Theory of irreversible processes in nonequilibrium quantum systems. I. General formalism

The theory of Van Hove for nonequilibrium quantum statistical mechanics is extensively reformulated in terms of a superspace (a kind of operator space). This reformulation enables us to introduce a diagrammatic method which makes it convenient to deal with practical problems in physical systems. In our formalism, quantum statistical effects are considered on the basis of a systematic rule for the contraction technique. A complicated statistical effect in boson or fermion systems can be treated by starting with a simple unsymmetrized formalism in the Boltzmann statistics.     

Spin density polarization effects in the presence of Coriolis force on ion acoustic waves in quantum plasma

Ion acoustic solitary waves in a quantum plasma, which is slowly rotating around an axis at an angle θ with the direction of magnetic field, are investigated. Quantum hydrodynamic model is under consideration with the effects of rotations which are included via Coriolis force. Fermions are degenerate and have different spin density states, that is, up and down characterized via parameter α. Linear analysis is performed by applying Fourier transformation to derive dispersion relation. For nonlinear analysis, we apply reductive perturbation method to derive Korteweg de Vries equation (KdV). The effects of variations of Coriolis force, spin polarization, and quantum parameter on characteristics of solitary structure are discussed. These results are applicable to astrophysical and laboratory plasmas.     

Quantenstatistik des Hochtemperatur-Plasmas im thermodynamischen Gleichgewicht

High temperature plasmas are investigated on the basis of quantum theory. A new method for the calculation of the thermodynamic properties is developed. According to the method of Morita effective potentials are introduced. They permit the evaluation of the partition function with the well-known formalism of classical statistical mechanics. The first corrections added to Debye's limiting law of the free energy are expressed by the two particle Slater sum S₂(r). A Bloch equation for S₂(r) is derived and is solved in the Fourier representation by a development according to the interaction parameter ζ = e²/kTλ, λ being the thermal wavelength. The effects of symmetry are taken into account. The free energy is calculated explicitely up to the order of ζ². In the case of small concentrations our results agrees with that derived by Trubnikow and Elesin. Effects of symmetry neglected, up to the order of ζ² the formula of DeWitt is obtained.     

傅里叶变换法求解两类简单的薛定谔方程

自由粒子和一维无限深势阱的薛定谔方程的求解是量子力学较为基础的内容.本文采用傅里叶变换对这两类简单的薛定谔方程进行了求解讨论.通过偏微分方程的傅里叶变换解法和偏微分方程作分离变数成常微分方程后的傅里叶变换解法的深入讨论,均得到与有关教材一致的结果,并讨论了这两种方法之间的差别和联系.     

键长-键角和Radau坐标下哈密顿算符在傅里叶基组表象下的厄米性

在量子动力学计算中,有时候为了规避奇点问题或者节省计算量,我们经常需要对哈密顿量进行变换. 然而,在使用傅里叶基矢计算时,哈密顿量的变换形式容易导致哈密顿矩阵失去厄米性,进而有些情况下使数值计算变得不稳定. 本文主要讨论构建具有厄米性的哈密顿算符的方法. 以三原子分子为例,构建了键长—键角和Radau坐标下描述分子运动的各种形式的哈密顿量. 基于这些哈密顿量,采用含时波包方法计算了OClO分子的吸收光谱,讨论了非厄米性矩阵对计算结果的影响. 本文所得到的结论对基于基函数展开的量子动力学计算都是适用的.     

Quantum ion-acoustic solitary waves in weak relativistic plasma

Small amplitude quantum ion-acoustic solitary waves are studied in an unmagnetized two- species relativistic quantum plasma system, comprised of electrons and ions. The one-dimensional quantum hydrodynamic model (QHD) is used to obtain a deformed Korteweg–de Vries (dKdV) equation by reductive perturbation method. A linear dispersion relation is also obtained taking into account the relativistic effect. The properties of quantum ion-acoustic solitary waves, obtained from the deformed KdV equation, are studied taking into account the quantum mechanical effects in the weak relativistic limit. It is found that relativistic effects significantly modify the properties of quantum ion-acoustic waves. Also the effect of the quantum parameter H on the nature of solitary wave solutions is studied in some detail.     

Selection rules,causality, and unitarity in statistical and quantum physics

The integrodifferential equations satisfied by the statistical frequency functions for physical systems undergoing stochastic transitions are derived by application of a causality principle and selection rules to the Markov chain equations. The result equations can be viewed as generalizations of the diffusion equation, but, unlike the latter, they have a direct bearing onactive transport problems in biophysics andcondensation aggregation problems of astrophysics and phase transition theory. Simple specific examples of the effects of severe selection rules, such as the relaxational Boltzmann transport equation and the diffusion equation, are also given. Finally, partial differential equations for the probability amplitudes of quantum mechanics are derived, usingunitarity instead of causality, and a selection rule is applied directly to obtain ageneralization of the Dirac equation in which infinite transitions between states arenot allowed.     

各向异性海底地层海洋可控源电磁响应三维积分方程法数值模拟

采用积分方程法对各向异性海底地层海洋可控源电磁（MCSEM）响应进行三维数值模拟.首先利用算子理论给出各向异性海底地层中的压缩积分方程,由于满足压缩映射条件,该积分方程在任意参数下总是迭代收敛的.然后为了提高计算效率,引入分区域多重网格准线性近似技术.通过具体算例验证了所述算法在计算精度与效率方面的有效性.最后利用该算法考察并分析了海底地层的各向异性对MCSEM三维响应特征的影响. 关键词： 海洋可控源电磁法 各向异性 三维模拟 积分方程法     

Nonequilibrium statistical mechanics of mixed quantum classical ensembles: application to noncontact atomic force microscopy

Using the nonequilibrium statistical operator method, we suggest a new general method of treating dynamics of a combined system consisting of interacting classical and quantum parts. The method is illustrated on the tip dynamics in the noncontact atomic force microscopy (NC-AFM) where a macroscopic tip interacts with a quantum microscopic system (the surface and the nanotip). The derived general equation of motion for the tip and the Fokker-Planck equation, applicable even at low temperatures, contain memory effects and a friction term which should (at least partially) be responsible for the observed energy dissipation in NC-AFM experiments.     

Pauli's electron as a dynamic system

A dynamic systemS _P described by the Pauli equation for nonrelativistic electron is investigated merely as a distributed dynamic system. No quantum principles are used. This system is shown to be a statistical ensemble of nonrelativistic stochastic pointlike particles. The electron spin is shown to have a classical analog which is a collective (statistical) property of the ensemble (not a property of a single electron). The magnetic moment of the electron is a quantum property which has no classical analog. The magnetic moment is parallel to the spin only in the stationary state. In the arbitrary state the magnetic moment is not connected with the spin direction.     

Renormalization-Group Method for Reduction of Evolution Equations; Invariant Manifolds and Envelopes

The renormalization group (RG) method as a powerful tool for reduction of evolution equations is formulated in terms of the notion of invariant manifolds. We start with derivation of an exact RG equation which is analogous to the Wilsonian RG equations in statistical physics and quantum field theory. It is clarified that the perturbative RG method constructs invariant manifolds successively as the initial value of evolution equations, thereby the meaning to set t₀=t is naturally understood where t₀ is the arbitrary initial time. We show that the integral constants in the unperturbative solution constitutes natural coordinates of the invariant manifold when the linear operator A in the evolution equation is semi-simple, i.e., diagonalizable; when A is not semi-simple and has a Jordan cell, a slight modification is necessary because the dimension of the invariant manifold is increased by the perturbation. The RG equation determines the slow motion of the would-be integral constants in the unperturbative solution on the invariant manifold. We present the mechanical procedure to construct the perturbative solutions hence the initial values with which the RG equation gives meaningful results. The underlying structure of the reduction by the RG method as formulated in the present work turns out to completely fit to the universal one elucidated by Kuramoto some years ago. We indicate that the reduction procedure of evolution equations has a good correspondence with the renormalization procedure in quantum field theory; the counter part of the universal structure of reduction elucidated by Kuramoto may be Polchinski's theorem for renormalizable field theories. We apply the method to interface dynamics such as kink–anti-kink and soliton–soliton interactions in the latter of which a linear operator having a Jordan-cell structure appears.     

Measurement in Bohm's versus Everett's quantum theory

The interpretations of measurements in Bohm's and Everett's quantum theories are compared. Since both theories are based on the assumption of a universally valid Schrödinger equation, they face the common problem of how to explain that arrow of time, which in conventional quantum theory is represented by the collapse of the wave function. Its solution requires, in a statistical sense, a very improbable initial condition for thetotal wave function of the universe. The historical importance of Bohm's quantum theory is pointed out.     

Quantenstatistische Zustandsgleichung von stark ionisierten Wasserstoffplasmen mittlerer Dichte

A quantum statistical formula for the equation of state of fully ionized plasmas in the mean temperature and mean density region is given. The complete second virial coefficient for the case of Debye screening is taken into account. The binary Slater sums are represented by interpolation between the analytical expressions for small and for large radii respectively. Using the interpolation formulae the equation of state is calculated numerically for the temperatures T = 20000 K and T = 80000 K. The resulting curves are in good agreement with the analytical equation of state for very small densities.     

On probability theory and probabilistic physics—Axiomatics and methodology

A new formulation involving fulfillment of all the Kolmogorov axioms is suggested for acomplete probability theory. This proves to be not a purely mathematical discipline. Probability theory deals with abstract objects—images of various classes of concrete objects—whereas experimental statistics deals with concrete objects alone. Both have to be taken into account. Quantum physics and classical statistical physics prove to be different aspects ofone probabilistic physics. The connection of quantum mechanics with classical statistical mechanics is examined and the origin of the Schrödinger equation is elucidated. Attention is given to the true meaning of the wave-corpuscle duality, and the incompleteness of nonrelativistic quantum mechanics is explained.     

Statistical reconstruction of the quantum states of three-level optical systems

The procedure of measurement followed by the reconstruction of the quantum state of a three-level optical system is implemented for a frequency-and spatially degenerate two-photon field. The method of statistical estimation of the quantum state from a solution to the likelihood equation and the analysis of the statistical properties of the obtained estimators is developed. Using the root method of estimating quantum states, the initial two-photon (qutrit) wave function is reconstructed from the measured fourth-order field moments.     

Quantum Boltzmann equation solved by Monte Carlo method for nano-scale semiconductor devices simulation

A two-dimensional (2D) full band self-consistent ensemble Monte Carlo (MC) method for solving the quantum Boltzmann equation, including collision broadening and quantum potential corrections, is developed to extend the MC method to the study of nano-scale semiconductor devices with obvious quantum mechanical (QM) effects. The quantum effects both in real space and momentum space in nano-scale semiconductor devices can be simulated. The effective mobility in the inversion layer of n and p channel MOSFET is simulated and compared with experimental data to verify this method. With this method 50nm ultra thin body silicon on insulator MOSFET are simulated. Results indicate that this method can be used to simulate the 2D QM effects in semiconductor devices including tunnelling effect.     

Mean-field theory and statistical treatment of residual interactions

The time-dependent Hartree-Fock theory is generalized in order to include two-body collisions. Using a method similar to the projection method of quantum statistics an exact and closed equation is derived for the one-particle density matrix which takes into account both the mean field and the two-body residual interactions. In the limit of long mean free path of the particles, the exact equation for the one-particle density matrix is simplified by introducing some general statistical properties of the residual interactions. An equation is obtained for the expectation values of operators which includes both the mean-field contributions and the correlations due to the residual interactions. The effects of the correlations on the fluctuations of collective variables are discussed. It is shown that the statistical approximations introduced do not destroy the energy and the momentum conservations.     

Calculation of norms of Bethe wave functions

A class of two dimensional completely integrable models of statistical mechanics and quantum field theory is considered. Eigenfunctions of the Hamiltonians are known for these models. Norms of these eigenfunctions in the finite box are calculated in the present paper. These models include in particular the quantum nonlinear Schrödinger equation and the HeisenbergXXZ model.     

Densities of states and the complex partition function

A Fourier transform technique based on the method of Dubner and Abate is here applied to the energy-level density problem for quantum partition functions. It is shown that, for typical molecular systems at realistic energies, the integrated level count, W(E), is obtainable to near ‘spectroscopic’ accuracy and in reasonable computing time. A smooth Fourier resolution of the simple density function, g(E), is likewise possible. Though somewhat less economical, this procedure entirely removes the slight discrepancies found in the steepest-descent method and can reasonably be claimed to dispose once and for all of the problem of quantum level-density computations in molecular rate theory.     

PT-Symmetric Quantum Field Theories and the Langevin Equation

Many non-Hermitian but PT-symmetric theories are known to have a real, positive spectrum, and for quantum-mechanical versions of these theories, there exists a consistent probabilistic interpretation. Since the action is complex for these theories, Monte Carlo methods do not apply. In this paper a field-theoretic method for numerical simulations of PT-symmetric Hamiltonians is presented. The method is the complex Langevin equation, which has been used previously to study complex Hamiltonians in statistical physics and in Minkowski space. We compute the equal-time one-point and two-point Green's functions in zero and one dimension, where comparisons to known results can be made. The method should also be applicable in four-dimensional space-time. This approach may grant insight into the formulation of a probabilistic interpretation for path integrals in PT-symmetric quantum field theories.