 5 (1991)

    Article  ADS  MathSciNet  Google Scholar 

  37. Valentini, A.: Signal-locality, uncertainty, and the subquantum h-theorem. II. Phys. Lett. A 158, 1 (1991)

    Article  ADS  MathSciNet  Google Scholar 

  38. Valentini, A., Westman, H.: Dynamical origin of quantum probabilities. Proc. R. Soc. A 461, 253 (2005)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  39. Tzemos, A., Contopoulos, G.: Bohmian quantum potential and chaos. Chaos Sol. Fract. 160, 112151 (2022)

    Article  MathSciNet  Google Scholar 

  40. Contopoulos, G., Efthymiopoulos, C., Harsoula, M.: Order and chaos in quantum mechanics. Nonlinear Phen Comput. Syst. 11, 107 (2008)

    MathSciNet  Google Scholar 

Download references

Acknowledgements

This research was conducted in the framework of the program of the RCAAM of the Academy of Athens “Study of the dynamical evolution of the entanglement and coherence in quantum systems”.

Funding

There are no ethical issues nor external funding associated with this work.

Author information

Authors and Affiliations

  1. Research Center for Astronomy and Applied Mathematics of the Academy of Athens, Soranou Efessiou 4, Athens, 11527, Attica, Greece

    Athanasios C. Tzemos & George Contopoulos

Authors
  1. Athanasios C. Tzemos
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. George Contopoulos
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Athanasios C. Tzemos.

Ethics declarations

Conflict of interest

On behalf of all authors, the corresponding author states that there is no conflict of interest.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Appendix 1: Finding the Nodal Points

Appendix 1: Finding the Nodal Points

The nodal points are of fundamental importance in the study of Bohmian chaos. These points are mathematical singularities of the Bohmian flow. Quantum particles close to a nodal point have large velocities, forming spiral vortices around it for some time. However, later the particles escape from the neighbourhood of the node. These esapes happen in two cases

  1. 1.

    when the nodal point acquires a large velocity when going or coming from infinity;

  2. 2.

    when a moving nodal point approaches and collides with a fixed (non moving) nodal point. This second mechanism appeared for the first time in the present paper.

After the escape the particle wanders around the configuration space until it comes close to the same or another nodal point and so on.

The evolution of the nodal points is not dictated by the Bohmian equations of motion but from their defining set of equations

$$\begin{aligned} \Psi _{Real}(x, y,t)=\Psi _{Imaginary}(x, y,t)=0. \end{aligned}$$
(34)

The solutions of these equations become singular from time to time. These are the times where the nodal points go to or come from infinity.

However, the solutions of (34) can be found analytically only if the quantum numbers m and n are small. But if m and n are large, then in most cases the nodal points are found only numerically. Namely, one needs first to detect graphically where the velocities of the Bohmian flow form vortices (as in Fig. 2) and then to plot successively the Bohmian velocity field by gradually increasing the time and spot the moving nodal points at the centers of the vortices.

In the present paper we followed this method and put the successive figures into video simulations, where the trajectories of the nodal points became evident. Furthermore, in the same way we found the positions of the X-points by observing the points from which emanate the two stable and two unstable eigendirections.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Tzemos, A.C., Contopoulos, G. Bohmian Chaos in Multinodal Bound States. Found Phys 52, 85 (2022). https://doi.org/10.1007/s10701-022-00599-1

Download citation

  • Received:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1007/s10701-022-00599-1

Keywords

Search

Navigation