# Quantum Classical Transition for Mixed States: The Scaled Von Neumann Equation

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## Abstract

**:**

## 1. Introduction

## 2. Theory

#### 2.1. Pure Ensembles in the de Broglie–Bohm Approach

#### 2.2. Mixed Ensembles in the de Broglie–Bohm Framework

#### 2.3. The Scaled Von Neumann Equation: A Proposal for Quantum Classical Transition

#### 2.4. The Scaled Wigner–Moyal Approach

## 3. Results and Discussion

## 4. Discussions and Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 1.**Scaled probability density plots for the superposition of two Gaussian wave packets, Equation (72), for different regimes: $\u03f5=1$ (

**left top panel**), $\u03f5=0.5$ (

**right top panel**), $\u03f5=0.1$ (

**left bottom panel**), and $\u03f5=0.01$ (

**right bottom panel**). We used as initial parameters, $m=1$, $\hslash =1$, ${p}_{0b}=-{p}_{0a}=2$, ${\sigma}_{0b}={\sigma}_{0a}=1$, ${x}_{0a}=-5$ and ${x}_{0b}=-15$.

**Figure 3.**A selection of scaled trajectories for the scaled pure $\tilde{\psi}(x,t)$ (

**left column**) and the scaled mixed state $\tilde{\rho}(x,y,t)$ (

**right column**) for the quantum regime $\u03f5=1$ (

**top panels**) and the nearly classical regime $\u03f5=0.01$ (

**bottom panels**). The same initial parameters were used as in previous figures.

**Figure 4.**Scaled arrival time distribution (79) at the detector location $X=-30$ for the pure state (72) (

**left top panel**) and the mixed state (73) (

**left bottom panel**) for different regimes: $\u03f5=1$ (green curve), $\u03f5=0.1$ (red curve), and $\u03f5=0.01$ (black curve). (

**Right top**,

**Right bottom**) panel depicts the scaled mean (uncertainty in) arrival time versus the transition parameter for the pure state (orange circled) and the mixed state (violet triangle up). The same initial parameters were used as in previous figures.

**Figure 5.**Expectation value of position operator (

**left top panel**), uncertainty in position (

**right top panel**), expectation value of momentum operator (

**left bottom panel**), and the product of uncertainties (

**right bottom panel**) for the scaled mixed state $\tilde{\rho}(x,y,t)$ for three different dynamical regimes: $\u03f5=1$ (green curves), $\u03f5=0.5$ (red curves), and $\u03f5=0.01$ (black curves). The same initial parameters were used as in previous figures.

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**MDPI and ACS Style**

Mousavi, S.V.; Miret-Artés, S.
Quantum Classical Transition for Mixed States: The Scaled Von Neumann Equation. *Symmetry* **2023**, *15*, 1184.
https://doi.org/10.3390/sym15061184

**AMA Style**

Mousavi SV, Miret-Artés S.
Quantum Classical Transition for Mixed States: The Scaled Von Neumann Equation. *Symmetry*. 2023; 15(6):1184.
https://doi.org/10.3390/sym15061184

**Chicago/Turabian Style**

Mousavi, S. V., and S. Miret-Artés.
2023. "Quantum Classical Transition for Mixed States: The Scaled Von Neumann Equation" *Symmetry* 15, no. 6: 1184.
https://doi.org/10.3390/sym15061184