在均匀强磁场中氢原子塞曼效应久期方程的简化公式

在均匀强磁场中,当氢原子的哈密顿量中B2项不能忽略时,氢原子的库仑场对称性遭到破坏,能级简并被全部解除.在应用变分法和数值法计算氢原子的能级过程中,计算十分复杂,而应用微扰法求解氢原子的能级,存在解久期方程的n2高阶行列式的困难.本文应用简并态微扰理论和球谐函数的性质,得到久期方程中非零微扰矩阵元普遍表达式.根据非零微扰矩阵元普遍表达式的性质,可以将氢原子塞曼效应久期方程的n2高阶行列式分解成1阶到n阶共n个低阶行列式的乘积,得到氢原子塞曼效应久期方程的简化公式,使得求解均匀强磁场中氢原子塞曼效应能级过程简化.而且由该公式可以得到氢原子在低能态时塞曼效应能级的解析解.根据该久期方程的简化公式计算了n=3氢原子塞曼效应一级近似能级.     

无反射势阱的非齐次Schrdinger方程的精确解

无反射势阱描述一种双原子分子的吸引势。将Rayleigh Schr dinger展开法应用到微扰无反射势阱 ,得到一组非齐次Schr dinger方程。利用最近报道的微扰方法 ,导出这些方程的积分形式的精确解。以静电场与基态系统的相互作用为例 ,通过精确解的有界及其不能在Hilbert空间展开为有限项的性质 ,论证了该微扰系统束缚态的存在性及通常量子力学发散困难的起因。     

无反射势阱的非齐次Schr(o)dinger方程的精确解

无反射势阱描述一种双原子分子的吸引势。将Ｒａｙｌｅｉｇｈ－Ｓｃｈｒｏｄｉｎｇｅｒ展开法应用到微扰无反射势阱,得以一组非齐次Ｓｃｈｒｏｄｉｎｇｅｒ方程。利用最近报道的微扰方法,导出这些方程的积分形式的精确解。以静电场与基态系统的相互作用为例,通过精确解的有界及其不能在Ｈｉｌｂｅｒｔ空间展开为有限项的性质,论证了该微扰系统束缚态的存在性及通常量子力学发散困难的起因。     

基于蒙特卡罗模拟的三维氢原子电子云可视化

本文具体讨论了使用蒙特卡罗模拟对氢原子三维“电子云”进行可视化的方法.根据氢原子中电子的波函数,利用Mathematica软件和蒙特卡罗模拟方法,实现了不同能级的氢原子三维电子云的可视化,使人们对氢原子中电子的行为有更加直观的认识.同时本文也给出了基于蒙特卡罗方法的研究一般原子系统,包括有微扰系统的电子云图像的普适算法,可用于量子力学可视化课程资源的进一步开发.     

德拜势中类氢原子能级近似解析式与幂级数求解

对德拜势(Debye)中类氢原子的能级问题采用Rayleigh-Schrdinger微扰展开,给出了能级的一阶修正与原子能级的近似解析式.同时,采用波函数幂级数解法,求得了德拜势下相关的递推关系.在此基础上,利用能量自洽法,求出了相当于二阶修正的德拜势下类氢原子的能级值,并就其计算结果与数值解进行了比较.同时,讨论了相应的临界束缚能态与截断条件.     

切换奇异系统的有限时间稳定

本文研究了一类连续切换奇异系统的有限时间稳定和状态反馈控制问题.首先,讨论了连续切换奇异系统解的存在条件,然后给出连续切换奇异系统有限时间稳定和有限时间有界的概念;其次,利用模型依赖平均驻留时间方法和Lyapunov函数方法,分别给出切换奇异系统是正则、脉冲自由且有限时间稳定和有限时间有界的充分条件,并设计状态反馈控制器,使得闭环系统有限时间稳定和有限时间有界且具有H∞性能指标γ;最后通过数值算例验证了本文方法的有效性.     

一类大气非线性动力热力学反应-扩散系统解的稳定性态

研究了一类大气非线性系统. 利用大气非线性热力动力学理论, 讨论了大气非线性强迫耗散系统的速度、温度和湿度所满足的非线性反应-扩散系统, 再利用微分方程Lyapunov稳定性理论, 得到了大气非线性反应-扩散系统在均匀定态解邻域内的微扰解. 最后由系统的控制参数的变化, 得到了大气非线性强迫耗散系统有关的物理量的有序-无序-有序状态的转化过程, 从而可以预报和预测相应的局部大气非线性强迫耗散系统的稳定性态.     

含非线性微扰项的二阶动力学系统的一阶近似守恒量的一种新求法

从一阶近似守恒量的性质出发,把受微扰系统视为未受微扰系统与微扰项的迭加,提出一种分三步求得一阶近似守恒量的新方法:先选择合适的方法求得未受微扰系统的守恒量I0,再考虑微扰项对守恒量I0的影响,最后利用一阶近似守恒量的性质求得一阶近似守恒量.用该方法研究了一实际的受非线性微扰作用的两自由度动力学系统,得到4个稳定的一阶近似守恒量.用坐标变换法和微扰法得到系统一阶近似解的表达式,并讨论4种特殊情况下的一阶近似解.     

Josephson结中混沌孤振子的超辐射

考虑两个耦合的长Josephson结，运用直接微扰法，讨论周期微扰下混沌孤振子的超辐射功率，结果表明：周期微扰下的孤子作混沌运动，当孤子逐渐与周期微扰同步时，辐射功率不断增大，其上限虽受限的有界条件制约，但通常比孤子对的辐射功率更大，理论分析和数值模拟均很好地说明了这一点，这为混沌孤振子的实际应用提供理论依据。     

三维氢原子斯塔克效应中能量一级修正公式的求解

进一步简化了三维氢原子斯塔克效应微扰矩阵元的普遍公式,并给出了三维氢原子斯塔克效应中任意态能量的一级修正公式.     

Existence and Asymptotic Stability of Periodic Solutions of the Reaction–Diffusion Equations in the Case of a Rapid Reaction

A singularly perturbed periodic in time problem for a parabolic reaction-diffusion equation in a two-dimensional domain is studied. The case of existence of an internal transition layer under the conditions of balanced and unbalanced rapid reaction is considered. An asymptotic expansion of a solution is constructed. To justify the asymptotic expansion thus constructed, the asymptotic method of differential inequalities is used. The Lyapunov asymptotic stability of a periodic solution is investigated.     

Chaotic rotations of a liquid-filled solid

The disturbed Hamiltonian equations of a solid filled with a rotating ellipsoidal mass of a liquid and subjected to small-applied moments are revisited using Deprit's variables. We investigate the chaotic dynamics of the orbiting liquid-filled solid and of the liquid-filled solid sliding and rolling on a perfectly smooth plane, in either energy-conservative or energy-dissipative conditions, when appropriately perturbed. Criteria for the judgment of potential chaotic rotations of the perturbed system are formulated by means of Melnikov-Holmes-Marsden (MHM) integrals. Strategies for the solution of heteroclinic orbits of the symmetrical liquid-filled solid under torque-free conditions are outlined theoretically. Physical parameters that will probably trigger the onset of chaotic motions can be determined accordingly. Results from MHM algorithms are crosschecked with Poincare sections together with Lyapunov characteristic exponents.     

Some comments on quasi-bound-state calculations

Quasi-bound-state energy calculations are made for a perturbed oscillator and a perturbed hydrogen atom. The theory of the least-squares method is analysed. A simple numerical variant of the well-known stabilization method is tested, and the results are compared with those from the least-squares and perturbation-series methods.     

Complex dynamical behaviors of compact solitary waves in the perturbed mKdV equation

In this paper, we give a detailed discussion about the dynamical behaviors of compact solitary waves subjected to the periodic perturbation. By using the phase portrait theory, we find one of the nonsmooth solitary waves of the mKdV equation, namely, a compact solitary wave, to be a weak solution, which can be proved. It is shown that the compact solitary wave easily turns chaotic from the Melnikov theory. We focus on the sufficient conditions by keeping the system stable through selecting a suitable controller. Furthermore, we discuss the chaotic threshold for a perturbed system. Numerical simulations including chaotic thresholds, bifurcation diagrams, the maximum Lyapunov exponents, and phase portraits demonstrate that there exists a special frequency which has a great influence on our system; with the increase of the controller strength, chaos disappears in the perturbed system. But if the controller strength is sufficiently large, the solitary wave vibrates violently.     

Stability of trapped Bose-Einstein condensates in one-dimensional tilted optical lattice potential

Using the direct perturbation technique,this paper obtains a general perturbed solution of the Bose-Einstein condensates trapped in one-dimensional tilted optical lattice potential. We also gave out two necessary and sufficient conditions for boundedness of the perturbed solution. Theoretical analytical results and the corresponding numerical results show that the perturbed solution of the Bose-Einstein condensate system is unbounded in general and indicate that the Bose-Einstein condensates are Lyapunov-unstable. However,when the conditions for boundedness of the perturbed solution are satisfied,then the Bose-Einstein condensates are Lyapunov-stable.     

Periodic and chaotic behaviour in a reduced form of the perturbed generalized Korteweg–de Vries and Kadomtsev–Petviashvili equations

The dynamical behaviour of a reduced form of the perturbed generalized Korteweg–de Vries and Kadomtsev–Petviashvili equations (extension of the Korteweg–de Vries equation to two space variables) are studied in this paper. Harmonic solutions of non-resonance and primary resonance are obtained using the perturbation method. Chaotic motion under harmonic excitations is studied using the Melnikov method.A wide range of solutions for the reduced perturbed generalized Korteweg–de Vries equations, in which non-linear phenomena appearing within transition from regular harmonic response (periodic solutions) to chaotic motion, are obtained using the time integration Runge–Kutta method. When chaos is found, it is detected by examining the phase plane, the Poincaré map, the sensitivity solution of the solution to initial conditions, and by calculating the largest Lyapunov exponent.     

Perturbation of the excited states of the hydrogen atom and hydrogen-like ions by a neutral atom

Owing to the degeneracy of the energy levels, the wavefunction of the electron in the excited states of the hydrogen atom and hydrogen-like ions perturbed by a neutral atom B is significantly different from the wavefunction of the unperturbed state. The perturbed function has a wide high maximum in the region of atom B, which is explained by multiple collisions of the electron with atom B, because the classical trajectories in the Coulomb field are closed and the size of atom B is much smaller than the size of the excited-state orbit. The radiative lifetimes of the excited states are much larger than those of unperturbed states. The orbital angular momentum L of the excited electron is strongly changed in collisions with atom B owing to the quantum interference or mixing of the temporal phases of adiabatic wavefunctions. The cross sections for such a change in the orbital angular momentum are several orders of magnitude larger than the cross sections found in early investigations in the approximation of the single collision of the electron with atom B.     

Highly excited atomic states in a static electric field

We have performed exact numerical calculations for a hydrogen atom perturbed by a uniform electric field for some Stark sublevels with n = 12, 13, 14, 15. The results are compared with recent experimental data for excited states of sodium.     

Periodicity and chaos in strongly perturbed classical orbitals for Coulomb interactions

Within the framework of classical mechanics two prototypes of strongly perturbed orbitals, the diamagnetism in hydrogen and electronic double excitation, are analyzed near critical phase space points (fixed points). The motion of the hydrogen electron under the joint influence of the Coulomb field and the magnetic field is periodic for any field strengths. For a two-electron atom however we find a chaotic time evolution of the electron pair correlation, causing presumably irregular spectral patterns.     

Molecular Dynamics Study on Microstructure of Potassium Dihydrogen Phosphates Solution

用分子动力学模拟方法研究了不同温度下磷酸二氢钾(KDP)水溶液的内能和微结构. 水分子被看作简单点电荷模型,磷酸二氢根被看作七节点模型. 另外系统地研究了溶液的内能和径向分布函数. 在饱和温度附近,局域粒子数密度有很大的波动;饱和溶液的比热要比不饱和溶液的比热高表明溶液在低于饱和温度的过程中经历了结晶过程;水中的氧原子跟磷酸二氢根中的氢原子的径向分布函数表明二者之间形成很强的氢键结构;在KDP水溶液中,钾阳离子跟磷酸二氢根中的氧原子有很强的相互作用,并且在饱和溶液中形成的离子对数量更多.