Solving the Schrodinger Equation for the Feynman Quantum Computer

Feynman showed that a closed, locallyinteracting quantum system is capable of performingdeterministic computation. For a finite-size version ofsuch a computer, the Schrodinger equation is solvedanalytically. The probability that the computer yields afinal result upon measurement is derived.     

Quantum Invariants for Solving the Time-Dependent Schrödinger Equation in One Dimension

Extending the work of Lewis and Leach on classical invariants for solving the classical equation of motion in one-dimensional system, the quantum invariants in polynomial form of momentum are obtained. The involved Hamiltonian is time-dependent and quadratic in momentum.     

Quantum Statistics for General Quadratic System

By the generalized linear quantum transformation theory, we concisely derive the analytic expressions of partition function for general quadratic systems of multi-mode boson and fermion Fock space without any information for the energy spectrum. Under a general condition, we firstly acquire the exact expressions of energy spectra for these systems.     

Regarding Nonstationary Quadratic Quantum Systems

Russian Physics Journal - With the help of the evolution operator method, we have established unitary connection between quadratic systems, namely between a free particle with variable mass M(t) ,...     

Quantum Liouville Equation

The symmetrized product of quantum observables is defined. It is seen as consisting of ordinary multiplication followed by application of the superoperator that orders the operators of coordinate and momentum. This superoperator is defined in the way that allows obstruction free quantization of algebra of observables as well as introduction of operator version of the Poisson bracket. It is shown that this bracket has all properties of the Lie bracket and that it can substitute the commutator in the von Neumann equation leading to quantum Liouville equation.     

An Approach for Solving Short-Wave Models for Camassa-Holm Equation and Degasperis-Procesi Equation

In this paper, to construct exact solution of nonlinear partial differential equation, an easy-to-use approach is proposed. By means of the transformation of the independent variables and the travelling wave transformation, the partial differential equation is reduced to an ordinary differential equation. To solve the ordinary differential equation, we assume the soliton solution in the explicit expression and obtain the travelling wave solution. By the transformation back to the original independent variables, the soliton solution of the original partial differential equation is derived. We investigate the short wave model for the Camassa-Holm equation and the Degasperis-Procesi equation respectively. One-cusp soliton solution of the Camassa-Flolm equation is obtained. One-loop soliton solution of the Degasperis- Procesi equation is also obtained, the approximation of which in a closed form can be obtained firstly by the Adomian decomposition method. The obtained results in a parametric form coincide perfectly with those given in the present reference. This illustrates the efficiency and reliability of our approach.     

An Approach for Solving Short-Wave Models for Camassa-Holm Equation and Degasperis-Procesi Equation

In this paper, to construct exact solution of nonlinear partial differential equation, an easy-to-use approach is proposed. By means of the transformation of the independent variables and the travelling wave transformation, the partial differential equation is reduced to an ordinary differential equation. To solve the ordinary differential equation, we assume the soliton solution in the explicit expression and obtain the travelling wave solution. By the transformation back to the original independent variables, the soliton solution of the original partial differential equation is derived. We investigate the short wave model for the Camassa-Holm equation and the Degasperis-Procesi equation respectively. One-cusp soliton solution of the Camassa-Holm equation is obtained. One-loop soliton solution of the Degasperis-Procesi equation is also obtained, the approximation of which in a closed form can be obtained firstly by the Adomian decomposition method. The obtained results in a parametric form coincide perfectly with those given in the present reference. This illustrates the efficiency and reliability of our approach.     

量子噪音与量子Langevin方程

介绍了量子噪音和量子Langevin方程 ,并与经典噪音和经典Langevin方程进行了比较 .量子噪音来源于两种途径 ,第一种与经典噪音相似 ,第二种则起源于Heisengberg测不准原理 .同时 ,也简略给出了量子Langevin方程的推导.The properties of quantum noise and Langevin equation are discussed. Comparisons between the quantum noise and Langevin eqution and the classic one are presented. A brief derivation for quantum Langevin equation is showed. The quantum noise comes from two ways, namely, the way as same as that of classic noise and the Heisenberg uncertainty.     

Invariant Quantum States of Quadratic Hamiltonians

The problem of finding covariance matrices that remain constant in time for arbitrary multi-dimensional quadratic Hamiltonians (including those with time-dependent coefficients) is considered. General solutions are obtained.     

Nonlinear Quantum Dynamical Equation for Self-Acting Electron

Abstract

From the action principle, the quantum dynamical equation is obtained both relativistically and gauge invariant, which is analogous to the Dirac equation and describes behaviour of an arbitrary number of self-acting charged particles. It is noted that solutions of this equation are indicative of the soliton nature of an electron and allow to determine the internal energy, dimensions and geometric shape of the electron in different quantum states. The theory proposed represents a synthesis of the standard QED and ideas of the self-organization theory of physical systems.     

Quantum Lattice-Gas Model for the Burgers Equation

A quantum algorithm is presented for modeling the time evolution of a continuous field governed by the nonlinear Burgers equation in one spatial dimension. It is a microscopic-scale algorithm for a type-II quantum computer, a large lattice of small quantum computers interconnected in nearest neighbor fashion by classical communication channels. A formula for quantum state preparation is presented. The unitary evolution is governed by a conservative quantum gate applied to each node of the lattice independently. Following each quantum gate operation, ensemble measurements over independent microscopic realizations are made resulting in a finite-difference Boltzmann equation at the mesoscopic scale. The measured values are then used to re-prepare the quantum state and one time step is completed. The procedure of state preparation, quantum gate application, and ensemble measurement is continued ad infinitum. The Burgers equation is derived as an effective field theory governing the behavior of the quantum computer at its macroscopic scale where both the lattice cell size and the time step interval become infinitesimal. A numerical simulation of shock formation is carried out and agrees with the exact analytical solution.     

Quantum Lattice KdV Equation

A quantum theory is developed for a difference-difference system which can serve as a toy model of the quantum Korteweg--de Vries equation.     

Separating Solution of a Quadratic Recurrent Equation

In this paper we consider the recurrent equation

$\Lambda_{p+1}=\frac{1}{p}\sum_{q=1}^pf\bigg(\frac{q}{p+1}\bigg)\Lambda _{q}\Lambda_{p+1-q}$

for p≥1 with f∈C[0,1] and Λ₁=y>0 given. We give conditions on f that guarantee the existence of y ⁽⁰⁾ such that the sequence Λ _p with Λ₁=y ⁽⁰⁾ tends to a finite positive limit as p→∞.

     

Two-Dimensional Legendre Wavelets for Solving Time-Fractional Telegraph Equation

In this paper, we develop an accurate and efficient Legendre wavelets method for numerical solution of the well known time-fractional telegraph equation. In the proposed method we have employed both of the operational matrices of fractional integration and differentiation to get numerical solution of the time-telegraph equation. The power of this manageable method is confirmed. Moreover, the use of Legendre wavelet is found to be accurate, simple and fast.     

Solutions of the Schrodinger Equation with Quantum Mechanical Gravitational Potential Plus Harmonic Oscillator Potential

The solutions of the Schrodinger equation with quantum mechanical gravitational potential plus harmonic oscillator potential have been presented using the parametric Nikiforov-Uvarov method. The bound state energy eigen values and the corresponding un-normalized eigen functions are obtained in terms of Laguerre polynomials. Also a special case of the potential has been considered and its energy eigen values are obtained.     

Scaling Laws for the Multidimensional Burgers Equation with Quadratic External Potential

The reordering of the multidimensional exponential quadratic operator in coordinate-momentum space (see X. Wang, C.H. Oh and L.C. Kwek (1998). J. Phys. A.: Math. Gen. 31:4329–4336) is applied to derive an explicit formulation of the solution to the multidimensional heat equation with quadratic external potential and random initial conditions. The solution to the multidimensional Burgers equation with quadratic external potential under Gaussian strongly dependent scenarios is also obtained via the Hopf-Cole transformation. The limiting distributions of scaling solutions to the multidimensional heat and Burgers equations with quadratic external potential are then obtained under such scenarios. AMS Subject Classifications: 60G60, 60G15, 62M15, 60H15     

Quantum Fluctuation Relations for the Lindblad Master Equation

An open quantum system interacting with its environment can be modeled under suitable assumptions as a Markov process, described by a Lindblad master equation. In this work, we derive a general set of fluctuation relations for systems governed by a Lindblad equation. These identities provide quantum versions of Jarzynski-Hatano-Sasa and Crooks relations. In the linear response regime, these fluctuation relations yield a fluctuation-dissipation theorem (FDT) valid for a stationary state arbitrarily far from equilibrium. For a closed system, this FDT reduces to the celebrated Callen-Welton-Kubo formula.     

A Collocation Method for Solving Fractional Riccati Differential Equation

In this article, we have introduced a Taylor collocation method, which is based on collocation method for solving fractional Riccati differential equation. The fractional derivatives are described in the Caputo sense. This method is based on first taking the truncated Taylor expansions of the solution function in the fractional Riccati differential equation and then substituting their matrix forms into the equation. Using collocation points, the systems of nonlinear algebraic equation are derived. We further solve the system of nonlinear algebraic equation using Maple 13 and thus obtain the coefficients of the generalized Taylor expansion. Illustrative examples are presented to demonstrate the effectiveness of the proposed method.     

求解对流扩散方程的紧致修正方法

提出了求解对流扩散方程的紧致修正方法,该方法是在低阶离散格式的源项中,引入紧致修正项,从而构造高阶紧致修正格式,并进行求解.采用紧致修正方法对典型的对流扩散方程进行计算.结果表明,紧致修正方法虽然与二阶经典差分方法建立在相同的结点数上,但紧致修正方法的精度与紧致方法的精度相同,均具有四阶精度.所以紧致修正方法可以在少网...     

On Kaup and Newell＇s Method for Solving DNLS Equation

Kaup and Newell＇s revised inverse scattering transform for the derivative nonlinear Schrodinger （DNLS） equation is investigated. We compared it with a more reasonable approach proposed recently, which is rigorously proven by the Liouville theorem. It is conduded that Kanp and Newell＇s revision is only suitable for giving single-soliton solution and can not be generalized to multi-soliton case.