# Diffusion Effect in Quantum Hydrodynamics

^{1}

^{2}

^{3}

^{*}

^{†}

## Abstract

**:**

## 1. Introduction

**real**distribution function $\mathrm{T}(\overrightarrow{r},t)$ due to a random process—where D is a

**real**diffusion coefficient and $\mathsf{\Delta}$ is the Laplace-operator—and the following time-dependent Schrödinger equation (particularly for the free motion, i.e., potential, $V=0$)

**complex**state function $\psi (\overrightarrow{r},t)$ whose absolute value squared, i.e., $\rho (\overrightarrow{r},t)=\psi \ast \psi $, is a quantum mechanical probability distribution and $\mathrm{i}\frac{\hslash}{2m}$ (with $\mathrm{i}$ and the reduced Planck constant $\hslash =\frac{h}{2\pi}$) now being a purely

**imaginary**coefficient. Despite the formal similarities, both equations describe very different processes.

**complex**function and the imaginary unit $\mathrm{i}$ appears explicitly in Equation (2). (These fundamental differences and their mathematical and physical consequences were already pointed out explicitly by Schrödinger himself in [2]—see also [3] for an English translation and comments.) However, this is an important—if not the most important—difference between classical and quantum physics, as stressed by C. N. Yang in their lecture on the occasion of Schrödinger’s 100th anniversary (see [4]), since “complex numbers become a conceptual element of the very foundations of physics”. This is reflected by the fact that $\mathrm{i}$ occurs explicitly in the fundamental equations of quantum mechanics not only in Schrödinger’s Equation (2), but also in Heisenberg’s commutation relation $pq-qp=-\mathrm{i}\hslash $ in their matrix mechanics (with $q=$ position and $p=$ momentum operators).

**not shrink**to its initial width. (It should be pointed out that this continuation of the spreading is not connected with any perturbance due to the interaction with an environment that might be assumed to take place in order to reverse the motion. The time reversal is purely a “Gedankenexperiment”, where no interaction with any environment is taken into account, only the Schrödinger equation for an isolated system is considered.) ${\sigma}_{0}$!

## 2. Conventional Quantum Hydrodynamics

**complex**Schrödinger equation is equivalent to a set of

**two real**hydrodynamic equations.

**complete departure**from classical mechanics, or rather (using the viewpoint of wave theory)

**from the kinematics underlying this mechanics**, that even for the simplest quantum theoretical problems the velocity of classical mechanics simply cannot be maintained.”

**complex**nature, since their origin is kinematic, whilst $\rho $, not being of a kinematic quantity, must remain a positive real quantity.

## 3. Complex Hydrodynamical Formulation

#### 3.1. Notation

#### 3.2. Continuity Equation

**imaginary**diffusion coefficient, $D=-\mathrm{i}\hslash /2m$, leading to an equation that has the form of an irreversible Fokker–Planck-type equation, in a position space of a so-called Smoluchowski equation:

#### 3.3. Example

#### 3.4. Euler Equation

## 4. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## References

- Mita, K. Schrödinger’s equation as a diffusion equation. Am. J. Phys.
**2021**, 89, 500–510. [Google Scholar] [CrossRef] - Schrödinger, E. Über die Umkehrung der Naturgesetze; Verlag der Akademie der wissenschaften: Berlin, Germany, 1931; p. 148. [Google Scholar]
- Chetrite, R.; Muratore-Ginanneschi, P.; Schwieger, K.; Schrödinger’s, E. 1931 paper “On the Reversal of the Laws of Nature” [Über die Umkehrung der Naturgesetze”, Sitzungsberichte der preussischen Akademie der Wissenschaften, physikalisch-mathematische Klasse, 8 N9 144–153]. Eur. Phys. J. H
**2021**, 46, 1–29. [Google Scholar] [CrossRef] - Yang, C.N. Square root of minus one, complex phases and Erwin Schrödinger. In Schrödinger: Centenary Celebration of a Polymath; Kilmister, C.W., Ed.; Cambridge Univ. Press: Cambridge, UK, 1987; pp. 53–64. [Google Scholar]
- Renou, M.O.; Trillo, D.; Weilenmann, M.; Le, T.P.; Tavakoli, A.; Gisin, N.; Acín, A.; Navascués, M. Quantum Theory based on real numbers can be experimentally falsified. Nature
**2021**, 600, 625–629. [Google Scholar] [CrossRef] - Li, Z.D.; Mao, Y.L.; Weilenmann, M.; Tavakoli, A.; Chen, H.; Feng, L.; Yang, S.-J.; Renou, M.-O.; Trillo, D.; Fan, J.; et al. Testing real Quantum Theory in an optical quantum network. Phys. Rev. Lett.
**2022**, 128, 040402. [Google Scholar] [CrossRef] [PubMed] - Bohr, N. On the notions of causality and complementarity. Dialectica
**1948**, 2, 312. [Google Scholar] [CrossRef] - Ehrenfest, P. Bemerkung über die angenäherte Gültigkeit der klassischen Mechanik innerhalb der Quantenmechanik. Z. Phys.
**1927**, 45, 455. [Google Scholar] [CrossRef] - Schrödinger, E. Quantisierung als Eigenwertproblem (Erste Mitteilung). Ann. Phys.
**1926**, 79, 361–376. [Google Scholar] [CrossRef] - Przibram, K. Briefe zur Wellenmechanik; Springer: Wien, Austria, 1963; p. 52. [Google Scholar]
- Schrödinger, E. Quantisierung als Eigenwertproblem (Vierte Mitteilung). Ann. Phys.
**1926**, 81, 109–139. [Google Scholar] [CrossRef] - Baylis, W.E.; Huschilt, J.; Wei, J. Why i? Am. J. Phys.
**1992**, 60, 788–797. [Google Scholar] [CrossRef] - Karam, R. Why are complex numbers needed in quantum mechanics? Some answers for the introductory level. Am. J. Phys.
**2020**, 88, 39. [Google Scholar] [CrossRef] - Karam, R. Schrödinger’s original struggles with a complex wave function. Am. J. Phys.
**2020**, 88, 433. [Google Scholar] [CrossRef] - Callender, C. What is ‘The Problem of the Direction of time’? Philos. Sci.
**1997**, 64, S223–S234. [Google Scholar] [CrossRef] - Callender, C. Is time ‘handed’ in a quantum world? Proc. Aristot. Soc.
**2000**, 100, 247–269. [Google Scholar] [CrossRef] - Callender, C. Quantum Mechanics: Keeping It Real? The British Journal for the Philosophy of Science; The University of Chicago Press: Chicago, IL, USA, 2021. [Google Scholar]
- Roberts, B.W. Three Myths About Time Reversal in Quantum Theory. Philos. Sci.
**2017**, 84, 315–334. [Google Scholar] [CrossRef] [Green Version] - Ardakani, R.M. Time Reversal Invariance in Quantum Mechanics. Master’s Thesis, Texas Tech. Univ., Lubbock, TX, USA, 2017. [Google Scholar]
- Wigner, E.P. Gruppentheorie und ihre Anwendung auf die Quantenmechanik der Atomspektren; Vieweg und Sohn: Braunschweig, Germany, 1931; pp. 251–254, (English Edition: Group Theory, Academic Press, New York, USA, 1958). [Google Scholar]
- Madelung, E. Quantentheorie in Hydrodynamischer Form. Z. Phys.
**1927**, 40, 322. [Google Scholar] [CrossRef] - Takabayasi, T. On the Formulation of Quantum Mechanics associated with Classical Pictures. Progr. Theor. Phys.
**1952**, 8, 143. [Google Scholar] [CrossRef] - Bohm, D. A Suggested Interpretation of the Quantum Theory in Terms of “Hidden” Variables I. Phys. Rev.
**1952**, 85, 166. [Google Scholar] [CrossRef] - Bohm, D. A Suggested Interpretation of the Quantum Theory in Terms of “Hidden” Variables II. Phys. Rev.
**1952**, 85, 180. [Google Scholar] [CrossRef] - Bonilla-Licea, M.; Schuch, D. Bohmian Trajectories as Borders of Regions of Constant Probability. Found. Phys.
**2022**, 52, 1–21. [Google Scholar] [CrossRef] - Benseny, A.; Albareda, G.; Sanz, Ṡ.; Mompart, J.; Oriols, X. Applied Bohmian Mechanics. Eur. Phys. J. D
**2014**, 68, 286. [Google Scholar] [CrossRef] - Sanz, Ṡ.; Miret-Artés, S. Setting up tunneling conditions by means of Bohmian mechanics. J. Phys. A Math. Theor.
**2011**, 44, 485301. [Google Scholar] [CrossRef] - Heisenberg, W. Über quantentheoretische Umdeutung kinematischer und mechanischer Beziehungen. Z. Phys.
**1925**, 33, 879. [Google Scholar] [CrossRef] - Bonilla-Licea, M.; Schuch, D. Quantum hydrodynamics with complex quantities. Phys. Lett. A
**2021**, 392, 127171. [Google Scholar] [CrossRef] - John, M.V. Modified de Broglie–Bohm Approach to Quantum Mechanics. Found. Phys. Lett.
**2002**, 15, 329. [Google Scholar] [CrossRef] [Green Version] - John, M.V. Probability and complex quantum trajectories. Ann. Phys.
**2009**, 324, 220. [Google Scholar] [CrossRef] [Green Version] - Dey, S.; Fring, A. Bohmian quantum trajectories from coherent states. Phys. Rev. A
**2013**, 88, 022116. [Google Scholar] [CrossRef] [Green Version] - Young, C.-D. Trajectory interpretation of the uncertainty principle in 1D systems using complex Bohmian mechanics. Phys. Lett. A
**2008**, 372, 6240. [Google Scholar] [CrossRef] - Chou, C.-C.; Wyatt, R.E. Considerations on the probability density in complex space. Phys. Rev. A
**2008**, 78, 044101. [Google Scholar] [CrossRef] - Chou, C.-C.; Wyatt, R.E. Complex-extended Bohmian mechanics. J. Che. Phys.
**2010**, 132, 134102. [Google Scholar] [CrossRef] - Poirier, B. Flux continuity and probability conservation in complexified Bohmian mechanics. Phys. Rev. A
**2008**, 77, 022114. [Google Scholar] [CrossRef] - Goldfarb, Y.; Degani, J.; Tannor, D.J. Bohmian mechanics with complex action: A new trajectory-based formulation of quantum mechanics. J. Chem. Phys.
**2006**, 125, 231103. [Google Scholar] [CrossRef] [Green Version] - Sanz, Á.S.; Miret-Artés, S. Comment on “Bohmian mechanics with complex action: A new trajectory-based formulation of quantum mechanics”. J. Chem. Phys.
**2007**, 127, 197101. [Google Scholar] [CrossRef] [Green Version] - Goldfarb, Y.; Tannor, D.J. Interference in Bohmian mechanics with complex action. J. Chem. Phys.
**2007**, 127, 161101. [Google Scholar] [CrossRef] [Green Version] - Sanz, Á.S.; Borondo, F.; Miret-Artés, S. Particle diffraction studied using quantum trajectories. J. Phys. Condens. Matter
**2002**, 14, 6109. [Google Scholar] [CrossRef] - Nelson, E. Derivation of the Schrödinger Equation from Newtonian Mechanics. Phys. Rev.
**1966**, 150, 1079. [Google Scholar] [CrossRef] - Bohm, D.; Hiley, B.J. Non-locality and Locality in the Stochastic Interpretation of Quantum Mechanics. Phys. Rep.
**1989**, 172, 93–122. [Google Scholar] [CrossRef] - Fürth, R. Über einige Beziehungen zwischen klassischer Statistik und Quantenmechanik. Z. Phys.
**1933**, 81, 143. [Google Scholar] [CrossRef] - Fényes, I. Eine wahrscheinlichkeitstheoretische Begründung und Interpretation der Quantenmechanik. Z. Phys.
**1952**, 132, 81–106. [Google Scholar] [CrossRef] - Bonilla-Licea, M.; Schuch, D. Bohmian mechanics in momentum representation and beyond. Phys. Lett. A
**2020**, 384, 126671. [Google Scholar] [CrossRef] - Schuch, D. Quantum Theory from a Nonlinear Perspective; Riccati Equations in Fundamental Physics. Fundamental Theories of Physics; Springer International: New York, NY, USA, 2018; Volume 191. [Google Scholar]

Publisher’s Note: MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affiliations. |

© 2022 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Bonilla-Licea, M.; Schuch, D.; Bonilla Estrada, M.
Diffusion Effect in Quantum Hydrodynamics. *Axioms* **2022**, *11*, 552.
https://doi.org/10.3390/axioms11100552

**AMA Style**

Bonilla-Licea M, Schuch D, Bonilla Estrada M.
Diffusion Effect in Quantum Hydrodynamics. *Axioms*. 2022; 11(10):552.
https://doi.org/10.3390/axioms11100552

**Chicago/Turabian Style**

Bonilla-Licea, Moise, Dieter Schuch, and Moises Bonilla Estrada.
2022. "Diffusion Effect in Quantum Hydrodynamics" *Axioms* 11, no. 10: 552.
https://doi.org/10.3390/axioms11100552