Truncated harmonic oscillator and parasupersymmetric quantum mechanics

We discuss in detail the parasupersymmetric quantum mechanics of arbitrary order where the parasupersymmetry is between the normal bosons and those corresponding to the truncated harmonic oscillator. We show that even though the parasusy algebra is different from that of the usual parasusy quantum mechanics, still the consequences of the two are identical. We further show that the parasupersymmetric quantum mechanics of arbitrary orderp can also be rewritten in terms ofp supercharges (i.e. all of which obeyQ _i ² =0). However, the Hamiltonian cannot be expressed in a simple form in terms of thep supercharges except in a special case. A model of conformal parasupersymmetry is also discussed and it is shown that in this case, thep supercharges, thep conformal supercharges along with HamiltonianH, conformal generatorK and dilatation generatorD form a closed algebra.     

一维谐振子量子运动中的经典特性

利用量子力学的态叠加原理和算符劈裂法,对处于一维谐振子势中的初始态为高斯波包的中心位置的量子运动进行了研究.结果表明:其中心位置的量子运动呈现出经典谐振子的运动特性;波包的初始位置和初始时刻所加动量对波包中心位置量子动力学的影响与经典谐振子类似条件对运动的影响有相同的性质.本结果对理解复杂量子运动中的高斯波方法有一定的启示作用.     

Information entropic measures of a quantum harmonic oscillator in symmetric and asymmetric confinement within an impenetrable box

Information‐based uncertainty measures like Shannon entropy, Onicescu energy and Fisher information (in position and momentum space) are employed to understand the effect of symmetric and asymmetric confinement in a quantum harmonic oscillator. Also, the transformation of the Hamiltonian into a dimensionless form gives an idea of the composite effect of force constant and confinement length (x_c). In the symmetric case, a wide range of x_c has been taken up, whereas asymmetric confinement is dealt with by shifting the minimum of the potential from the origin keeping box length and boundary fixed. Eigenvalues and eigenvectors for these systems are obtained quite accurately via an imaginary‐time propagation scheme. For asymmetric confinement, a variation‐induced exact diagonalization procedure is also introduced, which produces very high quality results. One finds that, in symmetric confinement, after a certain characteristic x_c, all these properties converge to respective values of a free harmonic oscillator. In the asymmetric situation, excited‐state energies always pass through a maximum. For this potential, the classical turning point decreases, whereas well depth increases with the strength of asymmetry. A study of these uncertainty measures reveals that localization increases with an increase of the asymmetry parameter.

     

1-D harmonic oscillator in MONDified inertia

In this paper we study the dynamics of a harmonic oscillator with laws of motion prescribed by MOND (Modified Newtonian Dynamics) in its modified inertia formulation. A differential equation for a 1D harmonic oscillator is obtained and several features of its solution are analyzed. Particular attention is given to the deep MOND limit regime, where the equations of motion are significantly different from the Newtonian one.     

Rashba自旋-轨道相互作用对一维谐振子势场中电子性质的影响

分析了Rashba自旋-轨道相互作用对一维谐振子势场中电子性质的影响,发现该自旋-轨道相互作用能够导致能级之间的自旋翻转,并且自旋翻转的性质明显依赖于自旋-轨道耦合系数和参考系坐标之间的关系.     

Equity prices as a simple harmonic oscillator with noise

The centred return on the London Stock Exchange's FTSE All Share Index is modelled as a simple harmonic oscillator with noise over the period from 1 January, 1994 until 30 June 2006. Our empirical results are compatible with the hypothesis that there is a period in the FTSE All Share Index of between two and two and one half years. This means the centred return will on average continue to increase for about a year after reaching the minimum in its oscillatory cycle; alternatively, it will continue on average to decline for about a year after reaching a maximum. Our analysis also shows that there is potential to exploit the harmonic nature of the returns process to earn abnormal profits. Extending our analysis to the low energy states of a quantum harmonic oscillator is also suggested.     

三量子比特系统中单个谐振子诱导的量子关联

我们考虑初始无关联并且与由一个谐振子构成的环境之间互相耦合的三量子比特系统。通过研究量子比特-环境的耦合强度以及量子比特初始态对量子关联的影响,我们发现环境可以诱导量子关联,提出并证明了四个命题阐述谐振子如何调控三个量子比特中量子关联的分布。给出了产生量子关联的条件。特别地,对于弱耦合,我们不但能够获得很多的量子关联,而且还使量子比特系统和环境始终处于分离态。一般地,量子关联动力学是很复杂 的,这是由于环境起着两个互相竞争的作用：一方面诱导出各个比特之间的量子关联;另一方面又使它们发生消相干,从而破坏量子关联。     

三量子比特系统中单个谐振子诱导的量子关联

我们考虑初始无关联并且与由一个谐振子构成的环境之间互相耦合的三量子比特系统。通过研究量子比特-环境的耦合强度以及量子比特初始态对量子关联的影响,我们发现环境可以诱导量子关联,提出并证明了四个命题阐述谐振子如何调控三个量子比特中量子关联的分布。给出了产生量子关联的条件。特别地,对于弱耦合,我们不但能够获得很多的量子关联,而且还使量子比特系统和环境始终处于分离态。一般地,量子关联动力学是很复杂 的,这是由于环境起着两个互相竞争的作用：一方面诱导出各个比特之间的量子关联;另一方面又使它们发生消相干,从而破坏量子关联。     

The effect of singular potentials on the harmonic oscillator

We address the problem of a quantum particle moving under interactions presenting singularities. The self-adjoint extension approach is used to guarantee that the Hamiltonian is self-adjoint and to fix the choice of boundary conditions. We specifically look at the harmonic oscillator added of either a δ-function potential or a Coulomb potential (which is singular at the origin). The results are applied to Landau levels in the presence of a topological defect, the Calogero model and to the quantum motion on the noncommutative plane.     

Minimal coupling in oscillator models of quantum dissipation

The dissipative harmonic oscillator has two representations. In the first representation the central oscillator couples with its position to an oscillator bath. In the second one it couples with its momentum to the bath. Both representations are related by a unitary transformation. If the central oscillator couples with its position and momentum to two independent baths, no such unitary transformation exists. We discuss two possible models of this type and their physical relevance.     

Equal-time second-order moments of a harmonic oscillator with stochastic frequency and driving force

Using a simple matrix method, we have obtained exact second-order equilibrium moments for a linearly damped harmonic oscillator with a fluctuating frequency (t) and driven by a fluctuating forcef(t). We have assumed each of the fluctuating quantities to be delta-correlated. We demonstrate that the final answers are identical whetherf(t) and (t) are statistically independent or delta-correlated. We have also established the region of parameter space in which the oscillator is energetically stable. The results are shown to be completely determined by the coefficients of the first and second cumulants of the fluctuations.Supported in part by the Office of Naval Research, NSF Grant # CHE 78-21460, and by a grant from Charles and Renée Taubman.     

一维受迫谐振子量子运动的经典特性

对经典一维受迫谐振子量子化,求解量子化后体系的时间演化算符.应用相空间准概率分布函数,研究了体系的量子特性.研究结果表明,初始为真空态,经过时间演化,系统波函数是一个二维高斯波包;波包中心的振幅和相位受到作用力的调制,成为调幅、调相波,波包中心的运动与经典受迫谐振子的运动形式相同.     

一维线性谐振子哈密顿量改写中的非幺正变换

通过引入一非幺正算符实现了对一维线性谐振子系统哈密顿量的从坐标、动量表象(Q,P)到占有数表象(a,a+)的改写.     

Time-dependent harmonic oscillator in a magnetic field and an electric field on the non-commutative plane

In the non-commutative space, wave functions and geometric phases are derived for the time-dependent harmonic oscillator in external time-dependent magnetic and electric field. Explicit forms of the coherent states are also given, which are not the minimum uncertainty states for the coordinates and momenta.     

用自洽平均值法计算非线性谐振子本征值

提出了一种自洽平均值法计算非线性谐振子本征值的方法.研究结果表明,等效势能曲线、坐标算符的一次项和二次项的平均值及能量本征值均在主量子数n较小时与微扰量子理论相吻合,在n=0时两者完全一致.     

Quantum vibrational partition function in the non-extensive Tsallis framework

The quantum vibrational partition function has been obtained in the Tsallis statistics framework for the entropic index, q, between 1 and 2. The effect of non-extensivity on the population of states and thermodynamic properties have been studied and compared with their corresponding values obtained in the Boltzmann-Gibbs (BG) statistics. Our results show that the non-extensive partition function of harmonic oscillator at any temperature is larger than its corresponding values for an extensive system and that their differences increase with temperature and entropic index. Also, the number of accessible states increases with q but, compared to the BG statistics, the occupation number decreases for low energy levels while the population of the higher energy levels increases. The internal energy and heat capacity have also been obtained for the non-extensive harmonic oscillator system. Results indicate that the heat capacity is greater than its corresponding value in the extensive (BG) system at low temperatures but that this trend is reversed at higher temperatures.     

Harmonic balance approach to the periodic solutions of the (an)harmonic relativistic oscillator

The first-order harmonic balance method via the first Fourier coefficient is used to construct two approximate frequency-amplitude relations for the relativistic oscillator for which the nonlinearity (anharmonicity) is a relativistic effect due to the time line dilation along the world line. Making a change of variable, a new nonlinear differential equation is obtained and two procedures are used to approximately solve this differential equation. In the first the differential equation is rewritten in a form that does not contain a square-root expression, while in the second the differential equation is solved directly. The approximate frequency obtained using the second procedure is more accurate than the frequency obtained with the first due to the fact that, in the second procedure, application of the harmonic balance method produces an infinite set of harmonics, while in the first procedure only two harmonics are produced. Both approximate frequencies are valid for the complete range of oscillation amplitudes, and excellent agreement of the approximate frequencies with the exact one are demonstrated and discussed. The discrepancy between the first-order approximate frequency obtained by means of the second procedure and the exact frequency never exceeds 1.6%. We also obtained the approximate frequency by applying the second-order harmonic balance method and in this case the relative error is as low 0.31% for all the range of values of amplitude of oscillation A.     

基态球谐振子的空间“塌陷”

用幂级数方法研究球谐振子定态薛定谔方程解析解，发现波函数球系表示ψ(r,θ，φ)在r～0处可以无界，只要rψ(r,θ，φ)在r～0处有界，就不违背波函数的玻恩统计解释而且基态能量是ω/2, 而不是3ω/2，这是低能量条件下的振动系统空间“塌陷”现象. 关键词： 球谐振子 基态能量 波函数 空间塌陷     

二维耦合量子谐振子的本征值和本征函数

运用广义线性量子变换理论,给出一类二维耦合量子谐振子的能量本征值、本征函数、坐标和动量算符在能量表象中的矩阵元及演化算符.