# The Heisenberg Uncertainty Principle as an Endogenous Equilibrium Property of Stochastic Optimal Control Systems in Quantum Mechanics

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. The Evolution of the Transition Probability Density and the Stationary Distribution

#### 2.1. The Requirement that the Stationary Density is Real Leads to Born’s Rule

#### 2.2. The Heisenberg Uncertainty Principle

## 3. Discussion and Interpretation of the Result

## 4. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## References

- Lindgren, J.; Liukkonen, J. Quantum mechanics can be understood through stochastic optimization on spacetimes. Sci. Rep.
**2019**, 9, 1–8. [Google Scholar] [CrossRef] - Heisenberg, W. Über den anschaulichen inhalt der quantentheoretischen kinematik und mechanik. In Original Scientific Papers Wissenschaftliche Originalarbeiten; Springer: Berlin/Heidelberg, Germany, 1985; pp. 478–504. [Google Scholar]
- Ozawa, M. Physical content of Heisenberg’s uncertainty relation: Limitation and reformulation. Phys. Lett. A
**2003**, 318, 21–29. [Google Scholar] [CrossRef][Green Version] - Koide, T.; Kodama, T. Generalization of uncertainty relation for quantum and stochastic systems. Phys. Lett. A
**2018**, 382, 1472–1480. [Google Scholar] [CrossRef][Green Version] - Popper, K. Quantum Theory and the Schism in Physics; Psychology Press: Hove, UK, 1992. [Google Scholar]
- Ballentine, L.E. The statistical interpretation of quantum mechanics. Rev. Mod. Phys.
**1970**, 42, 358–381. [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] - Frederick, C. Stochastic space-time and quantum theory. Phys. Rev. D
**1976**, 13, 3183–3191. [Google Scholar] [CrossRef] - Pavliotis, G.A. Stochastic Processes and Applications, Diffusion Processes, the Fokker-Planck and Langevin Equations; Springer: New York, NY, USA, 2014. [Google Scholar]
- Rozema, L.A.; Darabi, A.; Mahler, D.H.; Hayat, A.; Soudagar, Y.; Steinberg, A.M. Violation of Heisenberg’s measurement-disturbance relationship by weak measurements. Phys. Rev. Lett.
**2012**, 109, 100404. [Google Scholar] [CrossRef][Green Version] - Erhart, J.; Sponar, S.; Sulyok, G.; Badurek, G.; Ozawa, M.; Hasegawa, Y. Experimental demonstration of a universally valid error–disturbance uncertainty relation in spin measurements. Nat. Phys.
**2012**, 8, 185–189. [Google Scholar] [CrossRef][Green Version]

**Figure 1.**The tradeoff between the localizations of four-position and four-momentum: (

**a**) position localized; (

**b**) momentum localized.

© 2020 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 (http://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Lindgren, J.; Liukkonen, J. The Heisenberg Uncertainty Principle as an Endogenous Equilibrium Property of Stochastic Optimal Control Systems in Quantum Mechanics. *Symmetry* **2020**, *12*, 1533.
https://doi.org/10.3390/sym12091533

**AMA Style**

Lindgren J, Liukkonen J. The Heisenberg Uncertainty Principle as an Endogenous Equilibrium Property of Stochastic Optimal Control Systems in Quantum Mechanics. *Symmetry*. 2020; 12(9):1533.
https://doi.org/10.3390/sym12091533

**Chicago/Turabian Style**

Lindgren, Jussi, and Jukka Liukkonen. 2020. "The Heisenberg Uncertainty Principle as an Endogenous Equilibrium Property of Stochastic Optimal Control Systems in Quantum Mechanics" *Symmetry* 12, no. 9: 1533.
https://doi.org/10.3390/sym12091533