Posted on Categories:Condensed Matter Physics, 凝聚态物理, 物理代写

物理代写|凝聚态物理代写Condensed Matter Physics代考|PH4101 The Born–Oppenheimer adiabatic approximation

avatest™

avatest™帮您通过考试

avatest™的各个学科专家已帮了学生顺利通过达上千场考试。我们保证您快速准时完成各时长和类型的考试，包括in class、take home、online、proctor。写手整理各样的资源来或按照您学校的资料教您，创造模拟试题，提供所有的问题例子，以保证您在真实考试中取得的通过率是85%以上。如果您有即将到来的每周、季考、期中或期末考试，我们都能帮助您！

•最快12小时交付

•200+ 英语母语导师

•70分以下全额退款

物理代写|凝聚态物理代写Condensed Matter Physics代考|The Born–Oppenheimer adiabatic approximation

The cores are much heavier than the electrons; for example, $M \sim 5 \times 10^{4} \mathrm{~m}$ for aluminum. At densities corresponding to typical solids, the cores behave like classical particles, but the electrons form a degenerate electron gas, leading to a difference of orders of magnitude in the kinetic energy of the two species. This difference in mass and kinetic energy implies that the cores are significantly more sluggish than the electrons; therefore the electrons can react almost instantaneously to any motion of the cores.

For most systems, it can be assumed that the electrons feel the instantaneous potential produced by the cores in their “frozen” or “fixed” positions. Therefore, we define an electronic Hamiltonian for fixed cores,
$$H_{e}=\sum_{i}\left[\frac{\mathbf{p}{i}^{2}}{2 m}+\sum{n} V_{n}\left(\mathbf{r}{i}-\mathbf{R}{n}\right)\right]+\frac{1}{2} \sum_{i j}^{\prime} \frac{e^{2}}{\left|\mathbf{r}{i}-\mathbf{r}{j}\right|}+H_{R},$$
and determine its energy spectrum $E_{e}^{l}\left(\left{\mathbf{R}{n}\right}\right)$, where $l$ labels the ith excited state which depends implicitly on the core positions $\left{\mathbf{R}{n}\right}$. In particular, with the electronic system in the ground state, the energy $E_{e}^{0}({\mathbf{R}})$ defines the potential $V_{\mathrm{ec}}({\mathbf{R}})$ within which the cores move. Remaining contributions of Eq. (2.1) can be assigned to the core part of the Hamiltonian,
$$H_{c}=\sum_{n} \frac{\mathbf{p}{n}^{2}}{2 M{n}}+\frac{1}{2} \sum_{n n^{\prime}}^{\prime} \frac{Z_{n} Z_{n^{\prime}} e^{2}}{\left|\mathbf{R}{n}-\mathbf{R}{n^{\prime}}\right|}+V_{\mathrm{ec}}\left(\left{\mathbf{R}{n}\right}\right) .$$ The term $V{\mathrm{ec}}\left(\left{\mathbf{R}{n}\right}\right)$, which is $E{e}^{0}\left(\left{\mathbf{R}_{n}\right}\right)$, represents an electron-core term which can be evaluated once Eq. (2.2) is solved and then used in Eq. (2.3) to determine the energy specturm for the cores.

The procedure described above is the Born-Oppenheimer ${ }^{1}$ adiabatic approximation. It provides the important step of separating the electronic and core degrees of freedom. The electronic part (Eq. (2.2)) leads primarily to the determination of the properties of the electrons, holes, excitons, plasmons, and magnons. The core part (Eq. (2.3)) is used to describe the core motions and phonons. When the phonons are coupled to the electrons, including terms in Eq. (2.1) that go beyond Eqs. (2.2) and (2.3), one can examine polarons, superconductivity, resistivity, and other properties of solids. At this point, we restrict ourselves to the electronic Hamiltonian (Eq. (2.2)).

物理代写|凝聚态物理代写Condensed Matter Physics代考|The mean-field approximation

Even though the Hamiltonian (Eq. (2.2)) does not include the cores as dynamical variables, it still contains a very large number of particles, namely, all the valence electrons in the solid. Another approximation is needed. The simplest commonly employed one is the Hartree $^{2}$ mean-field approximation, which assumes that each electron moves in the average or mean field created by the cores together with all the other electrons. In the Hartree approach, one assumes that the electronic wavefunction of all the valence electrons is approximated by a product of one-electron wavefunctions, and each one is characterized by some one-electron spatial and spin quantum numbers. The effects of the Pauli exclusion principle are taken into account by requiring that no pairs of one-electron wavefunctions (orbitals) in the product have an identical set of quantum numbers. As discussed in standard quantum textbooks, demanding that the ground state has the lowest energy for the electronic system results in a set of self-consistent Euler-Lagrange equations (the Hartree equations) for the one-electron orbitals and energies with potential $V\left(\mathbf{r},\left{\mathbf{R}_{n}\right}\right)$.

The Hartree mean-field approximation accomplishes the important task of separating the Hamiltonian (Eq. (2.2)) into a sum of one-electron Hamiltonians
$$H_{e}=\sum_{i} H\left(\mathbf{r}{i},\left{\mathbf{R}{n}\right}\right),$$
where
$$H\left(\mathbf{r},\left{\mathbf{R}{n}\right}\right)=\frac{p^{2}}{2 m}+V\left(\mathbf{r},\left{\mathbf{R}{n}\right}\right) .$$
Another approach involves the Hartree-Fock approximations, which is also a meanfield approach for the electronic ground state. The Hartree-Fock method approximates the wavefunction by a determinant of one-electron orbitals, and thus automatically satisfies all the symmetry requirements of the Pauli principle. However, although more accurate, this approach has several complicating features which often make it inconvenient for many applications. Further discussion of this point appears in Chapter $6 .$

物理代写儗聚态物理代写Condensed Matter Physics代考|The Born-Oppenheimer adiabatic approximation

$$H_{e}=\sum_{i}\left[\frac{\mathbf{p} i^{2}}{2 m}+\sum n V_{n}(\mathbf{r} i-\mathbf{R} n)\right]+\frac{1}{2} \sum_{i j}^{\prime} \frac{e^{2}}{|\mathbf{r} i-\mathbf{r} j|}+H_{R},$$

〈left 的分隔符缺失或无法识别 . 特别是，当电子系统处于基态时，能量 $E_{e}^{0}(\mathbf{R})$ 定义湝力 $V_{e c}(\mathbf{R})$ 核心在其中移 动。方程式的剩余贡献。(2.1) 可以分配给哈密顿量的核心部分，
Yleft 的分隔符缺失或无法识别

物理代写儗聚态物理代写Condensed Matter Physics代考|The mean-field approximation

Hartree 方法中，假设所有价电子的电子波函数近似为单电子波函数的乘积，并且每个波函数都由一些单电子空间数和自旋量子数 来表征。通过要求乘积中没有一对单电子波函数（轨道）具有相同的一组量子数，考虑了泡利不相容原理的影响。正如标准量子教 科书中所讨论的，要求基态具有电子系统的最低能量会导致一组自洽的欧拉-拉格朗日方程 (哈特里方程) 用于单电子轨道和具有 势能的能荲 left 的分隔符缺失或无法识别
Hartree 平均场近似完成了将哈密顿量 (Eq. (2.2)) 分蓠为单电子哈密顿量之和的重要任务
\left 的分隔符缺失或无法识别

、left 的分隔符缺失或无法识别

MATLAB代写

MATLAB 是一种用于技术计算的高性能语言。它将计算、可视化和编程集成在一个易于使用的环境中，其中问题和解决方案以熟悉的数学符号表示。典型用途包括：数学和计算算法开发建模、仿真和原型制作数据分析、探索和可视化科学和工程图形应用程序开发，包括图形用户界面构建MATLAB 是一个交互式系统，其基本数据元素是一个不需要维度的数组。这使您可以解决许多技术计算问题，尤其是那些具有矩阵和向量公式的问题，而只需用 C 或 Fortran 等标量非交互式语言编写程序所需的时间的一小部分。MATLAB 名称代表矩阵实验室。MATLAB 最初的编写目的是提供对由 LINPACK 和 EISPACK 项目开发的矩阵软件的轻松访问，这两个项目共同代表了矩阵计算软件的最新技术。MATLAB 经过多年的发展，得到了许多用户的投入。在大学环境中，它是数学、工程和科学入门和高级课程的标准教学工具。在工业领域，MATLAB 是高效研究、开发和分析的首选工具。MATLAB 具有一系列称为工具箱的特定于应用程序的解决方案。对于大多数 MATLAB 用户来说非常重要，工具箱允许您学习应用专业技术。工具箱是 MATLAB 函数（M 文件）的综合集合，可扩展 MATLAB 环境以解决特定类别的问题。可用工具箱的领域包括信号处理、控制系统、神经网络、模糊逻辑、小波、仿真等。