# Quantum Dynamics of Charged Fermions in the Wigner Formulation of Quantum Mechanics

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## Abstract

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## 1. Introduction

## 2. Wigner Quantum Dynamics

## 3. The Initial Wigner Function for Canonical Ensemble

## 4. Results of Numerical Calculations

**The momentum distribution functions**: We define the momentum distribution function for holes ($a=h$) and electrons ($a=e$) by the following expressions:

**Transport coefficients**: A natural way to obtain transport coefficients is making use of the quantum Green–Kubo relations [12]. These relations give the transport coefficients in terms of integrals of equilibrium time-dependent correlation functions. According to Equation (8), the electron conductivity $\sigma $ is the integral of the velocity–velocity autocorrelation function

## 5. Discussion

## Author Contributions

## Acknowledgments

## Conflicts of Interest

## Abbreviations

PIMC | Monte Carlo methods based on the path integral formulation of quantum mechanics |

WPIMC | Path integral Monte Carlo method in phase space |

WMD | Quantum generalization of the classical molecular dynamics methods |

MD | Maxwell distribution function |

FD | Fermi–Dirac momentum distribution function |

P8 | Quantum correction to the momentum distribution function proportional to $const/{p}^{8}$ |

P8MDEF | Quantum correction multiplied on Maxwell distributions with effective temperature |

VVACF | Velocity-velocity auto correlation function |

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**Figure 1.**(Color online) (

**Left**) The momentum distribution functions ${w}_{a}(|p|)$ for interacting electrons and two times heavier holes. The electronic momentum distributions are presented by Lines 1, 2, 3, 7, and 9, while analogous results for holes are presented by Lines 4, 5, 6, 8, and 10. The physical meaning of these lines is given in text. Degeneracy of electrons is equal to 4 ($T/{E}_{F}=0.261,\phantom{\rule{0.166667em}{0ex}}{k}_{F}{\lambda}_{e}=4.91$), while ${r}_{s}$ and the plasma classical coupling parameter is equal to 2. (

**Central**) The velocity–velocity autocorrelation function (Line 1) and its antiderivative function (Line 2) versus time in atomic units (${\tau}_{0}=\hslash /\mathrm{Ha}$) for ${r}_{s}=6$, $T=1.27\mathrm{Ha}$. (

**Right**) Electrical electron conductivity as function of the coupling parameter ${\Gamma}_{e}$ for different fixed densities of two component Coulomb system. Empty scatters 1–5 show results of this work for the fixed ${r}_{s}=6,4,3,2,1$, respectively, while related lines present results of interpolation formula for conductivity of fully ionized hydrogen plasma [26,27]. Line 6 restricts approximately from above the region of conductivity sharp drop due to arising bound states of many particle clusters.

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**MDPI and ACS Style**

Filinov, V.; Larkin, A.
Quantum Dynamics of Charged Fermions in the Wigner Formulation of Quantum Mechanics. *Universe* **2018**, *4*, 133.
https://doi.org/10.3390/universe4120133

**AMA Style**

Filinov V, Larkin A.
Quantum Dynamics of Charged Fermions in the Wigner Formulation of Quantum Mechanics. *Universe*. 2018; 4(12):133.
https://doi.org/10.3390/universe4120133

**Chicago/Turabian Style**

Filinov, Vladimir, and Alexander Larkin.
2018. "Quantum Dynamics of Charged Fermions in the Wigner Formulation of Quantum Mechanics" *Universe* 4, no. 12: 133.
https://doi.org/10.3390/universe4120133