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经典力学和量子力学之间存在着一定的对应关系,在某种极限条件下,量子力学的规律转变为经典力学的规律.对于谐振子而言,这种对应关系更为明显,更有意义. 经典振子的哈密顿量是坐标 和动量P的函数:由经典振子的哈密顿方程可得从(2)和(3)式可解出上两式中A和B是积分常数,由t=0时 相似文献
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题目:(2010年高考全国卷Ⅰ第21题)一简谐振子沿x轴振动,平衡位置在坐标原点.t=0时刻振子的位移x=-0.1m;t=4/3s时刻x=0.1m;t=4s时刻x=0.1m.该振子的振幅和周期可能为 相似文献
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以界面摩擦为研究对象,探讨了基于耦合振子模型(coupled-oscillator model)的滑动摩擦微观机理,分析了滑动过程中的能量耗散问题. 采用Maugis-Dugdal接触模型替代界面摩擦中的Lennard-Jones势能,并将该模型融入耦合振子模型之中,通过计算振子在一个周期内的能量增加值,推导出了界面摩擦力的理论计算公式. 理论分析表明,对于探针-试样接触系统,滑动摩擦力近似随着法向载荷的2/3次方增加,这与纳米摩擦学经典理论是相符的.理论计算结果与超高真空原子力显微镜镀铜探针在Cu(111)晶面扫描实验结果符合良好,表明本文提出的理论和方法可行.
关键词:
耦合振子模型
界面摩擦
摩擦力
法向载荷 相似文献
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一维振子的过阻尼与临界阻尼现象的分析 总被引:2,自引:0,他引:2
对于受到阻尼的一维振子,若阻力与速度成正比时,其运动微分方程写作(l) 其中β为阻尼因数ω。是无阻尼时派子的固有园频率、鉴于在现行力学教程中对过阻尼与临界阻尼现象的讨论较为简略,本文准备通过分析方程(1)的解,研究振子在这两种情形下处于不同初始条件时的运动情况. (一)当β2>ω02:时,出现过阻尼现象、(1)的特征方程有不同二实根,令 q=,故 β>q.(1)的通解是 若运动的初始条件是则可以确定 在此后的讨论中,为确定起见,令x0>0. 1)由于λ1与λ2皆为负值,式(2)的衰减性质是明显的.在t→∞时,x→0.表明振子在经过足够长的时间之后必停止于… 相似文献
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对于一个有阻尼的弹簧振子,振动物体的运动微分方程为式中m为振动物体的质量,K为弹簧的劲度系数,r为阻尼系数. 令(ω0为振子固有频率),=2β(β为阻尼因子),则运动微分方程变为上式的解有三种情形: A.当β2>ω01时,有过阻尼振动解 B.当β2=ω02时、有临界阻尼振动解 C.当β2<ω02时,有阻尼振动解 一、对x临(t)“临界性”的证明 当β→ω0时,从表面上看x过(t)和x阻(t)的临界形式似乎是C’1e-ω0t,为什么在x临(t)中会出现c’zte-“。‘项,对这个问题,我们作如下证明. ,设振动的初始条件X一回。,可以定出三种解中的常数,即得 Zgi() 证明1当B… 相似文献
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用积分法求出了对称非线性水平弹簧振子的振动周期的级数解,并对所得结果进行讨论,得到了线形弹簧振子的振动周期以及x~3振荡器的振荡周期. 相似文献
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Y. Ben-Aryeh 《Journal of Quantitative Spectroscopy & Radiative Transfer》1980,23(4):403-409
A model of multiphoton absorption of i.r. laser radiation by an anharmonic, molecular, vibrational mode is discussed. The multiphoton absorption is described as a stochastic process for which the reservoir, including rotation and other vibrational modes, modulates the frequency of the active oscillator stochastically. The treatment of the nonlinear oscillator is based on quantization of a classical, driven anharmonic oscillator. 相似文献
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The classical Duffing oscillator is a dissipative chaotic system, and so there is not a definite Hamiltonian. We quantize
the Duffing oscillator on the basis of quantum state diffusion (QSD) which is a formulation for open quantum systems and a
useful tool for analyzing nonlinear problems and classical limits. We can define a ‘Lyapunov exponent’, which corresponds
to the classical one in the proper limit, and investigate the behavior of the system by varying the Planck constant effectively.
We show that there exists a critical stage, where the behavior of the system crosses over from classical to quantum one. 相似文献
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The stochastic quantization method recently developed by Haba and Kleinert is extended to non-autonomous mechanical systems, in the case of the time-dependent harmonic oscillator. In comparison with the autonomous case, the quantization procedure involves the solution of a nonlinear, auxiliary equation. Using a rescaling transformation, the Schrö-din-ger equation for the time-dependent harmonic oscillator is obtained after averaging of a classical stochastic differential equation. 相似文献
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The stochastic quantization method recently developed by Haba and Kleinert is extended to non-autonomous mechanical systems, in the case of the time-dependent harmonic oscillator. In comparison with the autonomous case, the quantization procedure involves the solution of a nonlinear, auxiliary equation. Using a rescaling transformation, the Schrödinger equation for the time-dependent harmonic oscillator is obtained after averaging of a classical stochastic differential equation. 相似文献
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将扩展Prelle-Singer法(扩展P-S法)用于求x=Ф1(x,y),y=Ф2(x,y)类型的二阶非线性耦合动力学系统的守恒量,得到了积分乘子满足的微分方程与守恒量的一般形式,并讨论所得守恒量的Noether对称性与Lie对称性.最后用扩展P-S法求得了四次非谐振子系统的两个守恒量,并讨论了系统的对称性. 相似文献
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The generation of pulsed squeezed light using an optical parametric oscillator (OPO) is discussed. This mode-locked optical parametric oscillator consists of a nonlinear crystal in a cavity which is resonant for both signal and idler waves and which is synchronously pumped by the second-harmonic of an acousto-optically mode-locked cw Nd: YAG laser. The fundamental wavelength of the pump laser provides local oscillator pulses for balanced homodyne detection of squeezed vacuum pulses emitted by the oscillator when operated below oscillation threshold. Photocurrent noise reduction to 30% below the classical shot-noise limit is observed, corresponding to squeezing of the field to a level approximately a factor of two below the mean square vacuum noise. 相似文献
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《Physics letters. A》2001,286(4):261-276
An analytical solution up to the second order in the coupling constant λ is obtained for a classical quartic anharmonic oscillator by using Taylor series method. Our solution yields, as a special instance, the corresponding results obtained by using Laplace transform. With the help of correspondence principle, the classical solution is used to obtain the solution corresponding to a quantum quartic anharmonic oscillator. In the weak coupling regime (i.e., anharmonic constant λ⪡1), the so-called secular terms in classical and quantum solutions are tucked in (summed up) to avoid the nonconvergence. Both the classical and quantum solutions are used to obtain the frequency shifts of the quartic oscillators. It is found that these frequency shifts coincide exactly with those of the earlier results obtained by other methods. From the quantum field theoretic point of view, our solution exhibits the so-called Lamb shift. As an application of the solution for the quantum oscillator, we examine the possibility of getting squeezed states out of the input coherent light interacting with a nonlinear medium of inversion symmetry. 相似文献
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O. Jedrkiewicz R. Loudon J. Jeffers 《The European Physical Journal D - Atomic, Molecular, Optical and Plasma Physics》2006,39(1):129-140
Previous work on the retrodictive theory of direct
detection is extended to cover the homodyne detection of coherent optical
signal states
and
. The retrodictive input state probabilities are obtained by
the application of Bayes' theorem to the corresponding predictive
distributions, based on the probability operator measure (POM) elements for
the homodyne process. Results are derived for the retrodictive information
on the complex amplitude of the signal field obtainable from the difference
photocount statistics of both 4-port and 8-port balanced homodyne detection
schemes. The local oscillator is usually assumed much stronger than the
signal but the case of equal strengths in 4-port detection is also
considered. The calculated probability distributions and error rates are
illustrated numerically for values of signal and local oscillator strengths
that extend from the classical to the quantum regimes. 相似文献
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The probability structure of the response and energy harvested from a nonlinear oscillator subjected to white noise excitation is investigated by solution of the corresponding Fokker–Planck (FP) equation. The nonlinear oscillator is the classical double well potential Duffing oscillator corresponding to the first mode vibration of a cantilever beam suspended between permanent magnets and with bonded piezoelectric patches for purposes of energy harvesting. The FP equation of the coupled electromechanical system of equations is derived. The finite element method is used to solve the FP equation giving the joint probability density functions of the response as well as the voltage generated from the piezoelectric patches. The FE method is also applied to the nonlinear inductive energy harvester of Daqaq and the results are compared. The mean square response and voltage are obtained for different white noise intensities. The effects of the system parameters on the mean square voltage are studied. It is observed that the energy harvested can be enhanced by suitable choice of the excitation intensity and the parameters. The results of the FP approach agree very well with Monte Carlo Simulation (MCS) results. 相似文献
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T. K. Melik-Barkhudarov 《Journal of Contemporary Physics (Armenian Academy of Sciences)》2007,42(2):43-48
Principles of development of the quantum theory of nonlinear processes on the basis of Lagrangian formulation are discussed. It is shown that in the framework of this formulation it is possible to preserve succession from the classical theory and, in particular, use these methods for studies of quantum systems. The quantum dispersion of a nonlinear oscillator excited by an external source and of a parametric generator is calculated. Its role is established in the solution of the problem of stability of oscillations. 相似文献