Method of green's functions in the theory of quantum kinetic equations. II

The kinetic and antikinetic equations are obtained for the single-particle Wigner function in the context of the method of Green's time-temperature functions for an inhomogeneous system of weakly interacting particles situated in a time-dependent electric field. The kinetic equation is derived here from the equation of motion for Green's function, satisfying the causality condition.     

A new parametrized quantum distribution function and its time development

We discuss a new general class of quantum distribution functions characterized by an arbitrary parameter b. The values b = -1, 0, 1 correspond to the anti-standard (Kirkwood), the Wigner, and the standard distribution functions, respectively. An analytic form of the equation of motion is derived. We conclude that, for time-dependent problems involving a potential which is a function of coordinates only, the Wigner distribution function is the optimum one to use, from a simplicity standpoint.     

On Wigner functions and a damped star product in dissipative phase-space quantum mechanics

Dito and Turrubiates recently introduced an interesting model of the dissipative quantum mechanics of a damped harmonic oscillator in phase space. Its key ingredient is a non-Hermitian deformation of the Moyal star product with the damping constant as deformation parameter. We compare the Dito-Turrubiates scheme with phase-space quantum mechanics (or deformation quantization) based on other star products, and extend it to incorporate Wigner functions. The deformed (or damped) star product is related to a complex Hamiltonian, and so necessitates a modified equation of motion involving complex conjugation. We find that with this change the Wigner function satisfies the classical equation of motion. This seems appropriate since non-dissipative systems with quadratic Hamiltonians share this property.     

A Charged Particle in an Electric Field in the Probability Representation of Quantum Mechanics

The Green's function and linear integrals of motion for a charged particle moving in an electric field are discussed. The Wigner functions and tomograms of the stationary states of the charged particle are obtained. The relationship between the quantum propagators for the Schrödinger evolution equation, the Moyal evolution equation, and the evolution equation in the tomographic-probability representation for a charged particle moving in an electric field is discussed.     

On the quantum-mechanical Fokker-Planck and Kramers-Chandrasekhar equation

In the classical theory of Brownian motion we can consider the Langevin equation as an infinitesimal transformation between the coordinates and momenta of a Brownian particle, given probabilistically, since the impulse appearing is characterized by a Gaussian random process. This probabilistic infinitesimal transformation generates a streaming on the distribution function, expressed by the classical Fokker-Planck and Kramers-Chandrasekhar equations. If the laws obeyed by the Brownian particle are quantum mechanical, we can reinterpret the Langevin equation as an operator relation expressing an infinitesimal transformation of these operators. Since the impulses are independent of the coordinates and momenta we can think of them as c numbers described by a Gaussian random process. The so resulting infinitesimal operator transformation induces a streaming on the density matrix. We may associate, according to Weyl functions with operators. The function associated with the density matrix is the Wigner function. Expressing, then, these operator relations in terms of these functions we can express the streaming as a continuity equation of the Wigner function. We find that in this parametrization the extra terms which appear are the same as in the classical theory, augmenting the usual Wigner equation.     

Quantum corrections to the entropy in a driven quantum Brownian motion model

The quantum Brownian motion model is a typical model in the study of nonequilibrium quantum thermodynamics. Entropy is one of the most fundamental physical concepts in thermodynamics.In this work, by solving the quantum Langevin equation, we study the von Neumann entropy of a particle undergoing quantum Brownian motion. We obtain the analytical expression of the time evolution of the Wigner function in terms of the initial Wigner function. The result is applied to the thermodynamic equilibrium initial state, which reproduces its classical counterpart in the high temperature limit. Based on these results, for those initial states having well-defined classical counterparts, we obtain the explicit expression of the quantum corrections to the entropy in the weak coupling limit. Moreover, we find that for the thermodynamic equilibrium initial state, all terms odd in h are exactly zero. Our results bring important insights to the understanding of entropy in open quantum systems.     

Extension of trigonometric and hyperbolic functions to vectorial arguments and its application to the representation of rotations and Lorentz transformations

The use of the axial vector representing a three-dimensional rotation makes the rotation representation much more compact by extending the trigonometric functions to vectorial arguments. Similarly, the pure Lorentz transformations are compactly treated by generalizing a scalar rapidity to a vector quantity in spatial three-dimensional cases and extending hyperbolic functions to vectorial arguments. A calculation of the Wigner rotation simplified by using the extended functions illustrates the fact that the rapidity vector space obeys hyperbolic geometry. New representations bring a Lorentz-invariant fundamental equation of motion corresponding to the Galilei-invariant equation of Newtonian mechanics.     

A quasi-linear response theory of statistical heavy-ion collisions: (I). Basic formulation : Transport equations and roles of Green functions

We develop a time-dependent theory of heavy-ion collisions which consistently treats the relative and the intrinsic motions by coupled equations derived from the many-body von Neumann equation. The structure of the equations determining the mean trajectory and the fluctuations of the relative motion is the same as that of the corresponding equations in the known linear response theory. The present theory differs, however, from the linear response theory, in that it presumes neither weak coupling between the relative motion and the intrinsic excitations, nor the canonical distribution function for the density operator of the intrinsic motions. We apply the theory especially to deep inelastic collisions, where the relative motion couples to intrinsic excitations through a stochastic hamiltonian. Based on the stochastic assumption, we study the properties of the Green functions that take into account the higher order effects of the coupling hamiltonian. We then discuss, in particular, the effects of the Green functions on the time evolution of the intrinsic state, which is described in terms of a coarse-grained master equation, the friction tensor and fluctuation dissipation theorems.     

The Wigner distribution for classical systems

We present an explicit procedure for obtaining the equation of motion for the Wigner distribution when the underlying governing equation is a linear ordinary or partial differential equation. The cases of constant and variable coefficients are considered.     

The Wigner Functions for a Spin-1/2 Relativistic Particle in the Presence of Magnetic Field

This paper provides a study of Wigner functions for a spin-1/2 relativistic particle in the presence of magnetic field. Since the Dirac equation is described as a matrix equation, it is necessary to describe the Wigner function as a matrix function in phase space. What’s more, this function is then proved to satisfy the Dirac equation with ⋆-product. Finally, by solving the ⋆-product Dirac equation, the energy levels as well as the Wigner functions for a spin-1/2 relativistic particle in the presence of magnetic field are obtained.     

自旋为整数的Bargmann-Wigner方程的严格解

从自旋为任意整数的Bargmann-Wigner方程出发,导出了自旋为整数的场的易于求解的相对论性波动方程,在坐标表象中求解此方程,严格导出了自旋为整数的场的场函数.     

Wigner Functions for Non-Hamiltonian Systems on Noncommutative Space

We propose a modified form of Wigner functions for generic non-Hamiltonian systems on noncommutative space and prove that it satisfies the corresponding ＊-genvalue equation. In addition, as an example, we derive exact energy spectra and Wigner functions for a non-Hamiltonian toy model on the noncommutative space.     

Equal-time kinetic equations in a rotational field

We investigate quantum kinetic theory for a massive fermion system under a rotational field. From the Dirac equation in rotating frame we derive the complete set of kinetic equations for the spin components of the 8- and 7-dimensional Wigner functions. While the particles are no longer on a mass shell in the general case due to the rotation–spin coupling, there are always only two independent components, which can be taken as the number and spin densities. With help from the off-shell constraint we obtain the closed transport equations for the two independent components in the classical limit and at the quantum level. The classical rotation–orbital coupling controls the dynamical evolution of the number density, but the quantum rotation–spin coupling explicitly changes the spin density.     

Modified Form of Wigner Functions for Non-Hamiltonian Systems

Quantization of non-Hamiltonian systems （such as damped systems） often gives rise to complex spectra and corresponding resonant states, therefore a standard form calculating Wigner functions cannot lead to static quasiprobability distribution functions. We show that a modified form of the Wigner functions satisfies a ＊-genvalue equation and can be derived from deformation quantization for such systems.     

The quantum state of the universe from deformation quantization and classical-quantum correlation

In this paper we study the quantum cosmology of homogeneous and isotropic cosmology, via the Weyl–Wigner–Groenewold–Moyal formalism of phase space quantization, with perfect fluid as a matter source. The corresponding quantum cosmology is described by the Moyal–Wheeler-DeWitt equation which has exact solutions in Moyal phase space, resulting in Wigner quasiprobability distribution functions peaking around the classical paths for large values of scale factor. We show that the Wigner functions of these models are peaked around the non-singular universes with quantum modified density parameter of radiation.     

Spin-flip effects in a parallel-coupled double quantum dot molecule

We investigate theoretically the electronic transport through a parallel-coupled double quantum dot (DQD) molecule attached to metallic electrodes, in which the spin-flip scattering on each quantum dot is considered. Special attention is paid to the effects of the intradot spin-flip processes on the linear conductance by using the equation of motion approach for Green’s functions. When a weak spin-flip scattering on each quantum dot is present, the single Fano peak splits into two Fano peaks, and the Breit–Wigner resonance may be suppressed slightly. When the spin-flip scattering strength on each quantum dot becomes strong, the linear conductance spectrum consists of two Breit–Wigner peaks and two Fano peaks due to the quantum interference effects. The positions and shapes of these resonant peaks can be controlled by using the magnetic flux through the quantum device.     

自旋1/2非对易朗道问题的Wigner函数(英文)

Wigner函数在对量子体系状态的描述方面具有重要的意义。 讨论了自旋1/2非对易朗道问题的Wigner函数。首先回顾了对易空间中Wigner函数所服从的星本征方程, 然后给出了非对易相空间中自旋1/2朗道问题的Hamiltonian, 最后利用星本征方程（Moyal 方程）计算了非对易相空间中自旋1/2朗道问题具有矩阵表示形式的Wigner函数及其能级。With great significance in describing the state of quantum system, the Wigner function of the spin half non commutative Landau problem is studied in this paper. On the basis of the review of the Wigner function in the commutative space, which is subject to the *eigenvalue equation, Hamiltonian of the spin half Landau problem in the non commutative phase space is given. Then, energy levels and Wigner functions in the form of a matrix of the spin half Landau problem in the non commutative phase space are obtained by means of the _*-eigenvalue equation (or Moyal equation).     

Influence of a collision term on finite-size one-dimensional TDHF

We investigate numerically the effect of residual two-body collisions on one-dimensional TDHF results. A phenomenological collision term is added to the equation of motion for the Wigner transform of the one-body density. It contains a parameter τ which governs the relaxation towards the mean momentum. For finite-size slabs significant effects of the collision term on the Wigner function occur. In particular the system may fuse due to the action of the collision term.