On the description of center of mass motion in the Glauber model

The problem of center of mass motion of nucleus is discussed in the framework of the Glauber model, starting from a translation invariance condition. Theπ-⁴He coherent scattering amplitude is calculated to illustrate how predictions of the model are sensitive to ways of handling recoil corrections.     

Free Schrödinger equation analyzed in terms of the wave equation

The free time-dependent Schrödinger equation in three-dimensional space is analyzed as a special case of the wave equation in five-dimensional spacetime. This approach transforms the separation of variables in the parabolic Schrödinger equation into the separation of variables in a nonparabolic equation. Then one can solve the problem using the last theorem of V. N. Shapovalov (see Differents. Uravn., No. 10, 1864 (1980)) on the necessary and sufficient conditions for the complete separation of variables. Other advantages of this approach are also discussed.Translated from Izvestiya Vysshikh uchebnykh Zavedenii, Fizika, No. 7, pp. 59–64, July 1990.     

A self-consistent master equation and a new kind of cumulants

Generalized master equations correspond to different kinds of cumulants. Here, we discuss the recently introduced non-crossing cumulants from a physical point of view and propose the corresponding integro-differential master equation as a new type of equation for a self-consistent treatment of memory effects. We prove the cluster property of the non-crossing cumulants, and show that the Gaussian approximation of our equation is given by a random matrix process. As an instructive example for our expansion formula we treat the random frequency-modulated oscillator.     

A relation between cumulants of a thermodynamic variable and the state equation

A general relation between the cumulant of any order of a thermal variable and the thermal equilibrium state equation is pointed out. This relation has particular importance near the critical point of a phase transtion. This is discussed for standard examples.     

The radiation damping terms in Synge's equations of motion

Synge's [4] approximation method is developed to include the lowest-order radiation terms in the equations of motion. The method applies to systems which have been stationary at some finite time in the past, this restriction being necessary in order to ensure convergence of the integrals occurring in the higher orders of approximation. The results obtained are then compared with those of Chandrasekhar and Esposito [7] and of Anderson and Decanio [8].     

Solution of the Schrödinger equation in terms of classical paths

An expression in terms of classical paths is derived for the Laplace transform $\text{Ω(s)}$ of the Green function G of the Schrödinger equation with respect to $\text{1}\text{h}\text{?}$. For an analytic potential V(r), the function Ω is also analytic in the plane of the complex action variable s; its singularities lie at the values $\text{S}$ of the action along each possible (complex) classical path, including the paths which reflect from singularities of the potential. Accordingly, G may be written as a sum of terms, each of which is associated with such a classical path, and contains the factor $\text{exp}\text{(}\text{iS}\text{h}\text{?}\text{)}$. This expansion formally solves the problem of constructing waves out of the corresponding (complex) classical paths. A similar expression, in terms of closed paths, is derived for the density ? of eigenvalues of the Schrödinger equation. In situations when the eigenvalues are dense, ? is well approximated by the contributions of the shortest closed paths: while the path of vanishing length corresponds to the Thomas-Fermi approximation and its smooth corrections, the other paths yield contributions which oscillate and are damped as $\text{exp}\text{(}\text{iS}\text{h}\text{?}\text{)}$. This result also holds for nonanalytic potentials V(r). If the spectrum is continuous, closed classical paths yield oscillations in the scattering phase-shift. The analysis is also extended to multicomponent wave functions (describing, e.g., motion of particles with spin, or coupled channel scattering); along a classical path, the internal degree of freedom varies adiabatically, except through points at which it is not coupled to the potential, where it may undergo discrete changes.     

On the onset of chaotic motion in terms of generalized Langevin equations

New theoretical insights into the generalized Langevin approach to the onset of chaos are given. To make it possible to compare theory with “experiments” a fast computer algorithm is set up.     

含弛豫项的广义光学Bloch方程的近似解

本文用叠代法求得了含弛豫项的广义光学Bloch方程的近似解。与计算机给出的数值积分解的比较表明,一阶叠代解具有足够好的精度。由此得出了上能级占有几率随时间变化的解析表达式及多光子吸收、Bloch-Siegert频移等有用结果。     

Quantitative conditions for time evolution in terms of the von Neumann equation

The adiabatic theorem describes the time evolution of the pure state and gives an adiabatic approximate solution to the Schr ¨odinger equation by choosing a single eigenstate of the Hamiltonian as the initial state. In quantum systems, states are divided into pure states(unite vectors) and mixed states(density matrices, i.e., positive operators with trace one). Accordingly, mixed states have their own corresponding time evolution, which is described by the von Neumann equation. In this paper, we discuss the quantitative conditions for the time evolution of mixed states in terms of the von Neumann equation. First, we introduce the definitions for uniformly slowly evolving and δ-uniformly slowly evolving with respect to mixed states, then we present a necessary and sufficient condition for the Hamiltonian of the system to be uniformly slowly evolving and we obtain some upper bounds for the adiabatic approximate error. Lastly, we illustrate our results in an example.     

关于Young-Laplace公式的吉布斯自由能证明

对于为论证球形液滴附加压强的Young-Laplace公式而设计的一个理想实验,有文献试图借助吉布斯自由能函数进行证明,本文给出符合这一条件的证明方法.     

Effective action and the quantum equation of motion

We carefully analyze the use of the effective action in dynamical problems, in particular the conditions under which the equation can be used as a quantum equation of motion and illustrate in detail the crucial relation between the asymptotic states involved in the definition of and the initial state of the system. Also, by considering the quantum-mechanical example of a double-well potential, where we can get exact results for the time evolution of the system, we show that an approximation to the effective potential in the quantum equation of motion that correctly describes the dynamical evolution of the system is obtained with the help of the wilsonian RG equation (already at the lowest order of the derivative expansion), while the commonly used one-loop effective potential fails to reproduce the exact results.Received: 1 April 2004, Published online: 2 July 2004     

Quantum equations of motion and the Liouville equation

Equations of motion for explicitly time-dependent operators in the Heisenberg and Schrödinger pictures, respectively, are reviewed. A simple transformation reduces the related equation in the Heisenberg picture to closed form. An algorithm is introduced for the classical limit which, in either picture, returns the classical equation of motion for dynamical functions. Applications of this algorithm to the equation of motion for the density matrix reduces it to the classical Liouville equation. A property related to this algorithm is established for the commutator of any two analytic functions of finite polynomials of conjugate variables. Thus, if \(\hat F\) and ? are two such functions and \(\hat F^\prime \) and ?′ contain an arbitrary permutation of variables, then $$[\hat F,\hat G] = [\hat F^\prime ,\hat G^\prime ] + \hbar ^2 \hat{A}$$ where  is a remainder commutator. In the classical limit [ \(\hat F\) , ?] is invariant to such permutation.     

The Dirac equation in the de Broglie-Bohm theory of motion

We discuss the application of the de Broglie-Bohm theory of relativistic spin-1/2 particles to the Klein paradox andzitterbewegung.     

Bloch equations with diffusion terms for rotational motion in fluids

The phenomenological Bloch equations for magnetic resonance are supplemented by terms for rotational diffusion in fluids. The equations are used to describe damping of magnetization components and the line shape of nonresonance absorption in parallel fields for ions with an anisotropic g factor and an effective spin of 1/2. In both cases, the contributions of the diffusion terms are comparable to the contribution from longitudinal and transverse relaxation times (T₁, T₂).Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 1, pp. 42–46, January, 1973.The authors are indebted to B. M. Kozyrev for his interest in the work and discussions of the results.     

Separable solutions of the sine-Gordon equation in terms of a Painlevé transcendent

We show that, when expressed in characteristic coordinates, the two-dimensional sine-Gordon equation may be reduced to a separable form via the use of a fifth Painlevé transcendent.     

Fractional differential equations of motion in terms of combined Riemann—Liouville derivatives

In this paper,we focus on studying the fractional variational principle and the differential equations of motion for a fractional mechanical system.A combined Riemann-Liouville fractional derivative operator is defined,and a fractional Hamilton principle under this definition is established.The fractional Lagrange equations and the fractional Hamilton canonical equations are derived from the fractional Hamilton principle.A number of special cases are given,showing the universality of our conclusions.At the end of the paper,an example is given to illustrate the application of the results.     

Chaos in the perturbed Korteweg-de Vries equation with nonlinear terms of higher order

The dynamical behaviour of the generalized Korteweg-de Vries (KdV) equation under a periodic perturbation is investigated numerically. The bifurcation and chaos in the system are observed by applying bifurcation diagrams, phase portraits and Poincaré maps. To characterise the chaotic behaviour of this system, the spectra of the Lyapunov exponent and Lyapunov dimension of the attractor are also employed.     

Glauber dynamics of the Little-Hopfield model

The Glauber dynamics of the Little-Hopfield model is studied analytically with the help of path-integrals recently introduced by Sommers. The results of Amit, Gutfreund and Sompolinsky concerning the statics of the Little-Hopfield model near saturation are dynamically verified. Their replica-symmetric solution corresponds exactly to the self-consistency equations of the long-time limits of some time-dependent functions emerging in our dynamical treatment. We also found their generalized Almeida-Thouless line where the solution becomes unstable. The time-dependent evolution of the macroscopic overlaps for a finite number of stored patterns is investigated and the problems arising for an infinite number of them are indicated.Work performed within the research program of the Sonderforschungsbereich 125, Aachen-Jülich-Köln