一维量子非线性晶格的动力学研究：含时变分原理

线性体系与非线性体系的本质区别在于线性体系的解满足简单的叠加原理而非线性体系的解一般不满足线性体系中的简单叠加原理也就是说对于线性体系的任意两个解的线性组合仍然是体系的解而非线性体系的解一般不具有这样的性质
湘潭大学 硕士学位论文 一维量子非线性晶格的动力学研究：含时变分原理 姓名：钟红伟 申请学位级别：硕士 专业：理论物理 指导教师：唐翌
The thesis consists of four chapters. Chapter one gives an introduction to the 1D classical and quantum nonlinear lattice models and their leading dynamic characters. In chapter two, we introduce the TDVP and some single-particle wave functions.
本文共分四章。第一章中简单介绍了一维经典和量子非线性晶格模型及其主 要动力学特征。第二章中，我们介绍了含时变分原理和一些单粒子波函数。
在第三章，我们采用含时变分法研究了公度量子 Frenkel-Kontorova（FK）模 型的声子色散关系。体系的波函数采用 Hartree 型多体试探波函数，单粒子态则 采用被冻结的 Jackiw-Kerman（JK）波函数。在最小测不准条件下，我们导出了 粒子的期望值所满足的运动方程，并由此得到了声子色散关系。结果表明：与对 应的经典模型相比，公度量子 FK 模型的格点势的强度和声子元激发带隙被量子 涨落削弱了。但是，变分法不能揭示体系在有限的临界质量时，声子元激发带隙 从有带隙相向无带隙相的转变。
In chapter four, a semiquantal approach to study the dynamics of 1D quantum nonlinear lattices is formulated. This approach is based on TDVP. The Hartree-type
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湘潭大学硕士毕业论文
摘要
自然界中存在大量的一维离散晶格体系；并且，纳米技术的飞速发展预示着 不久的将来会有更多的人造一维离散晶格体系出现。一维离散晶格体系的独特性 质使其具有潜在的和重要的应用价值。在一些一维离散晶格体系中，量子效应对 体系的性质具有至关重要的作用，应该加以考虑。在一维量子非线性晶格的研究 中，特别是动力学的研究中，求解多粒子体系的含时薛定谔方程是不可避免的。 求解含时薛定谔方程的标准方法由于其计算量与体系的自由度呈指数增长而大受 限制。众所周知，标准的方法只适用于不超过五、六个自由度的体系。对大体系 的长时间动力学的模拟，就必须求助于的近似方法。本文运用含时变分原理研究 了一维量子非线性晶格的动力学。其目的在于发展一种研究一维量子非线性晶格 动力学毕业论文
many-body trial wave function is used for the system of particles, and the single-particle state is taken to be the JK form. The quantum effects considered in this approach are the quantum fluctuations of lattices. As an application of this approach, we study the heat conduction in the quantum Fermi-Pasta-Ulam (FPU) model. Non-equilibrium molecular dynamics simulations show that the thermal conductivity increases with the system size under the semiquantal dynamics, and the Fourier’s law is not satisfied for the quantum FPU model. For the characteristic values of the effective Planck constant, the exponents of power-law divergence are about 0.40, which is almost equal to the value of the classical FPU model. Nevertheless, the thermal conductivity of quantum FPU model is larger than that of the classical counterpart, and the thermal conductivity increases with the quantum effects.
In chapter three, we study the phonon dispersion relation of the commensurate quantum Frenkel-Kontorova (FK) model by means of the time-dependent variational approach combined with a Hartree-type many-body trial wave function. The single-particle state is taken to be a frozen Jackiw-Kerman (JK) wave function. Under the condition of minimum uncertainty, equations of motion for the particle’s expectation values are derived to obtain the phonon dispersion relation. It is shown that the strength of the substrate potential and the phonon excitation gap are reduced due to the quantum fluctuations in comparison with those of the classical model. However, the variational approach can’t reveal the transition of the phonon excitation gap from the phase with a gap to the gapless phase at a finite critical mass.
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关键词：量子非线性晶格；动力学；含时变分原理；半量子方法；热传导
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湘潭大学硕士毕业论文
Abstract
There are many one-dimensional (1D) discrete lattice systems in the nature. Moreover, with the fast advancement of nanotechnology more and more man-made 1D discrete lattice systems are under expectation in the near future. Peculiar properties in the 1D discrete lattice systems exhibit potential and important applications. In some 1D discrete lattice systems, quantum effects play a crucial role to the properties of systems, and should be taken into account. In the study of 1D quantum nonlinear lattices, especially in the study of dynamics, it is unavoidable to solve the many-body time-dependent Schrodinger equation. The standard method solving the time-dependent Schrodinger equation is seriously restricted, by which the computational efforts scale exponentially with the number of degrees of freedom. It is well known that the standard method is capable of handling the systems with no more than five or six degrees of freedom. For simulating the long-time dynamics of large many-body systems, approximate approach is needed. In this thesis, we study the dynamics of 1D quantum nonlinear lattices by virtue of the time-dependent variational principle (TDVP). The purpose of this thesis is to develop an approximate approach to study the dynamics of 1D quantum nonlinear lattices.
在第四章，我们将半量子方法应用到一维量子非线性晶格的动力学研究中。 此方法在含时变分原理的基础上结合了 Hartree 型多体试探波函数。单粒子波函 数采用 JK 型波函数。这种方法中所考虑的量子效应是晶格的量子涨落。作为一 种具体应用，我们研究了量子 Fermi-Pasta-Ulam (FPU) 模型中的热传导。非平衡 分子动力学模拟的结果表明：在这个半量子动力学下，体系的热导率也随体系的 尺寸增加而增加的，傅立叶定律在量子 FPU 模型中也不成立；对于特征值范围之 内的有效普朗克常数，幂律发散的指数大约为 0.40，几乎与经典 FPU 模型中的值 相等；不过，量子 FPU 模型中的热导率要大于相应的经典模型的热导率，而且， 随着量子效应的增强，体系的热导率是增大的。
Finally, we present a conclusion for our work and some prospects for future works in this field. Key Words: quantum nonlinear lattices; dynamics; time-dependent variational
principle; semiquantal approach; heat conduction
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