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

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## Abstract

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## 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

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

**a**) position localized; (

**b**) momentum localized.

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**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