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• Statistical Inference 统计推断
• Statistical Computing 统计计算
• (Generalized) Linear Models 广义线性模型
• Statistical Machine Learning 统计机器学习
• Longitudinal Data Analysis 纵向数据分析
• Foundations of Data Science 数据科学基础

## 物理代写|费曼图代写Feynman diagram代考|one-dimensional lattice

Consider a chain of $N$ identical atoms. The equilibrium separation between the atoms is $a$. If $a$ is large. the chain will be a collection of isolated atoms. Each atom has its own orbitals: $1 \mathrm{~s}, 2 \mathrm{~s} .2 \mathrm{p}, \ldots$ with the lowest energy orbitals being occupied by electrons. As the atoms are brought closer together so that atomic wave functions begin to overlap. electrons tunnel from one atom to another. becoming delocalized. and the overlapping orbitals form bands. For example, whereas the 3 ss orbitals have a well-defined energy in isolated atoms, they broaden into a band when atoms are brought closer together. The one-electron Hamiltonian is $H=p^2 / 2 m+V(x)$, where $V(x)=V(x+a)$ is the periodic potential seen by the electron. This is sketched in Figure $2.6 .$

Now consider the band formed by the broadening of one type of atomic orbital. e.g.. the $3 \mathrm{~s}$ orbitals. Let $\left|\phi_m\right\rangle$ be the atomic orbital centcred on atom number $m$. located at $x=m a, m=1, \ldots . N$. We assume that $\left\langle\phi_m|H| \phi_m\right\rangle=\epsilon$, and that for $n \neq m .\left\langle\phi_n|H| \phi_m\right\rangle=-t \delta_{n, m \pm 1}$, i.e.. we assume that the overlap of atomic wave functions is appreciable only between nearest-neighbor atoms. We take $t$ to be real. Our goal is to tind the energy dispersion $E_k$ for this band.

We want to solve the eigenvalue equation $\left.H\left|\Psi_k\right\rangle=E_k \mid \Psi_k\right)$. We take the $N$ atomic orbitals centered on the $N$ atoms as the basis states in which $\left|\Psi_k\right\rangle$ is expanded.
$$\left|\Psi_k\right\rangle=\sum_m c_{m k}\left|\phi_m\right\rangle \Rightarrow \Psi_k(x)=\sum_{m=1}^N c_{m k} \phi(x-m a) .$$
The coefficients $c_{m k}$ are not arbitrary; they are chosen so that $\Psi_k(x)$ is a Bloch lunction, being a stationary state of an electron in a periodic potential. We thus require thal $\Psi_k(x+a)=e^{i k a} \Psi_k(x)$; this, in turn, implies that
$$\sum_m c_{m k} \phi(x+a-m a)=e^{i k a} \sum_m c_{m k} \phi(x-m a)$$

## 物理代写|费曼图代写Feynman diagram代考|Wannier states

For electrons subjected to the periodic potential produced by a lattice of ions, we have considered in Section $2.3$ the basis set of Bloch states $|n \mathbf{k} \sigma\rangle$ characterized by a band index $n$. wave vector $\mathbf{k} \in \mathrm{FBZ}$. and spin projection $\sigma$. These are modulated plane waves that extend throughout the crystal. Another basis set of states, consist ing of localized orbitals centered on lattice sites. may be constructed. For a given band index $n$, lattice site $\mathbf{R}i$, and spin projection $\sigma$. consider the states $$|n i \sigma\rangle=\frac{1}{\sqrt{N}} \sum{k \in F B Z} e^{-i k \cdot R_1}|n \mathbf{k} \sigma\rangle$$
These are called Wannier states; they have the following properties:

• The Wannier function $\phi_{n i \sigma}(\mathbf{r})=\langle\mathbf{r} \mid n i \sigma\rangle$ is centered on $\mathbf{R}i$; hence it is written as $\phi{n \sigma}\left(\mathbf{r}-\mathbf{R}_i\right)$
• The Wannier states form a complete, orthonormal set.
• The Wannier function $\phi_{n \sigma}\left(\mathbf{r}-\mathbf{R}i\right)$ is localized on the lattice site $i$. The tirst property follows directly from the second form of Bloch’s theorem. From the definition of the Wannier state. we can write $$\phi{n i \sigma}(\mathbf{r})=\frac{1}{\sqrt{N}} \sum_{\mathbf{k} \in \mathrm{FBZ}} e^{-i \mathbf{k} \cdot \mathbf{R}i} e^{i \mathbf{k} \cdot \mathbf{r}} u{n \mathbf{k}}(\mathbf{r})|\sigma\rangle$$
Since $u_{\text {,㿟 }}$ has the periodicity of the lattice, we can rewrite the above as

# 费曼图代考

## 物理代写|费曼图代写Feynman diagram代考|one-dimensional lattice

$$\left|\Psi_k\right\rangle=\sum_m c_{m k}\left|\phi_m\right\rangle \Rightarrow \Psi_k(x)=\sum_{m=1}^N c_{m k} \phi(x-m a) .$$

$$\sum_m c_{m k} \phi(x+a-m a)=e^{i k a} \sum_m c_{m k} \phi(x-m a)$$

## 物理代写|费曼图代写Feynman diagram代考|Wannier states

$$|n i \sigma\rangle=\frac{1}{\sqrt{N}} \sum k \in F B Z e^{-i k \cdot R_1}|n \mathbf{k} \sigma\rangle$$

• 万尼尔函数 $\phi_{n i \sigma}(\mathbf{r})=\langle\mathbf{r} \mid n i \sigma\rangle 以 \mathbf{R} i_i$ 因此它被写为 $\phi n \sigma\left(\mathbf{r}-\mathbf{R}_i\right)$
• Wannier 状态形成一个完整的正交集。
• 万尼尔函数 $\phi_{n \sigma}(\mathbf{r}-\mathbf{R} i)$ 定位在格点上 $i$. 第一个性质直接来自布洛赫定理的第二种形式。从Wannier状态的定义。我们可以写
$$\phi n i \sigma(\mathbf{r})=\frac{1}{\sqrt{N}} \sum_{\mathbf{k} \in \mathrm{FBZ}} e^{-i \mathbf{k} \cdot \mathbf{R} i} e^{i \mathbf{k} \cdot \mathbf{r}} u n \mathbf{k}(\mathbf{r})|\sigma\rangle$$

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## 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。

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