SU(1,1) Coherent States for the Generalized Two-Mode Time-Dependent Quadratic Hamiltonian System

The SU(1,1) coherent states, so-called Barut-Girardello coherent state and Perelomov coherent state, for the generalized two-mode time-dependent quadratic Hamiltonian system are investigated through SU(1,1) Lie algebraic formulation. Two-mode Schrödinger cat states defined as an eigenstate of $\hat{K}_{-}^{2}The SU(1,1) coherent states, so-called Barut-Girardello coherent state and Perelomov coherent state, for the generalized two-mode time-dependent quadratic Hamiltonian system are investigated through SU(1,1) Lie algebraic formulation. Two-mode Schr?dinger cat states defined as an eigenstate of are also studied. We applied our development to two-mode Caldirola-Kanai oscillator which is a typical example of the time-dependent quadratic Hamiltonian system. The time evolution of the quadrature distribution for the probability density in the coherent states are analyzed for the two-mode Caldirola-Kanai oscillator by plotting relevant figures.     

Time Evolution Caused by Hamiltonian Composed of Quadratic Combination of Canonical Operators and Time-Dependent Two-Mode Fresnel Operator

We show that the time-dependent two-mode Fresnel operator is just the time-evolutional unitary operator governed by the Hamiltonian composed of quadratic combination of canonical operators in the way of exhibiting SU（1,1） algebra. This is an approach for obtaining the time-dependent Hamiltonian from the preassigned time evolution in classical phase space, an approach which is in contrast to Lewis-Riesenfeld＇s invariant operator theory of treating timedependent harmonic oscillators.     

Canonical transformations and exact invariants for time‐dependent Hamiltonian systems

An exact invariant is derived for n‐degree‐of‐freedom non‐relativistic Hamiltonian systems with general time‐dependent potentials. To work out the invariant, an infinitesimalcanonical transformation is performed in the framework of the extended phase‐space. We apply this approach to derive the invariant for a specific class of Hamiltonian systems. For the considered class of Hamiltonian systems, the invariant is obtained equivalently performing in the extended phase‐space a finitecanonical transformation of the initially time‐dependent Hamiltonian to a time‐independent one. It is furthermore shown that the invariant can be expressed as an integral of an energy balance equation. The invariant itself contains a time‐dependent auxiliary function ξ (t) that represents a solution of a linear third‐order differential equation, referred to as the auxiliary equation. The coefficients of the auxiliary equation depend in general on the explicitly known configuration space trajectory defined by the system's time evolution. This complexity of the auxiliary equation reflects the generally involved phase‐space symmetry associated with the conserved quantity of a time‐dependent non‐linear Hamiltonian system. Our results are applied to three examples of time‐dependent damped and undamped oscillators. The known invariants for time‐dependent and time‐independent harmonic oscillators are shown to follow directly from our generalized formulation.     

An Alternative Approach to Obtain the Exact Wave Functions of Time-Dependent Hamiltonian Systems Involving Quadratic, Inverse Quadratic, and (1/x)p+p(1/x) Terms

By using the Lewis-Riesenfeld theory and algebraic method, we present an alternative approach to obtain the exact solution of time-dependent Hamiltonian systems involving quadratic, inverse quadratic and (1/x)p+p(1/x) terms. This solution is discussed and compared with that obtained by Choi, J. R. (2003). International Journal of Theoretical Physics 42, 853]. PACS: 03.65Ge; 03.65Fd; 03.65Bz     

用改进的格点哈密顿量计算1+2维U(1)规范场真空波函数

从格点U(1)规范场论中改进的哈密顿量出发,推导出截断本征方程.并对2+1维U(1)规范场真空波函数进行数值计算,验证了理论的预言:对格点哈密顿量进行改进能使真空波函数的标度行为明显地改善.     

Interference in Phase Space of Squeezed States for the Time-Dependent Hamiltonian System

We showed that the idea of Schleich and Wheeler (1987, Nature 326, 574) for the semiclassical approach of the interference in phase space of harmonic oscillator squeezed states can be extended to that of general time-dependent Hamiltonian system. The quantum phase properties of squeezed states for the general time-dependent Hamiltonian system are investigated by using the quantum distribution function. The weighted overlaps A _n and phases θ _n for the system are evaluated in the semiclassical limit.     

Comparison of Corrected Wave Functions Associated to Two Different Approaches for the Time-Dependent Hamiltonian Systems Involving $(1/\hat{x})\hat{p}+\hat{p}(1/\hat{x})$ Term

The quantum solutions which are the results of the research of Maamache et al. (Int. J. Theor. Phys. 45:2191–2198, 2006) and ours (Choi, J.R. in Int. J. Theor. Phys. 42:853–861, 2003) for the time-dependent Hamiltonian systems involving term are compared after performing some corrections from the original ones. We confirmed that the two corrected wave functions are completely the same each other.     

SU(1,1) Lie Algebra Applied to the General Time-dependent Quadratic Hamiltonian System

Exact quantum states of the time-dependent quadratic Hamiltonian system are investigated using SU(1,1) Lie algebra. We realized SU(1,1) Lie algebra by defining appropriate SU(1,1) generators and derived exact wave functions using this algebra for the system. Raising and lowering operators of SU(1,1) Lie algebra expressed by multiplying a time-constant magnitude and a time-dependent phase factor. Two kinds of the SU(1,1) coherent states, i.e., even and odd coherent states and Perelomov coherent states are studied. We applied our result to the Caldirola–Kanai oscillator. The probability density of these coherent states for the Caldirola–Kanai oscillator converged to the center as time goes by, due to the damping constant γ. All the coherent state probability densities for the driven system are somewhat deformed. PACS Numbers: 02.20.Sv, 03.65.-w, 03.65.Fd     

Constructing the time independent Hamiltonian from a time dependent one

In this paper we introduce a method for finding a time independent Hamiltonian of a given Hamiltonian dynamical system by canonoid transformation of canonical momenta. We find a condition that the system should satisfy to have an equivalent time independent formulation. We study the example of a damped harmonic oscillator and give the new time independent Hamiltonian for it, which has the property of tending to the standard Hamiltonian of the harmonic oscillator as damping goes to zero.      

Supersymmetric WKB Approximation of Anharmonic Potential V(r)=ar2+br-4+cr-6

This article uses the supersymmetric WKB approximation to obtain the approximate energy levels and wave functions of the anharmonic potential V(r) = ar2 br-4 cr-6 in order to tesify the correctness between [Phys.Lett. A 170 (1992) 335] and the paper written by M. Landtman [Phys. Lett. A 175 (1993) 147].     

Evolution and Generation of Dynamics of Two-Mode Squeezed States of Time-Dependent Coupled Boson System

The system of the Hamiltonian involving a driving part, a single mode part, and a two-mode squeezed one and a two-mode coupled one is discussed using the Lewis-Riesenfeld invariant theory. The time evolution operator is obtained. When the initial state is a coherent state, the quantum fluctuation of the system is calculated, and it is dependent on the squeezed part and the two-mode coupled part, but not dependent on the driving one.     

Evolution and Generation of Dynamics of Two-Mode Squeezed States of Time-Dependent Coupled Boson System

The system of the Hamiltonian involving a driving part, a single mode part, and a two-mode squeezed one and a two-mode coupled one is discussed using the Lewis-Riesenfeld invariant theory. The time evolution operator is obtained. When the initial state is a coherent state, the quantum fluctuation of the system is calculated, and it is dependent on the squeezed part and the two-mode coupled part, but not dependent on the driving one.     

Supersymmetric WKB Approximation of Anharmonic Potential V（r）=ar^2＋br^-4＋cr^-6

This article uses the supersymmetric WKB approximation to obtain the approximate energy levels and wave functions of the anharmonic potential V（r） = ar^2 ＋ br^-4 ＋ cr^-6 in order to tesify the correctness between [Phys. Left. A 170 （1992） 335] and the paper written by M. Landtman [Phys. Left. A 175 （1993） 147].     

2+1维格点QCD真空波函数的计算

用改进的格点哈密顿量和截断本征方程法计算2+1维QCD真空波函数,结果显示出良好的标度行为.     

三种方法求解外场驱动含时谐振子系统的异同

讨论了外场驱动含量谐振子系统的三种解法的异同，发现利用相干态平均方法与量子不变量理论所得结果一致，而利用海森伯运动方程所求结果与上述两种结果相关一个相位因子。     

Exact Wave Functions of Time-Dependent Hamiltonian Systems Involving Quadratic, Inverse Quadratic, and (1/\hat x)\hat p + \hat p(1/\hat x) Terms

We use the dynamical invariant method to derive quantum-mechanical solution of time-dependent Hamiltonian system consisting quadratic potential, inverse quadratic potential, and . The term in Hamiltonian containing gives the expression such as in coordinate space, which we can often meet in radial equation of quantum many body problem. The wave functions differed only a time-dependent phase factor from the eigenstates of the invariant operator Î and expressed in terms of an associated Laguerre function.     

Time-Dependent 2D Harmonic Oscillator in Presence of the Aharanov-Bohm Effect

We use the Lewis-Riesenfeld theory to determine the exact form of the wavefunctions of a two-dimensionnal harmonic oscillator with time-dependent mass and frequency in presence of the Aharonov-Bohm effect (AB). We find that the auxiliary equation is independent of the AB magnetic flux. In the particular case of quantized AB magnetic flux the wavefunctions coincide exactly with the wavefunctions of the 2D time-dependent harmonic oscillator. PACS: 03.65Ge; 03.65Fd; 03.65Bz     

与不变量有关的幺正变换方法和三次量子化宇宙波函数的演化

首先在推广了的量子不变量理论的基础上建立了与不变量有关的立正变换方法,并用此方法研究了三次量子化宇宙波函数的演化,求得了系统的相因子、波函数.并通过构造相干态求得了Wheeler—Dewitt方程的解以及由三次量子化引起的相对量子起伏.