Abstract
In an attempt to explore further the Madelung fluid-like representation of quantum mechanics, we derive the small perturbation equations of the fluid with respect to its basic states. The latter are obtained from the Madelung transform of the Schrödinger equation eigenstates. The fundamental eigenstates of de Broglie monochromatic matter waves are then shown to be mapped into the simple basic states of a fluid with constant density and velocity, where the latter is the de Broglie group velocity. The normal modes with respect to these basic states are derived and found to also satisfy the de Broglie dispersion relation. Despite being dispersive waves, their propagation mechanism is equivalent to that of sound waves in a classical ideal adiabatic gas. We discuss the physical interpretation of these results.
Similar content being viewed by others
Data availability
The manuscript has no associated data.
References
Madelung, E.: Quantentheorie in hydrodynamischer form. Z. Phys. A 40(1), 322–326 (1927)
Schrödinger, E.: An undulatory theory of the mechanics of atoms and molecules. Phys. Rev. 28, 1049–1070 (1926). https://doi.org/10.1103/PhysRev.28.1049
Vallis, G.K.: Atmospheric and Oceanic Fluid Dynamics, 2nd edn., p. 964. Cambridge University Press, Cambridge (2017)
Reddiger, M., Poirier, B.: Towards a mathematical theory of the Madelung equations. arXiv preprint arXiv:2207.11367 (2022)
Kundu, P.K., Cohen, I.M., Dowling, D.R.: Fluid Mechanics, 6th ed. (2016)
Heifetz, E., Cohen, E.: Toward a thermo-hydrodynamic like description of schrödinger equation via the Madelung formulation and fisher information. Foundations Phys. 45(11), 1514–1525 (2015)
Bohm, D.: A suggested interpretation of the quantum theory in terms of" hidden" variables I. Phys. Rev. 85(2), 166 (1952)
Tsekov, R., Heifetz, E., Cohen, E.: Relating quantum mechanics with hydrodynamic turbulence. Europhys. Lett. 122(4), 40002 (2018)
Holton, J.R.: An Introduction to Dynamic Meteorology, 4th edn., p. 535. Elsevier Academic Press, Burlington (2004)
Merzbacher, E.: Quantum Mechanics. John Wiley & Sons Inc, New York (1998)
Bühler, O.: Waves and Mean Flows. Cambridge University Press, Cambridge (2014)
Salmon, R.: Lectures on Geophysical Fluid Dynamics. Oxford University Press, Oxford (1998)
Heifetz, E., Plochotnikov, I.: Effective classical stochastic theory for quantum tunneling. Phys. Lett. A 384(21), 126511 (2020)
Heifetz, E., Plochotnikov, I.: Madelung transformation of the quantum bouncer problem. Europhys. Lett. 130(1), 10002 (2020)
Heifetz, E., Maas, L.R.M.: Zero absolute vorticity state in thermal equilibrium as a hydrodynamic analog of the quantum harmonic oscillator ground state. Phys. Fluids 33(3) (2021)
Heifetz, E., Maas, L.R., Mak, J.: Zero absolute vorticity plane couette flow as an hydrodynamic representation of quantum energy states under perpendicular magnetic field. Phys. Fluids 33(12), 127120 (2021)
Heisenberg, W.: Über stabilität und turbulenz von flüssigkeitsströmen. In: Original Scientific Papers Wissenschaftliche Originalarbeiten, pp. 31–81. Springer (1985)
Acknowledgements
Eyal Heifetz is grateful to Rachel Heifetz and Yair Zarmi for inspiring discussions.
Author information
Authors and Affiliations
Corresponding author
Additional information
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.
About this article
Cite this article
Heifetz, E., Guha, A. & Maas, L. de Broglie Normal Modes in the Madelung Fluid. Found Phys 53, 35 (2023). https://doi.org/10.1007/s10701-023-00676-z
Received:
Accepted:
Published:
DOI: https://doi.org/10.1007/s10701-023-00676-z