Hall Conductivity, Transition-State Theory and Their Effect on Entanglement and the Energy Density

Ahlem Abidi^1,2

¹Nanostructured Materials, Quantum and Nonlinear Optics Laboratory, Department of Physics, Faculty of Sciences of Tunis, Tunis El Manar University, Rommana, Tunisia.
²Higher Institute of Technological Studies of Jendouba, Jendouba, Tunisia.

DOI: 10.4236/oalib.1113749                

Abstract

Two isotropic harmonic oscillators in a magnetic field on the non-commutative phase space (NCPS) are studied in this paper. We derive the corresponding entanglement entropies of the vacuum state and we find analytically purity function, Hall conductivity and the energy density. Transition-state theory is also developed to examine the purity function and the energy density. By considering the action of the non-commutativity parameters, entanglement and Hall conductivity present a similar behavior. A large magnetic field acts as a disturbance of the accurate system information and the non-localization of the two harmonic oscillators in the atom by increasing its stored energy. The application of the transition-state theory increases entanglement and accelerates the non-localization phenomenon.

Keywords

Two Isotropic Harmonic Oscillators in a Magnetic Field, NCPS, Entanglement, Hall Conductivity, Energy Density, Transition-State Theory

1. Introduction

A great deal of interest has been recently given to the formulation and possible experimental consequences of the extension of the standard quantum physical formalism to accommodate the non-commutativity of phase space operators [1]-[5]. The idea was inspired by quantum field theory and string theory [6]-[8]. The crucial difference with the standard quantum theory is to replace the usual product with the Moyal product [9] [10]. Therefore, they allow us to better understand various phenomena. So far, many examples have been studied intensively, such as the spectrum of the hydrogen atom [11] [12], the harmonic oscillator [13] [14], the Aharonov-Bohm effect under the action of a magnetic field [15]-[17], the Landau problem [18] [19], etc. We specify the particular system of two isotropic harmonic oscillators in a magnetic field, in the framework of non-commutative quantum mechanics, accordingly, because of the magnetic field application considering the Hall effect. This can be stated as follows: a semiconductor material through which an electric current flows perpendicular to the movement of charge carriers, a voltage is produced from the latter, so called Hall voltage, which has been attributed to the Hall effect. The Hall conductivity connected to the Hall effect has been taken into consideration in this work. Also, in the presence of the magnetic field $B$ , we discuss the localized energy density, which is the amount of energy stored in a point of the material conductor. These two concepts are discussed in detail in this paper to clarify some properties of an entangled system. Entanglement concepts in non-commutative quantum mechanics are studied by many references, see for example [20]-[22] and we applied the transition-state theory to examine their effect on entanglement and the energy density.

2. Theoretical Framework (Definitions, Hamiltonian Diagonalization)

A harmonic oscillator can behave as an electron under a magnetic field of induction $B$ , in this framework, consider two isotropic harmonic oscillators with unit masses and are exposed to a magnetic field [23]. The Hamiltonian is written:

$H=\frac{1}{2}{\left(-i\hslash \frac{\partial }{\partial y_1}+\frac{e}{2}B{y}_2\right)}^2+\frac{1}{2}{\left(-i\hslash \frac{\partial }{\partial y_2}-\frac{e}{2}B{y}_1\right)}^2+\frac{1}{2}{\omega }^2\left(y_1^2+y_2^2\right).$ (2.1)

where $\omega$ is the common angular frequency of the two harmonic oscillators, $e$ is the electric charge. (2.1) can be reformulated as

$H=-\frac{1}{2}{\hslash }^2\frac{{\partial }^2}{\partial y_1^2}-\frac{1}{2}{\hslash }^2\frac{{\partial }^2}{\partial y_2^2}+\frac{1}{2}\zeta y_1^2+\frac{1}{2}\zeta y_2^2-\frac{1}{2}\beta \left[y_1\left(-i\hslash \frac{\partial }{\partial y_2}\right)-\left(-i\hslash \frac{\partial }{\partial y_1}\right)y_2\right],$ (2.2)

where $\zeta =\frac{e^2}{4}{B}^2+{\omega }^2$ and $\beta =eB$ . To diagonalize (2.2), we define the unitary operator

$\theta =\text{exp}\left[-{\gamma }_1{y}_1{y}_2-{\gamma }_2\frac{\partial }{\partial y_1}\frac{\partial }{\partial y_2}\right].$ (2.3)

where $\theta$ acts on Hamiltonian (2.2) as $H=\theta H{\theta }^{-}$ to get the form

$H=\frac{1}{2}\left({\sigma }_1{y}_1^2-{\sigma }_2\frac{{\partial }^2}{\partial y_1^2}\right)+\frac{1}{2}\left({\sigma }_1^{\text{*}}{y}_2^2+{\sigma }_2\frac{{\partial }^2}{y_2^2}\right),$ (2.4)

where

${\sigma }_1=\zeta -{\hslash }^2{\gamma }_1^2+i\hslash \beta {\gamma }_1\text{and}{\sigma }_2=i\hslash \beta {\gamma }_2.$ (2.5)

The solution (2.4) gives

${\gamma }_1=\frac{i\hslash \beta -2\zeta {\gamma }_2}{2{\hslash }^2+i\hslash \beta {\gamma }_2}\text{and}{\gamma }_2=\sqrt{\frac{{\hslash }^2}{\zeta }}.$ (2.6)

3. Methods

The non-commutativity of positions $y_k$ and momentums $-i\hslash \frac{\partial }{\partial y_l}$ operators in two-dimensional space is imposed by the relations [24] $\left[y_1,y_2\right]=i{\theta }_1$ , $\left[-i\hslash \frac{\partial }{\partial y_1},-i\hslash \frac{\partial }{\partial y_2}\right]=i{\theta }_2$ and $\left[y_k,-i\hslash \frac{\partial }{\partial y_l}\right]=i{\delta }_{kl}\hslash$ , where $\left(k=1,2\right)$ , $\left(l=1,2\right)$ and ${\theta }_1$ , ${\theta }_2$ are the non-commutativity variables. As usual, we assume after, in the numerical section that $\hslash =1$ and $e=1$ . Quantization deformation is the suitable method to study eignensolution of Hamiltonian (2.4) in non-commutative phase space [25]. $\star$ is the Moyal product introducing the non-commutativity, applied when we treat the classical quantities $\left(y,-i\hslash \frac{\partial }{\partial y}\right)$ to replacing the ordinary product, it is given as

$\star =\text{exp}\frac{i\hslash }{2}\left({\overrightarrow{\partial }}_{y_1}{\overrightarrow{\partial }}_{\frac{\partial }{\partial y_2}}-{\overrightarrow{\partial }}_{\frac{\partial }{\partial y_1}}{\overrightarrow{\partial }}_{y_2}\right)\text{exp}\frac{i{\theta }_1}{2}{\xi }_{kl}\left({\overrightarrow{\partial }}_{y_1}{\overrightarrow{\partial }}_{y_2}-{\overrightarrow{\partial }}_{y_2}{\overrightarrow{\partial }}_{y_1}\right)\text{exp}\frac{i{\theta }_2}{2}{\xi }_{kl}\left({\overrightarrow{\partial }}_{\frac{\partial }{\partial y_1}}{\overrightarrow{\partial }}_{\frac{\partial }{\partial y_2}}-{\overrightarrow{\partial }}_{\frac{\partial }{\partial y_2}}{\overrightarrow{\partial }}_{\frac{\partial }{\partial y_1}}\right),$ (3.1)

where ${\xi }_{kl}$ is defined as a matrix of dimension 2 and we have

${\xi }_{kl}=\left(\begin{array}{ll}0\hfill & 1\hfill \\ -1\hfill & 0\hfill \end{array}\right).$

Hamiltonian (2.4) can be reformulated as the sum of two Hamiltonians $H_1$ and $H_2$ such as

$\begin{array}{c}{H}_1={\left(i\frac{\partial }{\partial y_1}\sqrt{{\sigma }_2}\cos \left({\omega }_1\right)-y_2\sqrt{{\sigma }_1^{*}}\sin \left({\omega }_1\right)\right)}^2\\ +{\left(\frac{\partial }{\partial y_2}\sqrt{{\sigma }_2}\cos \left({\omega }_2\right)-y_1\sqrt{{\sigma }_1}\sin \left({\omega }_2\right)\right)}^2,\end{array}$ (3.2)

and

$\begin{array}{c}{H}_2={\left(i\frac{\partial }{\partial y_1}\sqrt{{\sigma }_2}\sin \left({\omega }_1\right)+y_2\sqrt{{\sigma }_1^{*}}\cos \left({\omega }_1\right)\right)}^2\\ +{\left(\frac{\partial }{\partial y_2}\sqrt{{\sigma }_2}\sin \left({\omega }_2\right)+y_1\sqrt{{\sigma }_1}\cos \left({\omega }_2\right)\right)}^2.\end{array}$ (3.3)

In expressions (3.2) and (3.3):

${\omega }_1=\frac{1}{2}\text{arctan}\left(-\frac{2\hslash \left(\frac{\sqrt{{\sigma }_2}}{\sqrt{{\sigma }_1^{\text{*}}}}-1\right){\theta }_2}{\vartheta }\right),$

${\omega }_2=\frac{1}{2}\text{arctan}\left(\frac{2\hslash \frac{\sqrt{{\sigma }_2}}{\sqrt{{\sigma }_1^{\text{*}}}}\left(\frac{\sqrt{{\sigma }_2}}{\sqrt{{\sigma }_1^{\text{*}}}}-1\right){\theta }_1}{\vartheta }\right),$ (3.4)

where

$\vartheta ={\hslash }^2{\left(\frac{\sqrt{{\sigma }_2}}{\sqrt{{\sigma }_1^{\text{*}}}}-1\right)}^2-\left(\frac{{\sigma }_2}{{\sigma }_1^{\text{*}}}{\theta }_1^2-{\theta }_2^2\right).$ (3.5)

The Moyal product acts between $H_1$ and $H_2$ commutation relation as

${\left[H_1,H_2\right]}_{\star }=H_1\star H_2-H_2\star H_1=0,$ (3.6)

and verify the ordinary relation

$H_1\star H_2=H_2\star H_1=H_2{H}_1.$ (3.7)

The eigenvalue solution of Hamiltonian (2.4) is given by solving the eigenequation

$H\star W_{n,m}=W_{n,m}\star H=E_{n,m}{W}_{n,m}.$ (3.8)

where $W_{n,m}$ is the Wigner function of the $\left(n,m\right)$ quantum states and $E_{n,m}$ is the corresponding eigenenergy. (3.8) corresponds to the standard two-dimensional Schrödinger equation, the solution is given by the following Wigner functions

$W_n=\frac{{\left(-1\right)}^n}{\pi {\delta }_1}{\text{e}}^{-\frac{H_1}{\hslash {\delta }_1\sqrt{{\sigma }_1{\sigma }_2}}}{L}_n\left(\frac{2H_1}{\hslash {\delta }_1\sqrt{{\sigma }_1{\sigma }_2}}\right),$ (3.9)

and

$W_m=\frac{{\left(-1\right)}^m}{\pi {\delta }_2}{\text{e}}^{-\frac{H_2}{\hslash {\delta }_2\sqrt{{\sigma }_1^{\text{*}}{\sigma }_2}}}{L}_m\left(\frac{2H_2}{\hslash {\delta }_2\sqrt{{\sigma }_1^{\text{*}}{\sigma }_2}}\right).$ (3.10)

${\delta }_1$ and ${\delta }_2$ in expressions (3.9) and (3.10) have the forms:

$\begin{array}{c}{\delta }_1=\hslash \sqrt{{\sigma }_2}\left(\sqrt{{\sigma }_1}\cos \left({\omega }_1\right)\sin \left({\omega }_2\right)-\sqrt{{\sigma }_1^{*}}\sin \left({\omega }_1\right)\cos \left({\omega }_2\right)\right)\\ +\sqrt{{\sigma }_2}\left({\theta }_1\sqrt{{\sigma }_1}\cos \left({\omega }_1\right)\cos \left({\omega }_2\right)-{\theta }_2\sqrt{{\sigma }_1^{*}}\sin \left({\omega }_1\right)\sin \left({\omega }_2\right)\right),\end{array}$

and

$\begin{array}{c}{\delta }_2=\hslash \sqrt{{\sigma }_2}\left(\sqrt{{\sigma }_1}\sin \left({\omega }_1\right)\cos \left({\omega }_2\right)-\sqrt{{\sigma }_1^{*}}\cos \left({\omega }_1\right)\sin \left({\omega }_2\right)\right)\\ +\sqrt{{\sigma }_2}\left({\theta }_1\sqrt{{\sigma }_1}\sin \left({\omega }_1\right)\sin \left({\omega }_2\right)-{\theta }_2\sqrt{{\sigma }_1^{*}}\cos \left({\omega }_1\right)\cos \left({\omega }_2\right)\right).\end{array}$ (3.11)

We write the Wigner function and the eigenenergy of expression (3.8) as

$W_{nm}=W_n\star W_m=W_n{W}_m,$ (3.12)

and

$E_{n,m}=E_n+E_m.$ (3.13)

Consequently, we get respectively

$W_{nm}=\frac{{\left(-1\right)}^{n+m}}{{\pi }^2{\delta }_1{\delta }_2}{\text{e}}^{-\frac{H_1}{\hslash {\delta }_1\sqrt{{\sigma }_1{\sigma }_2}}}{\text{e}}^{-\frac{H_2}{\hslash {\delta }_2\sqrt{{\sigma }_1^{\text{*}}{\sigma }_2}}}{L}_n\left(\frac{2H_1}{\hslash {\delta }_1\sqrt{{\sigma }_1{\sigma }_2}}\right)L_m\left(\frac{2H_2}{\hslash {\delta }_2\sqrt{{\sigma }_1^{\text{*}}{\sigma }_2}}\right),$ (3.14)

$E_{n,m}=\sqrt{\hslash {\delta }_1{\sigma }_1{\sigma }_2}\left(n+\frac{1}{2}\right)+\sqrt{\hslash {\delta }_2{\sigma }_1^{\text{*}}{\sigma }_2}\left(m+\frac{1}{2}\right).$ (3.15)

One can easily show that

${\displaystyle \int W_{nm}\left(y_1,\frac{\partial }{\partial y_1}:y_2,\frac{\partial }{\partial y_2}\right)\text{d}{y}_1\text{d}{y}_2\text{d}\left(\frac{\partial }{\partial y_1}\right)\text{d}\left(\frac{\partial }{\partial y_2}\right)}=1.$ (3.16)

Particularly the vacuum state, (3.14) becomes

$W_{00}=\frac{1}{{\pi }^2{\delta }_1{\delta }_2}{\text{e}}^{-\frac{H_1}{\hslash {\delta }_1\sqrt{{\sigma }_1{\sigma }_2}}}{\text{e}}^{-\frac{H_2}{\hslash {\delta }_2\sqrt{{\sigma }_1^{\text{*}}{\sigma }_2}}}.$ (3.17)

To calculate various entropies in non-commutative phase space, we need first to calculate the reduced Wigner functions of each of two harmonic oscillators. Using (3.17), one has

$\begin{array}{c}{W}_{00}\left(y_1,\frac{\partial }{\partial y_1}\right)={\displaystyle \int W_{00}}\left(y_1,\frac{\partial }{\partial y_1};y_2,\frac{\partial }{\partial y_2}\right)\text{d}{y}_2\text{d}\left(\frac{\partial }{\partial y_2}\right)\\ =\frac{\gamma }{\pi \hslash }{\text{e}}^{-\frac{1}{\hslash {\delta }_1{\delta }_2\sqrt{{\sigma }_1{\sigma }_2}\sqrt{{\sigma }_1^{\text{*}}{\sigma }_2}}\left(\frac{{\sigma }_1}{{\vartheta }_1}{y}_1^2+\frac{{\sigma }_2}{{\vartheta }_2}\frac{{\partial }^2}{\partial y_1^2}\right)},\end{array}$ (3.18)

and

$\begin{array}{c}{W}_{00}\left(y_2,\frac{\partial }{\partial y_2}\right)={\displaystyle \int W_{00}\left(y_1,\frac{\partial }{\partial y_1};y_2,\frac{\partial }{\partial y_2}\right)\text{d}{y}_1\text{d}\left(\frac{\partial }{\partial y_1}\right)}\\ =\frac{{\gamma }^{\prime }}{\pi \hslash }{\text{e}}^{-\frac{1}{\hslash {\delta }_1{\delta }_2\sqrt{{\sigma }_1{\sigma }_2}\sqrt{{\sigma }_1^{\text{*}}{\sigma }_2}}\left(\frac{{\sigma }_1^{\text{*}}}{{{\vartheta }^{\prime }}_1}{y}_2^2+\frac{{\sigma }_2}{{{\vartheta }^{\prime }}_2}\frac{{\partial }^2}{\partial y_2^2}\right)}.\end{array}$ (3.19)

In expressions (3.18) and (3.19),

$\gamma =\sqrt{\frac{1}{{\delta }_1^2{\delta }_2^2{\sigma }_1^{\text{*}}{\sigma }_2{\vartheta }_1{\vartheta }_2}},{\gamma }^{\prime }=\sqrt{\frac{1}{{\delta }_1^2{\delta }_2^2{\sigma }_1{\sigma }_2{{\vartheta }^{\prime }}_1{{\vartheta }^{\prime }}_2}},$

${\vartheta }_1=\frac{1}{{\delta }_1\sqrt{{\sigma }_1^{*}{\sigma }_2}}{\sin }^2\left({\omega }_1\right)+\frac{1}{{\delta }_2\sqrt{{\sigma }_1{\sigma }_2}}{\cos }^2\left({\omega }_1\right),$

${\vartheta }_2=\frac{1}{{\delta }_1\sqrt{{\sigma }_1^{\text{*}}{\sigma }_2}}{\text{cos}}^2\left({\omega }_2\right)+\frac{1}{{\delta }_2\sqrt{{\sigma }_1{\sigma }_2}}{\text{sin}}^2\left({\omega }_2\right),$

${{\vartheta }^{\prime }}_1=\frac{1}{{\delta }_1\sqrt{{\sigma }_1^{\text{*}}{\sigma }_2}}{\text{cos}}^2\left({\omega }_1\right)+\frac{1}{{\delta }_2\sqrt{{\sigma }_1{\sigma }_2}}{\text{sin}}^2\left({\omega }_1\right),$

and

${{\vartheta }^{\prime }}_2=\frac{1}{{\delta }_1\sqrt{{\sigma }_1^{\text{*}}{\sigma }_2}}{\text{sin}}^2\left({\omega }_2\right)+\frac{1}{{\delta }_2\sqrt{{\sigma }_1{\sigma }_2}}{\text{cos}}^2\left({\omega }_2\right).$ (3.20)

Since the system is entangled then expression (3.18) or (3.19) is sufficient to calculate entanglement entropies. The purity function is a derivation of the linear entropy and it also provides another form to examine entanglement, its expression is

$P={\displaystyle \iint \left[{\left(W_{00}\right)}_{\star }^2\right]\text{d}{y}_1\text{d}}\left(\frac{\partial }{\partial y_1}\right).$ (3.21)

Following problem in ref. [26] and by applying the Moyal product $\star$ on expression (3.18), we obtain

$\begin{array}{l}{\left(W_{00}\left(y_1,\frac{\partial }{\partial y_1}\right)\right)}_{\star }=\frac{1}{\pi \hslash }\frac{\gamma }{\sqrt{1-{\gamma }^2}}{\text{exp}}_{\star }\left[-\frac{1}{2{\hslash }^2}\chi \left(\frac{{\sigma }_1}{{\vartheta }_1}{y}_1^2+\frac{{\sigma }_2}{{\vartheta }_2}\frac{{\partial }^2}{\partial y_1^2}\right)\right],\hfill \end{array}$ (3.22)

where $\chi =\frac{1}{\gamma {\delta }_1{\delta }_2\sqrt{{\sigma }_1{\sigma }_2}\sqrt{{\sigma }_1^{\text{*}}{\sigma }_2}}\text{ln}\left(\frac{1+\gamma }{1-\gamma }\right)$ .

By substituting (3.22) in (3.21), we have

$\begin{array}{l}P=\frac{1}{{\pi }^2{\hslash }^2}\frac{{\gamma }^2}{1-{\gamma }^2}{\left(\frac{\pi }{\frac{1}{{\hslash }^2}\chi \frac{{\sigma }_2}{{\vartheta }_2}}\right)}^{\frac{1}{2}}{\displaystyle \iint {\text{exp}}_{\star }}\left[-\frac{1}{2{\hslash }^2}\chi \frac{{\sigma }_1}{{\vartheta }_1}\left(y_1^2+{y^{\prime }}_1^2\right)\right]\text{d}{y}_1\text{d}{y^{\prime }}_1.\hfill \end{array}$ (3.23)

Consequently, we can show that

$P=\frac{1}{\sqrt{\pi }{\hslash }^2}\frac{{\gamma }^2}{1-{\gamma }^2}\left(\frac{1}{\frac{1}{2{\hslash }^2}\chi \frac{{\sigma }_1}{{\vartheta }_1}}\right){\left(\frac{1}{\frac{1}{2{\hslash }^2}\chi \frac{{\sigma }_2}{{\vartheta }_2}}\right)}^{\frac{1}{2}}.$ (3.24)

For all integer $n\ge 1$ , the Rényi entropy writes

$S_n\left(W_{00}\right)=\frac{1}{1-n}\text{ln}\left({\left(\frac{1}{2\pi \hslash }\right)}^{1-n}{\displaystyle \int {\left(W_{00}\right)}_{\star }^n\text{d}{y}_1\text{d}\left(\frac{\partial }{\partial y_1}\right)}\right).$ (3.25)

In order $n$ ,

$\begin{array}{l}{\left(W_{00}\left(y_1,\frac{\partial }{\partial y_1}\right)\right)}_{\star }^n=\frac{2{\gamma }^n}{{\left(\pi \hslash \right)}^n\left[{\left(1+\gamma \right)}^n+{\left(1-\gamma \right)}^n\right]}\\ \times \exp \left[-\frac{1}{\hslash {\delta }_1{\delta }_2\sqrt{{\sigma }_1{\sigma }_2}\sqrt{{\sigma }_1^{\text{*}}{\sigma }_2}}\left(\frac{{\sigma }_1}{{\vartheta }_1}{y}_1^2+\frac{{\sigma }_2}{{\vartheta }_2}\frac{{\partial }^2}{\partial y_1^2}\right)\frac{\left[{\left(1+\gamma \right)}^n-{\left(1-\gamma \right)}^n\right]}{\gamma \left[{\left(1+\gamma \right)}^n+{\left(1-\gamma \right)}^n\right]}\right].\end{array}$ (3.26)

Insert (3.26) in (3.25), we have

$S_n\left(W_{00}\right)=\frac{n}{1-n}\text{ln}\left(2\gamma \right)-\frac{1}{1-n}\text{ln}\left[{\left(1+\gamma \right)}^n-{\left(1-\gamma \right)}^n\right].$ (3.27)

The particular case where $n\to 1$ , expression (3.27) reduces to the von Neumann entropy as

$S_1\left(W_{00}\right)={\displaystyle \int W_{00}}\star {\text{ln}}_{\star }\left(2\pi \hslash W_{00}\right)\text{d}{y}_1\text{d}\left(\frac{\partial }{\partial y_1}\right).$ (3.28)

From (3.28), we can obtain $S_1\left(W_{00}\right)$ as

$\begin{array}{l}{S}_1\left(W_{00}\right)=\frac{1}{2\gamma }\left[-\text{ln}\left(2\gamma \right)-\left(1-\gamma \right)\text{ln}\left(1-\gamma \right)+\left(1+\gamma \right)\text{ln}\left(1+\gamma \right)\right].\hfill \end{array}$ (3.29)

Such expressions (3.24), (3.27) and (3.29) are interesting because they show all the ingredients to investigate the system. In the ordinary case when ${\theta }_1={\theta }_2=0$ , purity function, Rényi and the von Neumann entropies vanish. Recently, the quantization deformation method is extended to three dimensional to study entanglement of three isotropic harmonic oscillators by ref. [27].

4. Hall Conductivity and Energy Density

A natural generalization of two harmonic oscillators under a magnetic field in the framework of non-commutative quantum mechanics is devoted to deriving some properties of these two concepts: Hall conductivity and energy density, from the aspect of quantum information (entangled system). To start, we calculate the current density. Suppose that from (2.4),

$H^{\prime }\left(y_1,\frac{\partial }{\partial y_1}\right)=\frac{1}{2}\left({\sigma }_1{y}_1^2-{\sigma }_2\frac{{\partial }^2}{\partial y_1^2}\right),$ (4.1)

and using expression (3.18), we have the current density as

$\begin{array}{c}J\left(y_1\right)={\displaystyle \int \frac{\partial }{\partial \left(\frac{\partial }{\partial y_1}\right)}{H}^{\prime }\left(y_1,\frac{\partial }{\partial y_1}\right)W_{00}\left(y_1,\frac{\partial }{\partial y_1}\right)\text{d}\left(\frac{\partial }{\partial y_1}\right)}\\ ={\displaystyle \int \frac{\partial }{\partial \left(\frac{\partial }{\partial y_1}\right)}}\left({\sigma }_1{y}_1^2-{\sigma }_2\frac{{\partial }^2}{\partial y_1^2}\right)\star {\left(W_{00}\left(y_1,\frac{\partial }{\partial y_1}\right)\right)}_{\star }\text{d}\left(\frac{\partial }{\partial y_1}\right).\end{array}$ (4.2)

By applying the Moyal product (3.1), we can write

$\begin{array}{l}\left({\sigma }_1{y}_1^2-{\sigma }_2\frac{{\partial }^2}{\partial y_1^2}\right)\star {\left(W_{00}\left(y_1,\frac{\partial }{\partial y_1}\right)\right)}_{\star }\\ =\frac{1}{2}({\sigma }_1{y}_1^2-{\sigma }_2\frac{{\partial }^2}{\partial y_1^2}-\frac{{\hslash }^2}{4}\left({\sigma }_1{\partial }_{\left(\frac{\partial }{\partial y_1}\right)}^2-{\sigma }_2{\partial }_{y_1}^2\right)\\ +i\hslash \left({\sigma }_1{y}_1{\partial }_{\left(\frac{\partial }{\partial y_1}\right)}+{\sigma }_2\frac{\partial }{\partial y_1}{\partial }_{y_1}\right)){\left(W_{00}\left(y_1,\frac{\partial }{\partial y_1}\right)\right)}_{\star }\end{array}$ (4.3)

Substituting (3.22) in (4.3), after some calculation we find expression (4.2) as:

$\begin{array}{c}J\left(y_1\right)=\frac{\gamma }{\pi \hslash \sqrt{1-{\gamma }^2}}{\left(\frac{1}{\frac{1}{2{\hslash }^2}\chi \frac{{\sigma }_2}{{\vartheta }_2}}\right)}^{\frac{1}{2}}\\ \times \left[\left(-{\sigma }_2+D_2\right){\left(\frac{1}{\frac{1}{2{\hslash }^2}\chi \frac{{\sigma }_2}{{\vartheta }_2}}\right)}^{\frac{1}{2}}-i\sqrt{\pi }{\sigma }_1{\sigma }_2\chi \left(\frac{1}{{\vartheta }_2}-\frac{1}{{\vartheta }_1}\right)y_1\right]\\ \times \text{exp}\left[-\frac{1}{2{\hslash }^2}\chi \frac{{\sigma }_1}{{\vartheta }_1}{y}_1^2\right].\end{array}$ (4.4)

The average of the current density operator $\langle J\left(y_1\right)\rangle$ is defined as

$\langle J\left(y_1\right)\rangle ={\displaystyle \int {\left|{\psi }_0\left(y_1\right)\right|}^2\star J\left(y_1\right)\text{d}{y}_1}.$ (4.5)

where ${\left|{\psi }_0\left(y_1\right)\right|}^2$ is the probability distribution in the phase space. It is defined with the integral of the Wigner function on the momentum space as

${\left|{\psi }_0\left(y_1\right)\right|}^2=\frac{1}{2\hslash }{\displaystyle \int {\left(W_{00}\left(y_1,\frac{\partial }{\partial y_1}\right)\right)}_{\star }\text{d}}\left(\frac{\partial }{\partial y_1}\right).$ (4.6)

We develop this expression using (3.22), we have

$\begin{array}{l}\frac{1}{2\hslash }{\displaystyle \int {\left(W_{00}\left(y_1,\frac{\partial }{\partial y_1}\right)\right)}_{\star }\text{d}\left(\frac{\partial }{\partial y_1}\right)}\\ =\frac{1}{2{\hslash }^2\sqrt{\pi }}\frac{\gamma }{\sqrt{1-{\gamma }^2}}{\left(\frac{1}{\frac{1}{2{\hslash }^2}\chi \frac{{\sigma }_2}{{\vartheta }_2}}\right)}^{\frac{1}{2}}\times \text{exp}\left[-\frac{1}{2{\hslash }^2}\chi \frac{{\sigma }_1}{{\vartheta }_1}{y}_1^2\right].\end{array}$ (4.7)

Using expression (4.7), we can then express (4.5) which interests us as

$\begin{array}{c}\langle J\left(y_1\right)\rangle =\frac{1}{2{\hslash }^3\pi }\frac{{\gamma }^2}{1-{\gamma }^2}\left(\frac{1}{\frac{1}{2{\hslash }^2}\chi \frac{{\sigma }_2}{{\vartheta }_2}}\right){\left(\frac{1}{\frac{1}{{\hslash }^2}\chi \frac{{\sigma }_1}{{\vartheta }_1}}\right)}^{\frac{1}{2}}[\left(-{\sigma }_2+D_2\right){\left(\frac{1}{\frac{1}{2{\hslash }^2}\chi \frac{{\sigma }_2}{{\vartheta }_2}}\right)}^{\frac{1}{2}}\\ -\frac{i}{2}{\sigma }_1{\sigma }_2\chi \left(\frac{1}{{\vartheta }_2}-\frac{1}{{\vartheta }_1}\right){\left(\frac{1}{\frac{1}{{\hslash }^2}\chi \frac{{\sigma }_1}{{\vartheta }_1}}\right)}^{\frac{1}{2}}],\end{array}$ (4.8)

where

$D_2=-\frac{1}{4{\hslash }^2}{\left(\frac{{\sigma }_2}{{\vartheta }_2}\right)}^2{\chi }^2{\zeta }_1.$ (4.9)

Hall conductivity is defined as the ratio of the current density $J\left(y_1\right)$ and the electric field $E\left(y_1\right)$ :

$\text{Γ}=\frac{\langle J\left(y_1\right)\rangle }{\langle E\left(y_1\right)\rangle }.$ (4.10)

From (4.1), we have

$V\left(y_1\right)=\frac{1}{2}{\sigma }_1{y}_1^2,$ (4.11)

consequently

${\left(V\left(y_1\right)\right)}_{\star }=\frac{1}{2}\left({\sigma }_1{y}_1^2-\frac{{\hslash }^2}{4}{\partial }_{\left(\frac{\partial }{\partial y_1}\right)}^2+i\hslash y_1{\partial }_{\left(\frac{\partial }{\partial y_1}\right)}\right).$ (4.12)

So, it is easy to verify that

${\left(E\left(y_1\right)\right)}_{\star }=-{\left(\frac{\text{d}V\left(y_1\right)}{\text{d}{y}_1}\right)}_{\star }=-{\sigma }_1{y}_1-\frac{1}{2}i\hslash {\partial }_{\left(\frac{\partial }{\partial y_1}\right)}.$ (4.13)

The average of (4.13)

$\begin{array}{c}\langle E\left(y_1\right)\rangle ={\displaystyle \int {\left|{\psi }_0\left(y_1\right)\right|}^2\star {\left(E\left(y_1\right)\right)}_{\star }\text{d}{y}_1}\\ =\frac{2{\sigma }_1}{\sqrt{\pi }}\frac{\gamma }{\sqrt{1-{\gamma }^2}}\left(\frac{1}{\frac{1}{2{\hslash }^2}\chi \frac{{\sigma }_1}{{\vartheta }_1}}\right)\left(\frac{1}{\frac{1}{2{\hslash }^2}\chi \frac{{\sigma }_2}{{\vartheta }_2}}\right).\end{array}$ (4.14)

Insert (4.8), (4.14) in (4.10), we obtain

$\begin{array}{c}\text{Γ}=\frac{1}{4{\hslash }^3}\frac{1}{\sqrt{\pi }{\sigma }_1}\frac{\gamma }{\sqrt{1-{\gamma }^2}}\frac{1}{{\left(\frac{1}{\frac{1}{2{\hslash }^2}\chi \frac{{\sigma }_1}{{\vartheta }_1}}\right)}^{\frac{1}{2}}}[\left(-{\sigma }_2+D_2\right){\left(\frac{1}{\frac{1}{2{\hslash }^2}\chi \frac{{\sigma }_2}{{\vartheta }_2}}\right)}^{\frac{1}{2}}\\ -\frac{i}{2}{\sigma }_1{\sigma }_2\chi \left(\frac{1}{{\vartheta }_2}-\frac{1}{{\vartheta }_1}\right){\left(\frac{1}{\frac{1}{{\hslash }^2}\chi \frac{{\sigma }_1}{{\vartheta }_1}}\right)}^{\frac{1}{2}}].\end{array}$ (4.15)

Equation above summarizes some important specificity of an entangled system that can be quantified using Hall effect. Similarly, Hall conductivity is calculated in three dimensions using the Kubo formula from the bulk-edge [28]. Note that we have studied from two harmonic oscillators, entanglement entropies, Hall conductivity. It would be interesting to study their energy density. Their calculation in this context represents an important vision. Energy density reads

$\epsilon \left(y_1\right)={\displaystyle \int H^{\prime }}\left(y_1,\frac{\partial }{\partial y_1}\right)\star {\left(W_{00}\left(y_1,\frac{\partial }{\partial y_1}\right)\right)}_{\star }\text{d}\left(\frac{\partial }{\partial y_1}\right).$ (4.16)

Go back expression (4.3), after integration, (4.16) becomes

$\begin{array}{c}\epsilon \left(y_1\right)=\frac{1}{\pi \hslash }\frac{\gamma }{\sqrt{1-{\gamma }^2}}{\left(\frac{1}{2{\hslash }^2\chi \frac{{\sigma }_2}{{\vartheta }_2}}\right)}^{\frac{1}{2}}[\sqrt{\pi }\left({\sigma }_1+D_1\right)y_1^2-\frac{i}{2}{\sigma }_1{\sigma }_2\chi \left(\frac{1}{{\vartheta }_2}-\frac{1}{{\vartheta }_1}\right){\left(\frac{1}{\frac{1}{2{\hslash }^2}\chi \frac{{\sigma }_2}{{\vartheta }_2}}\right)}^{\frac{1}{2}}{y}_1\\ +\frac{\sqrt{\pi }}{4}\left(-{\sigma }_2+D_2\right){\left(\frac{1}{2{\hslash }^2\chi \frac{{\sigma }_2}{{\vartheta }_2}}\right)}^{\frac{1}{2}}]\text{exp}\left[-\frac{1}{2{\hslash }^2}\chi \frac{{\sigma }_1}{{\vartheta }_1}{y}_1^2\right],\end{array}$ (4.17)

where

$D_1=-\frac{1}{4{\hslash }^2}{\left(\frac{{\sigma }_1}{{\vartheta }_1}\right)}^2{\chi }^2{\zeta }_2.$ (4.18)

The most famous examples that reflect the usefulness of both concepts are provided by the quantum Hall effect [29] [30], semiconductor quantum devices [31] [32], etc.

5. Application of Transition-State Theory

We will here briefly introduce some aspects of transition-state theory, focusing on the aspects that will be useful to us later. For more details, see references [33]-[35], from which this presentation is strongly inspired. We have performed a transition to the first excited state. Harmonic oscillator one moves into the first excited state, second harmonic oscillator in the ground state, so Wigner function (3.14) becomes

$W_{10}=-W_{00}+\frac{2}{{\pi }^2{\delta }_1^2{\delta }_2\sqrt{{\sigma }_1{\sigma }_2}}{H}_1{\text{e}}^{-\frac{H_1}{\hslash {\delta }_1\sqrt{{\sigma }_1{\sigma }_2}}}{\text{e}}^{-\frac{H_2}{\hslash {\delta }_2\sqrt{{\sigma }_1^{\text{*}}{\sigma }_2}}}.$ (5.1)

Consequently, of expression (5.1), the reduced Wigner function of the variables $\left(y_1,\frac{\partial }{\partial y_1}\right)$ , is written

$W_{10}\left(y_1,\frac{\partial }{\partial y_1}\right)=\frac{2}{\hslash \sqrt{{\sigma }_1{\sigma }_2}}\left({\zeta }_1{y}_1^2+{\zeta }_2\frac{{\partial }^2}{\partial y_1^2}+{\zeta }_3\right)W_{00}\left(y_1,\frac{\partial }{\partial y_1}\right).$ (5.2)

In expression (5.2),

${\zeta }_1=\frac{d^2}{4c^2}+\sqrt{{\sigma }_1{\sigma }_2}\frac{d}{c}\sin \left({\omega }_2\right)\cos \left({\omega }_2\right)+{\sigma }_1{\sin }^2\left({\omega }_2\right),$

${\zeta }_2=\frac{b^2}{4a^2}-\sqrt{{\sigma }_1{\sigma }_2}\frac{b}{a}\sin \left({\omega }_1\right)\cos \left({\omega }_1\right)-{\sigma }_1{\sin }^2\left({\omega }_1\right),$

and

${\zeta }_3=\frac{1}{2a}{\sigma }_1^{*}{\sin }^2\left({\omega }_1\right)+\frac{1}{2c}{\sigma }_2{\cos }^2\left({\omega }_2\right)-1,$ (5.3)

where

$a=\frac{1}{\hslash {\delta }_1\sqrt{{\sigma }_1{\sigma }_2}}{\sigma }_1^{*}{\sin }^2\left({\omega }_1\right)+\frac{1}{\hslash {\delta }_2\sqrt{{\sigma }_1^{*}{\sigma }_2}}{\sigma }_1^{*}{\cos }^2\left({\omega }_1\right),$

$b=2i\sqrt{{\sigma }_1^{*}{\sigma }_2}\sin \left({\omega }_1\right)\cos \left({\omega }_1\right)\left(-\frac{1}{\hslash {\delta }_1\sqrt{{\sigma }_1{\sigma }_2}}+\frac{1}{\hslash {\delta }_2\sqrt{{\sigma }_1^{*}{\sigma }_2}}\right),$

$c=\frac{1}{\hslash {\delta }_1\sqrt{{\sigma }_1{\sigma }_2}}{\sigma }_2{\cos }^2\left({\omega }_2\right)+\frac{1}{\hslash {\delta }_2\sqrt{{\sigma }_1^{*}{\sigma }_2}}{\sigma }_2{\sin }^2\left({\omega }_2\right),$

and

$d=2\sqrt{{\sigma }_1{\sigma }_2}\sin \left({\omega }_2\right)\cos \left({\omega }_2\right)\left(-\frac{1}{\hslash {\delta }_1\sqrt{{\sigma }_1{\sigma }_2}}+\frac{1}{\hslash {\delta }_2\sqrt{{\sigma }_1^{*}{\sigma }_2}}\right).$ (5.4)

Using ref. [36] and applying the Moyal product (3.1), we have

$\begin{array}{l}{\left(W_{10}\left(y_1,\frac{\partial }{\partial y_1}\right)\right)}_{\star }\\ =\left(\left({\zeta }_1+D_1\right)y_1^2+\left({\zeta }_2+D_2\right)\frac{{\partial }^2}{\partial y_1^2}-\frac{1}{{\hslash }^2}\chi \left({\zeta }_1\frac{{\sigma }_2}{{\vartheta }_2}+{\zeta }_2\frac{{\sigma }_1}{{\vartheta }_1}\right)y_1\frac{\partial }{\partial y_1}+{\zeta }_3\right)\\ \times {\left(W_{00}\left(y_1,\frac{\partial }{\partial y_1}\right)\right)}_{\star }.\end{array}$ (5.5)

At the barrier, the oscillator frequency is perturbed, defining thus the potential using (4.11) as

${\sigma }_1{y}_1^2\simeq {\sigma }_{1b}{y}_{1b}^2-{\sigma }_{1b}{y}_1^2,$ (5.6)

where ${\sigma }_{1b}=-i{\sigma }_1$ , (5.6) becomes

$y_1^2\simeq i\left(y_1^2-y_{1b}^2\right),$ (5.7)

Insert (5.7) in (5.5), we have

$\begin{array}{c}{\left({\left(W_{10}\left(y_1,\frac{\partial }{\partial y_1}\right)\right)}_b\right)}_{\star }=\frac{\gamma }{\pi \hslash \sqrt{1-{\gamma }^2}}(i\left({\zeta }_1+D_1\right)\left(y_1^2-y_{1b}^2\right)+\left({\zeta }_2+D_2\right)\frac{{\partial }^2}{\partial y_1^2}\\ -\frac{1}{{\hslash }^2}\chi \left({\zeta }_1\frac{{\sigma }_2}{{\vartheta }_2}+{\zeta }_2\frac{{\sigma }_1}{{\vartheta }_1}\right)\sqrt{i\left(y_1^2-y_{1b}^2\right)}\frac{\partial }{\partial y_1}+{\zeta }_3)\\ \times {\text{exp}}_{\star }\left[-\frac{1}{2{\hslash }^2}\chi \left(i\frac{{\sigma }_1}{{\vartheta }_1}\left(y_1^2-y_{1b}^2\right)+\frac{{\sigma }_2}{{\vartheta }_2}\frac{{\partial }^2}{\partial y_1^2}\right)\right].\end{array}$ (5.8)

At the barrier, we read the Wigner function as

${\left(W_{10}\left(y_{1b},\frac{\partial }{\partial y_1}\right)\right)}_{\star }={\displaystyle \int {\left({\left(W_{10}\left(y_1,\frac{\partial }{\partial y_1}\right)\right)}_b\right)}_{\star }\text{d}{y}_1}$ (5.9)

To compute analytically integral of expression (5.9), we use ref. [37] and we ended up with

$\begin{array}{l}{\left(W_{10}\left(y_{1b},\frac{\partial }{\partial y_1}\right)\right)}_{\star }\\ =\frac{\gamma }{\pi \hslash \sqrt{1-{\gamma }^2}}{\left(\frac{\pi }{\frac{1}{2{\hslash }^2}\chi \frac{{\sigma }_1}{{\vartheta }_1}}\right)}^{\frac{1}{2}}(\lambda \frac{\partial }{\partial y_1}\text{exp}\left[-\frac{1}{2{\hslash }^2}\chi \frac{{\sigma }_1}{{\vartheta }_1}{y}_{1b}^2\right]-i\left({\zeta }_1+D_1\right)y_{1b}^2\\ -i\left({\zeta }_2+D_2\right)\frac{{\partial }^2}{\partial y_1^2}+{\lambda }_1)f\left(y_{1b}\right){\text{exp}}_{\star }\left[-\frac{1}{2{\hslash }^2}\chi \left(\frac{{\sigma }_1}{{\vartheta }_1}{y}_{1b}^2+\frac{{\sigma }_2}{{\vartheta }_2}\frac{{\partial }^2}{\partial y_1^2}\right)\right].\end{array}$ (5.10)

$f\left(y_{1b}\right)$ is the Step function, it is defined as

$f\left(y_{1b}\right)=\{\begin{array}{ll}0\hfill & y_{1b}<0\hfill \\ 1\hfill & y_{1b}>0,\hfill \end{array}$ (5.11)

Now, we have just evaluated entanglement and the energy density via the transition-state theory, starting from expression

$P_b={\displaystyle \iint \left[{\left(W_{10}\left(y_{1b},\frac{\partial }{\partial y_1}\right)\right)}_{\star }^2\right]\text{d}{y}_{1b}\text{d}\left(\frac{\partial }{\partial y_1}\right)}$ (5.12)

This allows us to write

$\begin{array}{c}{P}_b={\displaystyle \iint \frac{1}{{\pi }^2{\hslash }^2}\frac{{\gamma }^2}{1-{\gamma }^2}}\left(\frac{\pi }{\frac{1}{2{\hslash }^2}\chi \frac{{\sigma }_1}{{\vartheta }_1}}\right){\left(\frac{\pi }{\frac{1}{{\hslash }^2}\chi \frac{{\sigma }_2}{{\vartheta }_2}}\right)}^{\frac{1}{2}}{\exp }_{\star }\left[-\frac{1}{2{\hslash }^2}\chi \frac{{\sigma }_1}{{\vartheta }_1}\left(y_{1b}^2+{y^{\prime }}_{1b}^2\right)\right]\\ \times [{\lambda }^2\left(\frac{1}{\frac{1}{{\hslash }^2}\chi \frac{{\sigma }_2}{{\vartheta }_2}}\right)\exp \left[-\frac{1}{2{\hslash }^2}\chi \frac{{\sigma }_1}{{\vartheta }_1}\left(y_{1b}^2+{y^{\prime }}_{1b}^2\right)\right]+\frac{i}{\sqrt{\pi }}{\left(\frac{1}{\frac{1}{{\hslash }^2}\chi \frac{{\sigma }_2}{{\vartheta }_2}}\right)}^{\frac{1}{2}}\left({\zeta }_1+D_1\right)\lambda \\ \times \left({y^{\prime }}_{1b}^2\exp \left[-\frac{1}{2{\hslash }^2}\chi \frac{{\sigma }_1}{{\vartheta }_1}{y}_{1b}^2\right]-y_{1b}^2\exp \left[-\frac{1}{2{\hslash }^2}\chi \frac{{\sigma }_1}{{\vartheta }_1}{y^{\prime }}_{1b}^2\right]\right)+i\lambda {\lambda }_2\left(\frac{1}{\frac{1}{{\hslash }^2}\chi \frac{{\sigma }_2}{{\vartheta }_2}}\right)\\ \times \left(\exp \left[-\frac{1}{2{\hslash }^2}\chi \frac{{\sigma }_1}{{\vartheta }_1}{y}_{1b}^2\right]-\exp \left[-\frac{1}{2{\hslash }^2}\chi \frac{{\sigma }_1}{{\vartheta }_1}{y^{\prime }}_{1b}^2\right]\right)+\frac{\lambda }{\sqrt{\pi }}\\ \times \left({\lambda }_1^{*}\exp \left[-\frac{1}{2{\hslash }^2}\chi \frac{{\sigma }_1}{{\vartheta }_1}{y}_{1b}^2\right]+{\lambda }_1\exp \left[-\frac{1}{2{\hslash }^2}\chi \frac{{\sigma }_1}{{\vartheta }_1}{y^{\prime }}_{1b}^2\right]\right)+\left({\zeta }_1+D_1\right)({\lambda }_2\left(y_{1b}^2+{y^{\prime }}_{1b}^2\right)\\ \begin{array}{c}\\ \\ \end{array}+\left({\zeta }_1+D_1\right)y_{1b}^2{y^{\prime }}_{1b}^2+i\left({\lambda }_1{y^{\prime }}_{1b}^2-{\lambda }_1^{*}{y}_{1b}^2\right))+{\lambda }_2\left(i\left({\lambda }_1-{\lambda }_1^{*}\right)+{\lambda }_2\right)+{\lambda }_1{\lambda }_1^{*}]\text{d}{y}_{1b}\text{d}{y^{\prime }}_{1b},\end{array}$ (5.13)

In expression (5.13),

$\begin{array}{l}\lambda =-\frac{\pi }{\text{Γ}\left(-\frac{1}{2}\right)}\frac{1}{{\hslash }^2}\chi \left({\zeta }_1\frac{{\sigma }_2}{{\vartheta }_2}+{\zeta }_2\frac{{\sigma }_1}{{\vartheta }_1}\right){\left(\frac{1}{2{\hslash }^2}\chi \frac{{\sigma }_1}{{\vartheta }_1}\right)}^2,\\ {\lambda }_1={\left(\frac{\pi }{\frac{1}{2{\hslash }^2}\chi \frac{{\sigma }_1}{{\vartheta }_1}}\right)}^{\frac{1}{2}}\left(\frac{i}{2}{\lambda }_3+{\zeta }_3\right),\end{array}$

${\lambda }_3=\frac{1}{\frac{1}{2{\hslash }^2}\chi \frac{{\sigma }_1}{{\vartheta }_1}}\left({\zeta }_1+D_1\right)\text{and}{\lambda }_2=\frac{1}{4}\frac{1}{\frac{1}{{\hslash }^2}\chi \frac{{\sigma }_2}{{\vartheta }_2}}\left({\zeta }_2+D_2\right).$ (5.14)

Expression (5.13) is a direct consequence of

$\begin{array}{l}{P}_b=\left(\frac{\pi }{\frac{1}{2{\hslash }^2}\frac{{\sigma }_1}{{\vartheta }_1}\chi }\right)[\left(\frac{1}{\frac{1}{{\hslash }^2}\frac{{\sigma }_2}{{\vartheta }_2}\chi }\right){\lambda }^2+i\left({\lambda }_1-{\lambda }_1^{\text{*}}\right)\left({\lambda }_2+{\lambda }_3\right)+{\lambda }_3\left({\lambda }_2+{\lambda }_3\right)\\ \begin{array}{c}\\ \\ \stackrel{}{}\end{array}+\frac{1}{\sqrt{2\pi }}\left({\lambda }_1^{\text{*}}+{\lambda }_1\right)+{\lambda }_1{\lambda }_1^{\text{*}}]P.\end{array}$ (5.15)

In expression (5.15), $P$ is the purity function of expression (3.24). Although with the transition performed to the first exited state and at the height of the barrier, the star product of (4.1) and (5.10) is written

$\begin{array}{l}{\left(H^{\prime }\left(y_1,\frac{\partial }{\partial y_1}\right)\right)}_b\star {\left({\left(W_{10}\left(y_1,\frac{\partial }{\partial y_1}\right)\right)}_b\right)}_{\star }\\ =\frac{1}{2}\left(i{\sigma }_1\left(y_1^2-y_{1b}^2\right)-{\sigma }_2\frac{{\partial }^2}{\partial y_1^2}\right)\star {\left({\left(W_{10}\left(y_1,\frac{\partial }{\partial y_1}\right)\right)}_b\right)}_{\star }\\ =\frac{1}{2}[i{\sigma }_1\left(y_1^2-y_{1b}^2\right)-{\sigma }_2\frac{{\partial }^2}{\partial y_1^2}+\frac{{\hslash }^2}{4}{\sigma }_2{\partial }_{y_1}^2\\ +i\hslash \left(i{\sigma }_1{y}_1{\partial }_{\left(\frac{\partial }{\partial y_1}\right)}-i{\sigma }_1{y}_{1b}{\partial }_{\left(\frac{\partial }{\partial y_1}\right)}+{\sigma }_2\frac{\partial }{\partial y_1}{\partial }_{y_1}\right)]\times {\left({\left(W_{10}\left(y_1,\frac{\partial }{\partial y_1}\right)\right)}_b\right)}_{\star }\end{array}$ (5.16)

Therefore, we generalize the energy density in the framework of transition-state theory as

$\left(\epsilon {\left(y_1\right)}_b\right)={\displaystyle \int {\left(H^{\prime }\left(y_1,\frac{\partial }{\partial y_1}\right)\right)}_b}\star {\left({\left(W_{10}\left(y_1,\frac{\partial }{\partial y_1}\right)\right)}_b\right)}_{\star }\text{d}\left(\frac{\partial }{\partial y_1}\right).$ (5.17)

More explicitly, we have

$\begin{array}{c}\left(\epsilon {\left(y_1\right)}_b\right)=\{y_1^4{\alpha }_1^2\sqrt{\pi {\alpha }_2}\left[{\sigma }_2{D}_1+i\kappa \right]+y_1^3{\alpha }_2[-i{\sigma }_1\left(\sqrt{{\zeta }_1{D}_2}+\sqrt{{\zeta }_2{D}_1}\right)+\frac{{\sigma }_2}{{\alpha }_1}\left({\zeta }_1-2i{D}_1\right)\\ \begin{array}{c}\\ \end{array}-{\sigma }_2{\alpha }_1^2\sqrt{{\zeta }_1{D}_2}]+2i{\sigma }_1{\zeta }_1{y}_{1b}^3+y_1^2[{\sigma }_2{D}_1\sqrt{\pi {\alpha }_2}\left(\frac{1}{{\alpha }_1}+{\alpha }_2+\frac{9}{2}{\hslash }^2\right)\end{array}$

$\begin{array}{c}+{\sigma }_2{D}_2\sqrt{\pi {\alpha }_2^5}\left(\frac{1}{{\alpha }_1^2}+\frac{1}{{\alpha }_1}-i{\alpha }_2\right)+{\sigma }_1\sqrt{{\zeta }_2{D}_1}\left(\frac{1}{4}\sqrt{\pi {\alpha }_2^3}-\frac{1}{2}\sqrt{\pi {\alpha }_2}-\sqrt{\frac{1}{{\alpha }_2}}\right)\\ +{\sigma }_1\sqrt{{\zeta }_1{D}_2}\left(\sqrt{\pi {\alpha }_2^3}-\frac{1}{2{\hslash }^2}\right)+i{\sigma }_1{\zeta }_3\sqrt{\pi {\alpha }_2}]+y_{1b}^2[\frac{i}{4}\kappa \sqrt{\pi {\alpha }_2^3}\\ +{\sigma }_1{\zeta }_1\left(1+\frac{1}{2{\alpha }_1}-\frac{1}{2}\sqrt{\pi {\alpha }_2}\right)+2\hslash {\sigma }_1\sqrt{{\zeta }_1{D}_2}+i{\sigma }_2{\zeta }_1]\\ +y_1[{\sigma }_2\sqrt{{\zeta }_1{D}_2}\left({\alpha }_2^2-\frac{1}{2}{\alpha }_2+2\frac{{\sigma }_1}{{\sigma }_2}\right)+\frac{3}{2}{\sigma }_2{\zeta }_2{\alpha }_2+\frac{3}{2}i{\sigma }_2{\zeta }_1\frac{1}{{\alpha }_2}+i\frac{1}{{\alpha }_1}{\sigma }_2{\zeta }_3{\alpha }_2\\ +2i{\sigma }_1{\zeta }_3\frac{1}{{\alpha }_1}]+y_{1b}\left[-i\frac{{\sigma }_1}{4}\sqrt{{\zeta }_2{D}_1}+{\sigma }_2{\zeta }_1{\alpha }_2-{\sigma }_1{\zeta }_3\right]-i{\sigma }_1{y}_1^2{y}_{1b}^2\sqrt{\pi {\alpha }_2}{D}_1\\ -i{\sigma }_1{\zeta }_1{y}_1^2{y}_{1b}+i{y}_{1b}^2{y}_1\left[{\sigma }_1\sqrt{{\zeta }_2{D}_1}{\alpha }_2+i\frac{1}{{\alpha }_1}{\alpha }_2{\sigma }_2{\zeta }_1-{\hslash }^2{\sigma }_1{\zeta }_1\right]\\ +y_1{y}_{1b}\left[2{\sigma }_1\sqrt{{\zeta }_2{D}_1}\left(1-\sqrt{\pi {\alpha }_2}\right)\right]+{\sigma }_1{\zeta }_2\left[{\alpha }_2^2-2{\alpha }_2+\frac{1}{{\alpha }_1}\right]\\ +{\sigma }_2{\zeta }_1\left[\frac{1}{{\alpha }_1}-\frac{i}{2}\sqrt{\pi {\alpha }_2^2}+\frac{i}{2}\right]+\frac{1}{4}{\sigma }_2{\zeta }_2\left[3\sqrt{\pi {\alpha }_2^5}+\frac{3}{{\alpha }_1}\sqrt{\pi {\alpha }_2^3}-\frac{1}{{\alpha }_1^2}\right]\\ +\frac{1}{4}{\sigma }_2{\zeta }_3\left[\sqrt{\pi {\alpha }_2^2}-2\frac{1}{{\alpha }_1}\right]\}{\exp }_{\star }\left[-\frac{1}{2{\hslash }^2}\chi i\frac{{\sigma }_1}{{\vartheta }_1}\left(y_1^2-y_{1b}^2\right)\right].\end{array}$ (5.18)

At the barrier, the energy density is written

$\epsilon \left(y_{1b}\right)={\displaystyle \int \left(\epsilon {\left(y_1\right)}_b\right)\text{d}{y}_1}$ (5.19)

which gives according to (5.18)

$\begin{array}{c}\epsilon \left(y_{1b}\right)=\{2i{\sigma }_1{\zeta }_1{y}_{1b}^3+y_{1b}^2[{\sigma }_1{\zeta }_1\left(1+\frac{1}{2{\alpha }_1}-\frac{i}{2}{\hslash }^2{\alpha }_1-\frac{1}{2}\sqrt{\pi {\alpha }_2}\right)+{\sigma }_2{\zeta }_1\left(i-\frac{1}{2}{\alpha }_2\right)\\ -\frac{i}{4}{\pi }^2{\sigma }_1{D}_1\sqrt{{\alpha }_1^3{\alpha }_2}+2\hslash {\sigma }_1\sqrt{{\zeta }_1{D}_2}+\frac{i}{2}{\sigma }_1\sqrt{{\zeta }_2{D}_1}{\alpha }_1{\alpha }_2+\frac{i}{4}\kappa \sqrt{\pi {\alpha }_2^3}]\\ +y_{1b}\left[{\sigma }_1\sqrt{{\zeta }_2{D}_1}\left({\alpha }_1-\sqrt{\pi {\alpha }_1^2{\alpha }_2}-\frac{i}{4}\right)+{\sigma }_2{\zeta }_1{\alpha }_2-{\sigma }_1{\zeta }_3-\frac{i}{4}{\sigma }_1\sqrt{\pi {\alpha }_1^3}\right]\\ +{\sigma }_1{\zeta }_2\left[\frac{1}{{\alpha }_1}+{\alpha }_2^2-2{\alpha }_2\right]+i{\sigma }_1{\zeta }_3\left[1+\frac{1}{4}\pi \sqrt{{\alpha }_1^3{\alpha }_2}\right]+{\sigma }_2{\zeta }_1[\frac{1}{2}{\alpha }_1{\alpha }_2+\frac{3}{4}i\frac{{\alpha }_1}{{\alpha }_2}\\ +\frac{1}{{\alpha }_1}-\frac{i}{2}\sqrt{\pi }{\alpha }_2+\frac{i}{2}]+\frac{1}{4}{\sigma }_2{\zeta }_2\left[-\frac{1}{{\alpha }_1^2}+3\sqrt{\pi }\frac{\sqrt{{\alpha }_2^3}}{{\alpha }_1}+3\sqrt{\pi {\alpha }_2^5}+3{\alpha }_1{\alpha }_2\right]\\ +\frac{1}{2}{\sigma }_2{\zeta }_3\left[\frac{5}{2}i{\alpha }_2-\frac{1}{{\alpha }_1}\right]+\frac{1}{2}\sqrt{{\zeta }_1{D}_2}[{\sigma }_1(-i{\alpha }_1^2{\alpha }_2+\frac{1}{2}\pi \sqrt{{\alpha }_1^3{\alpha }_2^3}\\ +\frac{1}{4{\hslash }^2}\sqrt{\pi {\alpha }_1^3}+2{\alpha }_1)-{\sigma }_2\left({\alpha }_1^4{\alpha }_2-{\alpha }_1{\alpha }_2^2+\frac{1}{2}{\alpha }_1{\alpha }_2\right)]\\ +\frac{1}{2}\sqrt{{\zeta }_2{D}_1}\left[{\sigma }_1\left(-i{\alpha }_1^2{\alpha }_2+\frac{1}{8}\pi \sqrt{{\alpha }_1^3{\alpha }_2^3}-\frac{1}{4}\pi \sqrt{{\alpha }_1^3{\alpha }_2}-\frac{1}{4}\sqrt{\pi \frac{{\alpha }_1^3}{{\alpha }_2}}\right)\right]\\ -\frac{1}{4}\pi {\sigma }_2{D}_2\left[\sqrt{{\alpha }_1^3{\alpha }_2^7}-\sqrt{{\alpha }_1{\alpha }_2^5}-\sqrt{\frac{{\alpha }_2^5}{{\alpha }_1}}\right]+\frac{3}{4}i\pi {\alpha }_1^4\sqrt{{\alpha }_1{\alpha }_2}\kappa \}{\exp }_{\star }\left[\frac{1}{2{\hslash }^2}\chi i\frac{{\sigma }_1}{{\vartheta }_1}{y}_{1b}^2\right],\end{array}$ (5.20)

where

$\kappa ={\sigma }_2{\zeta }_1-{\sigma }_1{\zeta }_2,{\alpha }_1=\frac{1}{\frac{1}{2{\hslash }^2}}\chi \text{and}{\alpha }_2=\frac{1}{\frac{1}{2{\hslash }^2}\chi \frac{{\sigma }_2}{{\vartheta }_2}}.$ (5.21)

6. Some Particular Cases

Let us interpret the results obtained in (3.24), (3.27), (3.29), (4.15), (4.17), (5.15) and (5.20) in terms of the non-commutativity variables ${\theta }_1$ , ${\theta }_2$ and the magnetic field $B$ .

6.1. The First Case

If we assume from expression (3.4) that

${\omega }_1={\omega }_2=\frac{\pi }{4},$ (6.1)

(3.11) become

${\delta }_1={\delta }_2=\frac{\hslash }{2}{\sigma }_2\left({\sigma }_1-{\sigma }_1^{\text{*}}\right)+{\sigma }_2\left({\theta }_1{\sigma }_1-{\theta }_2{\sigma }_1^{\text{*}}\right)$ (6.2)

As a consequence, (3.20) read

$\gamma =\sqrt{\frac{1}{{\delta }_1^4{\sigma }_1^{\text{*}}{\sigma }_2{\vartheta }_1^2}},$ (6.3)

and

${\vartheta }_1={\vartheta }_2=\frac{1}{2{\delta }_1}\left(\frac{1}{\sqrt{{\sigma }_1^{\text{*}}{\sigma }_2}}+\frac{1}{\sqrt{{\sigma }_1{\sigma }_2}}\right).$ (6.4)

where ${\theta }_1$ and ${\theta }_2$ are fields and are functions of position and momentum, we consider from Figure 1, the evolution of the von Neumann entropy and the Hall conductivity with respect ${\theta }_1$ and ${\theta }_2$ . These two concepts of quantum mechanics have a similar behavior and they increase with ${\theta }_1$ and ${\theta }_2$ . This explains that non-commutativity introduces an intrinsic correlation into the system, affecting both the global quantum state (entanglement) and the electrical response (Hall conductivity). This behavior shows a duality between the quantities defined in the momentum space and those defined in the position space. The fields ${\theta }_1$ and ${\theta }_2$ that appear naturally in the expression of von Neumann entropy and Hall conductivity have a singularity at the origin of the coordinate position and momentum. Figure 2 shows that the transition in the quantum state performs clearly increases entanglement.

6.2. The Second Case

The second case is considered as follows: ${\theta }_2=0$ which gives from (3.4), ${\omega }_1=0$ . We define ${\lambda }_2$ in expression (3.4) such as ${\omega }_2=\frac{1}{2}\text{arctan}\left({\lambda }_2\right)$ and

Figure 1. von Neumann entropy $S_1\left(W_{00}\right)$ (3.29) and Hall conductivity $\text{Γ}$ (4.15) for $B=0.1$ and $\omega =1.2$ .

Figure 2. Purity functions $P$ (3.24) and $P_b$ (5.15) for $B=0.1$ and $\omega =0.7$ .

${\lambda }_2=\frac{2\hslash \left(\sqrt{\frac{{\sigma }_2}{{\sigma }_1^{\text{*}}}}-1\right){\theta }_1}{{\hslash }^2{\left(\sqrt{\frac{{\sigma }_2}{{\sigma }_1^{\text{*}}}}-1\right)}^2-\frac{{\sigma }_2}{{\sigma }_1^{\text{*}}}{\theta }_1^2}$ . This formalism leads us to generalize the evolution of the Rényi entropy and the energy density by considering a field that depends on the position non-commutativity parameter ${\theta }_1$ and the magnetic field $B$ . (3.11) and (3.20) become

${\delta }_1=\sqrt{{\sigma }_1{\sigma }_2}\left(\hslash \sin \left({\omega }_2\right)+{\theta }_1\cos \left({\omega }_2\right)\right),$ (6.5)

${\delta }_2=-\sqrt{{\sigma }_1^{*}{\sigma }_2}\hslash \sin \left({\omega }_2\right),$ (6.6)

and

${\vartheta }_1=\frac{1}{2{\delta }_2}\frac{1}{\sqrt{{\sigma }_1{\sigma }_2}}.$ (6.7)

Let us now evaluate the Rényi entropy on another basis according to ${\lambda }_2$ and $n$ parameters. $n=2$ represents the lowest state, $n=8$ represents a more excited state.

We set ${\theta }_1=0.5$ and $\omega =1.4$ .

Observe Figure 3 as the magnetic field increases, Rényi entropy values of the highest excited states appear together and merge into one curve to a single value close to zero. The transition performs clearly increases entanglement, acts as an information-relevant perturbation. Let’s see how the energy density of an entangled system spreads according to the magnetic field $B$ and the non-commutativity parameter ${\theta }_1$ .

Figure 3. Rényi entropy (3.27) for different values of $n$ ; $n=2$ (blue solid line), $n=4$ (brown solid line), $n=6$ (red solid line), $n=8$ (black solid line).

We set $\omega =1.4$ .

The coupling ${\theta }_1$ of the position operator expression to the non-commutative property implies the non-localization of the two harmonic oscillators to the atom. Figure 4 and Figure 5 show that, an increase of $B$ increases the energy density, so it increases the energy stored in the masses of the system. Consequently, it accelerates the non-localization to the atom. By comparing the two (Figure 4 and Figure 5), we prove that at the height of the barrier, a harmonic oscillator in a magnetic field ${\theta }_1$ , which is a priori a function of the position $y_{1b}$ , has involuntarily and considerably increased its stored energy, so it increases its instabilization inside the atom. The effect of the ${\theta }_1$ -field then only manifests in the presence of a position-dependent potential $y_1$ and $y_{1b}$ .

Figure 4. Energy density $\epsilon \left(y_1\right)$ of (4.17).

Figure 5. Energy density $\epsilon \left(y_{1b}\right)$ of (5.20).

7. Summary

In this work, we have examined quantum entanglement, as well as we have found the Hall conductivity and the energy density of two isotropic harmonic oscillators under a magnetic field in non-commutative phase space. The transition-state theory is applied to calculate the purity function and the energy density. Entanglement and Hall conductivity show a similar pattern because of the non-commutative parameters. A strong magnetic field disturbs the exact information of the highest excited states, and it accelerates the non-localization of the system in the atom by increasing its energy density. Transition-state theory makes the system more entangled and more instable.

Acknowledgements

We thank Prof. Hichem Eleuch for useful discussions.

Conflicts of Interest

The author declares no conflicts of interest.

Conflicts of Interest

The author declares no conflicts of interest.

References

[1]	Gamboa, J., Méndez, F., Loewe, M. and Rojas, J.C. (2001) The Landau Problem and Noncommutative Quantum Mechanics. Modern Physics Letters A, 16, 2075-2078. https://doi.org/10.1142/s0217732301005345
[2]	Motavalli, H. and Akbarieh, A.R. (2010) Klein-Gordon Equation for the Coulomb Potential in Noncommutative Space. Modern Physics Letters A, 25, 2523-2528. https://doi.org/10.1142/s0217732310033529
[3]	Santos, W.O. and Souza, A.M.C. (2014) Phenomenology of Noncommutative Phase Space via the Anomalous Zeeman Effect in Hydrogen Atom. International Journal of Modern Physics A, 29, Article ID: 1450177. https://doi.org/10.1142/s0217751x14501772
[4]	Maireche, A. (2017) Effects of Two-Dimensional Noncommutative Theories on Bound States Schrödinger Diatomic Molecules under New Modified Kratzer-Type Interactions. International Letters of Chemistry, Physics and Astronomy, 76, 1-11. https://doi.org/10.56431/p-91lat5
[5]	Couch, J., Eccles, S., Fischler, W. and Xiao, M. (2018) Holographic Complexity and Noncommutative Gauge Theory. Journal of High Energy Physics, 3, Article No. 108. https://doi.org/10.1007/jhep03(2018)108
[6]	Seiberg, N. and Witten, E. (1999) String Theory and Noncommutative Geometry. Journal of High Energy Physics, 9, Article No. 32. https://doi.org/10.1088/1126-6708/1999/09/032
[7]	Hirshfeld, A.C. and Henselder, P. (2003) Star Products and Quantum Groups in Quantum Mechanics and Field Theory. Annals of Physics, 308, 311-328. https://doi.org/10.1016/s0003-4916(03)00144-1
[8]	Ho, P. and Kao, H. (2002) Noncommutative Quantum Mechanics from Noncommutative Quantum Field Theory. Physical Review Letters, 88, Article ID: 151602. https://doi.org/10.1103/physrevlett.88.151602
[9]	Nadaud, F. (1998) Generalised Deformations, Koszul Resolutions, Moyal Products. Reviews in Mathematical Physics, 10, 685-704. https://doi.org/10.1142/s0129055x98000215
[10]	Kammerer, J. (1986) Analysis of the Moyal Product in a Flat Space. Journal of Mathematical Physics, 27, 529-535. https://doi.org/10.1063/1.527400
[11]	Chaichian, M., S-Jabbari, M.M. and Tureanu, A. (2002) Comments on the Hydrogen Atom Spectrum in the Noncommutative Space. Chinese Physics Letters, 23, 1122-1123.
[12]	Kupriyanov, V.G. (2013) A Hydrogen Atom on Curved Noncommutative Space. Journal of Physics A: Mathematical and Theoretical, 46, Article ID: 245303. https://doi.org/10.1088/1751-8113/46/24/245303
[13]	Hatzinikitas, A. and Smyrnakis, I. (2002) The Noncommutative Harmonic Oscillator in More than One Dimension. Journal of Mathematical Physics, 43, 113-125. https://doi.org/10.1063/1.1416196
[14]	Smailagic, A. and Spallucci, E. (2002) Noncommutative 3D Harmonic Oscillator. Journal of Physics A: Mathematical and General, 35, L363-L368. https://doi.org/10.1088/0305-4470/35/26/103
[15]	Li, K. and Dulat, S. (2006) The Aharonov-Bohm Effect in Noncommutative Quantum Mechanics. The European Physical Journal C, 46, 825-828. https://doi.org/10.1140/epjc/s2006-02538-2
[16]	Gangopadhyay, S. and Scholtz, F.G. (2014) Path Integral Action of a Particle in a Magnetic Field in the Noncommutative Plane and the Aharonov-Bohm Effect. Journal of Physics A: Mathematical and Theoretical, 47, Article ID: 075301. https://doi.org/10.1088/1751-8113/47/7/075301
[17]	Ma, K., Wang, J. and Yang, H. (2016) Time-Dependent Aharonov-Bohm Effect on the Noncommutative Space. Physics Letters B, 759, 306-312. https://doi.org/10.1016/j.physletb.2016.05.094
[18]	Dayi, Ö.F. and Kelleyane, L.T. (2002) Wigner Functions for the Landau Problem in Noncommutative Spaces. Modern Physics Letters A, 17, 1937-1944. https://doi.org/10.1142/s0217732302008356
[19]	Dulat, S. and Kang, L. (2008) Landau Problem in Noncommutative Quantum Mechanics. Chinese Physics C, 32, 92-95. https://doi.org/10.1088/1674-1137/32/2/003
[20]	Adhikari, S., Chakraborty, B., Majumdar, A.S. and Vaidya, S. (2009) Quantum Entanglement in a Noncommutative System. Physical Review A, 79, Article ID: 042109. https://doi.org/10.1103/physreva.79.042109
[21]	Bastos, C., Bernardini, A.E., Bertolami, O., Dias, N.C. and Prata, J.N. (2013) Entanglement Due to Noncommutativity in Phase Space. Physical Review D, 88, Article ID: 085013. https://doi.org/10.1103/physrevd.88.085013
[22]	Nascimento, J.P.G., Aguiar, V. and Guedes, I. (2017) Entropy and Information of a Harmonic Oscillator in a Time-Varying Electric Field in 2D and 3D Noncommutative Spaces. Physica A: Statistical Mechanics and its Applications, 477, 65-77. https://doi.org/10.1016/j.physa.2017.02.018
[23]	Harms, B. and Micu, O. (2007) Noncommutative Quantum Hall Effect and Aharonov-Bohm Effect. Journal of Physics A: Mathematical and Theoretical, 40, 10337-10347. https://doi.org/10.1088/1751-8113/40/33/024
[24]	Zhang, J. (2004) Fractional Angular Momentum in Non-Commutative Spaces. Physics Letters B, 584, 204-209. https://doi.org/10.1016/j.physletb.2004.01.049
[25]	Lin, B., Jing, S. and Heng, T. (2008) Deformation Quantization for Coupled Harmonic Oscillators on a General Noncommutative Space. Modern Physics Letters A, 23, 445-456. https://doi.org/10.1142/s0217732308023992
[26]	Lin, B., Xu, J. and Heng, T. (2019) Induced Entanglement Entropy of Harmonic Oscillators in Non-Commutative Phase Space. Modern Physics Letters A, 34, Article ID: 1950269. https://doi.org/10.1142/s0217732319502699
[27]	Abidi, A. (2025) Purity Function of Three Isotopic Harmonic Oscillators. Modern Physics Letters A, Article ID: 2550137.
[28]	Chang, M., Ma, R. and Sheng, L. (2023) Chern Number and Hall Conductivity in Three-Dimensional Quantum Hall Effect in Weyl Semimetals. Physical Review B, 108, Article ID: 165416. https://doi.org/10.1103/physrevb.108.165416
[29]	Jellal, A. and Houça, R. (2009) Noncommutative Description of Spin Hall Effect. International Journal of Geometric Methods in Modern Physics, 6, 343-360. https://doi.org/10.1142/s0219887809003539
[30]	Markowich, P.A., Mauser, N.J. and Poupaud, F. (1994) A Wigner-Function Approach to (Semi)classical Limits: Electrons in a Periodic Potential. Journal of Mathematical Physics, 35, 1066-1094. https://doi.org/10.1063/1.530629
[31]	Rosati, R., Dolcini, F., Iotti, R.C. and Rossi, F. (2013) Wigner-Function Formalism Applied to Semiconductor Quantum Devices: Failure of the Conventional Boundary Condition Scheme. Physical Review B, 88, Article ID: 035401. https://doi.org/10.1103/physrevb.88.035401
[32]	Jungwirth, T., Niu, Q. and MacDonald, A.H. (2002) Anomalous Hall Effect in Ferromagnetic Semiconductors. Physical Review Letters, 88, Article No. 20. https://doi.org/10.1103/physrevlett.88.207208
[33]	Pollak, E. (1986) Transition-State Theory for Tunneling in Dissipative Media. Physical Review A, 33, 4244-4252. https://doi.org/10.1103/physreva.33.4244
[34]	Hansen, N.F. and Andersen, H.C. (1994) A New Formulation of Quantum Transition State Theory for Adiabatic Rate Constants. The Journal of Chemical Physics, 101, 6032-6037. https://doi.org/10.1063/1.467318
[35]	Miller, W.H. (1976) Importance of Nonseparability in Quantum Mechanical Transition-State Theory. Accounts of Chemical Research, 9, 306-312. https://doi.org/10.1021/ar50104a005
[36]	Hirshfeld, A.C. and Henselder, P. (2002) Deformation Quantization in the Teaching of Quantum Mechanics. American Journal of Physics, 70, 537-547. https://doi.org/10.1119/1.1450573
[37]	Prudnikov, A.P., Brychkov, Yu.A. and Marichev, O.I. (1986) Integrals and Series: Special Functions. Gordon and Breach Science, OPA (Amsterdam), 2.