Flow equations for band-matrices

Continuous unitary transformations can be used to diagonalize or approximately diagonalize a given Hamiltonian. In the last four years, this method has been applied to a variety of models of condensed matter physics and field theory. With a new generator for the continuous unitary transformation proposed in this paper one can avoid some of the problems of former applications. General properties of the new generator are derived. It turns out that the new generator is especially useful for Hamiltonians with a banded structure. Two examples, the Lipkin model, and the spin-boson model are discussed in detail. Received: 2 February 1998 / Accepted: 17 March 1998     

Correlation effects in the electronic structure of high Tc superconductors

We model Cu-O high T_c superconductors by a generalized Hubbard Hamiltonian. We obtain the ground state for finite linear chain systems exactly. This allows us to calculate physical quantities to show that: a) those systems can change from a highly ionic to a highly metallic state as holes are added. b) In the metallic state holes tend to form pairs in next nearest neighbour sites.     

Adiabatic switching approach to multidimensional supersymmetric quantum mechanics for several excited states

We present a time-dependent method for determining several approximate excited-state energies and wave functions using a vectorial approach to multidimensional supersymmetric quantum mechanics. First, a vectorial approach is used to generate the tensor sector two Hamiltonian, which is isospectral with the original scalar sector one Hamiltonian above the ground state of the sector one Hamiltonian. We construct a time-dependent Hamiltonian interpolating between the scalar sector one Hamiltonian and the tensor sector two Hamiltonian. Then, we can adiabatically switch from the ground state of the sector one Hamiltonian to the ground state of the sector two Hamiltonian by solving the time-dependent Schrödinger equation. In addition, by employing an initial wave packet orthogonal to that leading to the ground state of sector two, we also obtain the first-excited state of sector two. Construction of the orthogonal sector one states is trivial due to the tensor nature of sector two. The ground and first-excited states of the sector two Hamiltonian can be used with the charge operator to obtain the first two excited state wave functions of the sector one Hamiltonian. Excellent computational results are obtained for two-dimensional nonseparable degenerate and nondegenerate systems.     

Fresnel Operator for Deriving the Propagator of 2D Harmonic Oscillator with Cross Coupling

Using the identity of operator decomposition we obtain a normal ordered form of the time-evolution operator for cross coupling quantum harmonic oscillator Hamiltonian system in two dimensions, which is just a special two-mode Fresnel operator. The Feynman propagator for the Hamiltonian system is found by a direct calculation by means of the method deriving the matrix element of two-mode Fresnel operator in the entangled state representation. The technique of integration within an ordered product (IWOP) of operators is employed to derive the matrix elements of the operator in the coherent state and the entangled state representations.     

Quantum-mechanical oscillator in a uniform force field

The symmetry group of an isotropic oscillator explaining additional degeneracy of its energy levels was for the first time considered by Demkov [1–3]. He directed attention to the fact that harmonic oscillator Hamiltonian symmetry was beyond the scope of usual geometric concepts and could be expressed using canonical transformations relating coordinates and momenta. We show in this paper how the symmetry properties of the harmonic oscillator Hamiltonian can be used to solve the dynamic problem of an oscillator in a uniform force field.     

Lie symmetries and non-Noether conserved quantities for Hamiltonian canonical equations

This paper focuses on studying Lie symmetries and non-Noether conserved quantities of Hamiltonian dynamical systems in phase space. Based on the infinitesimal transformations with respect to the generalized coordinates and generalized momenta, we obtain the determining equations and structure equation of the Lie symmetry for Hamiltonian dynamical systems. This work extends the research of non-Noether conserved quantity for Hamilton canonical equations, and leads directly to a new type of non-Noether conserved quantities of the systems. Finally, an example is given to illustrate these results.     

Ground State Properties for Bose-Condensed Gas in Anisotropic Traps

Based on the energy functional and variational method, we present a new method to investigate the ground state properties for a weakly interacting Bose-condensed gas in an anisotropic harmonic trap at zero temperature. With this method we are able to find the analytic expression of the ground-state wavefunction and to explore the relevant quantities, such as energy, chemical potential, and the aspect ratio of the velocity distribution. These results agree well with previous ground state numerical solutions of the Gross-Pitaevskii equation given by Dalfovo et al. [Phys. Rev. A 53 （1996） 2477] This new method is simple compared to other methods used to solve numerically the Gross-Pitaevskii equation, and one can obtain analytic and reliable results.     

Covariant formulation of non-equilibrium statistical thermodynamics

The Fokker Planck equation is considered as the master equation of macroscopic fluctuation theories. The transformation properties of this equation and quantities related to it under general coordinate transformations in phase space are studied. It is argued that all relations expressing physical properties should be manifestly covariant, i.e. independent of the special system of coordinates used. The covariance of the Langevin-equations and the Fokker Planck equation is demonstrated. The diffusion matrix of the Fokker Planck equation is used as a contravariant metric tensor in phase space. Covariant drift vectors associated to the Langevin- and the Fokker Planck equation are found. It is shown that special coordinates exist in which the covariant drift vector of the Fokker Planck equation and the usual non-covariant drift vector are equal.The physical property of detailed balance and the equivalent potential conditions are given in covariant form. Finally, the covariant formulation is used to study how macroscopic forces couple to a system in a non-equilibrium steady state. A general fluctuation-dissipation theorem for the linear response to such forces is obtained.     

向列型液晶中短程序的计算

向列型液晶中的短程序有重要的物理效应。本文在格胞理论的基础上,用新的数值方法计算这些效应。不用任何函数展开,通过平衡态方程送代求解,得到精确的取向分布函数。从而计算各种物理量,给出了相变点序参数、熵变等量的精确数值。     

QUANTUM BROWNIAN MOTION AND NUCLEAR FISSION

Based on a simplified Hamiltonian model the transition from quantum to classical diffusion behaviours for a system has been shown.The approximation of a locally harmonic oscillator and reduced density operator in the harmonic oscillator are generalized to calculate the escape tate over the barrier for the system in a fission potential consisting of a ground state well and barrier.     

A set of Lie symmetrical non－Noether conserved quantity for the relativistic Hamiltonian systems

For the relativistic Hamiltonian system, a new type of Lie symmetrical non-Noether conserved quantities are given. On the basis of the theory of invariance of differential equations under infinitesimal transformations, and introducing special infinitesimal transformations for q_s and p_s, we construct the determining equations of Lie symmetrical transformations of the system, which only depend on the canonical variables. A set of non-Noether conserved quantities are directly obtained from the Lie symmetries of the system. An example is given to illustrate the application of the results.     

Basic Theory of Fractional Conformal Invariance of Mei Symmetry and its Applications to Physics

In this paper, we present a basic theory of fractional dynamics, i.e., the fractional conformal invariance of Mei symmetry, and find a new kind of conserved quantity led by fractional conformal invariance. For a dynamical system that can be transformed into fractional generalized Hamiltonian representation, we introduce a more general kind of single-parameter fractional infinitesimal transformation of Lie group, the definition and determining equation of fractional conformal invariance are given. And then, we reveal the fractional conformal invariance of Mei symmetry, and the necessary and sufficient condition whether the fractional conformal invariance would be the fractional Mei symmetry is found. In particular, we present the basic theory of fractional conformal invariance of Mei symmetry and it is found that, using the new approach, we can find a new kind of conserved quantity; as a special case, we find that an autonomous fractional generalized Hamiltonian system possesses more conserved quantities. Also, as the new method’s applications, we, respectively, find the conserved quantities of a fractional general relativistic Buchduhl model and a fractional Duffing oscillator led by fractional conformal invariance of Mei symmetry.     

Density matrix for an electron confined in quantum dots under uniform magnetic field and static electrical field

Using unitary transformations, this paper obtains the eigenvalues and the common eigenvector of Hamiltonian and a new-defined generalized angular momentum (L_z) for an electron confined in quantum dots under a uniform magnetic field (UMF) and a static electric field (SEF). It finds that the eigenvalue of L_z just stands for the expectation value of a usual angular momentum l_z in the eigen-state. It first obtains the matrix density for this system via directly calculating a transfer matrix element of operator \exp( -\beta H) in some representations with the technique of integral within an ordered products (IWOP) of operators, rather than via solving a Bloch equation. Because the quadratic homogeneity of potential energy is broken due to the existence of SEF, the virial theorem in statistical physics is not satisfactory for this system, which is confirmed through the calculation of thermal averages of physical quantities.     

A Set of Lie Symmetrical Conservation Law for Rotational Relativistic Hamiltonian Systems

For the rotational relativistic Hamiltonian system, a new type of the Lie symmetries and conserved quantities are given. On the basis of the theory of invariance of differential equations under infinitesimal transformations, and introducing infinitesimal transformations for generalized coordinates qs and generalized momentums ps, the determining equations of Lie symmetrical transformations of the system are constructed, which only depend on the canonical variables.A set of non-Noether conserved quantities are directly obtained from the Lie symmetries of the system. An example is given to illustrate the application of the results.     

非对易相空间中谐振子体系热力学性质的探讨

2000年以来, 有关非对易空间的各种物理问题一直是研究的热点, 并在量子力学、场论、凝聚态物理、天体物理等各领域中已被广泛地探讨. 采用统计物理方法讨论非对易效应对谐振子体系热力学性质的影响. 先以对易相空间中确定二维和三维谐振子的配分函数求出谐振子体系的热力学函数; 非对易相空间中的坐标和动量通过坐标-坐标和动量-动量之间的线性变换而以对易相空间中的坐标和动量来表示; 最终以非对易相空间中求出配分函数来讨论非对易效应对谐振子体系热力学性质的影响. 结果显示, 在非对易相空间中谐振子体系的配分函数和熵表达式均包含因非对易引起的修正项. 从分析结果得出如下结论: 非对易效应对谐振子的配分函数和熵函数等微观状态函数有一定的影响, 但对谐振子体系的内能、热容量等宏观热力学函数没有影响. 研究结果只是对应于满足玻尔兹曼统计的经典体系, 对于满足费米-狄拉克和玻色-爱因斯坦统计的量子体系需进一步推广研究.     

Ground-state phase diagram of the 1-D Holstein–Hubbard model

The Holstein–Hubbard model is investigated in one-dimension at half filling employing a series of unitary transformations taking into account the coherence and correlation of phonons. To treat the phonon subsystem more accurately a new squeezing transformation is introduced to incorporate the electron-density-dependent onsite phonon correlations to lower the energy further. The effective electronic Hamiltonian is next obtained by averaging the transformed Hamiltonian with respect to the zero-phonon state and the resulting effective electronic Hamiltonian is solved exactly using the method of Bethe ansatz. Finally the ground state is obtained by minimizing the energy with respect to all the variational parameters. The present method gives better results for the ground state energy of the system and also suggests the existence of a wider intermediate metallic phase at the charge-density-wave–spin-density-wave crossover region, which was first predicted by Takada and Chatterjee and later supported by Krishna and Chatterjee.     

Hamiltonian approach to nonlinear oscillators

A Hamiltonian approach to nonlinear oscillators is suggested. A conservative oscillator always admits a Hamiltonian invariant, H, which keeps unchanged during oscillation. This property is used to obtain approximate frequency-amplitude relationship of a nonlinear oscillator with acceptable accuracy. Two illustrating examples are given to elucidate the solution procedure.     

A simple generation of exactly solvable anharmonic oscillators

An elementary finite difference algorithm shortens the Darboux method, permitting an easy generation of families of anharmonic potentials almost isospectral to the harmonic oscillator. Against common belief, it is possible to associate a SUSY partner to a given Hamiltonian H using a factorization energy greater than the ground state energy of H. The explicit 3-SUSY partners of the oscillator potential are found and discussed.     

Canonical structure of soliton equations. I

The canonical structure of the nonlinear evolution equations in 1 + 1 dimensions solvable in terms of an N × N inverse scattering problem is discussed. The simplest form of the scattering problems, that is those containing the spectral parameter linearly, is considered. It applies to most of the known soliton equations, like the Korteweg-de Vries eq., the sine-Gordon eq. and the Boussinesq eq. Discussion of various possible reductions of the number of dependent variables by imposing constraints consistent with the Hamiltonian flows is given together with the canonical structure of the reduced systems. A direct proof of the involutive character of the infinite number of conserved quantities is given for the general case as well as the reduced case. The relation between the conserved quantities and symmetry transformations (Lie-Bäcklund transformations) becomes very simple in this framework.     

Role of surface integrals in the Hamiltonian formulation of general relativity

It is shown that if the phase space of general relativity is defined so as to contain the trajectories representing solutions of the equations of motion then, for asymptotically flat spaces, the Hamiltonian does not vanish but its value is given rather by a nonzero surface integral. If the deformations of the surface on which the state is defined are restricted so that the surface moves asymptotically parallel to itself in the time direction, then the surface integral gives directly the energy of the system, prior to fixing the coordinates or solving the constraints. Under more general conditions (when asymptotic Poincaré transformations are allowed) the surface integrals giving the total momentum and angular momentum also contribute to the Hamiltonian. These quantities are also identified without reference to a particular fixation of the coordinates. When coordinate conditions are imposed the associated reduced Hamiltonian is unambiguously obtained by introducing the solutions of the constraints into the surface integral giving the numerical value of the unreduced Hamiltonian. In the present treatment there are therefore no divergences that cease to be divergences after coordinate conditions are imposed. The procedure of reduction of the Hamiltonian is explicity carried out for two cases: (a) Maximal slicing, (b) ADM coordinate conditions.A Hamiltonian formalism which is manifestly covariant under Poincaré transformations at infinity is presented. In such a formalism the ten independent variables describing the asymptotic location of the surface are introduced, together with corresponding conjugate momenta, as new canonical variables in the same footing with the g_ij, π^ij. In this context one may fix the coordinates in the “interior” but still leave open the possibility of making asymptotic Poincaré transformations. In that case all ten generators of the Poincaré group are obtained by inserting the solution of the constraints into corresponding surface integrals.