Posted on Categories:固体物理, 物理代写

# 物理代写|固体物理代写Solid Physics代考|Bloch’sTheorem

avatest™

avatest固体物理Solid Physics代写，免费提交作业要求， 满意后付款，成绩80\%以下全额退款，安全省心无顾虑。专业硕 博写手团队，所有订单可靠准时，保证 100% 原创。avatest™， 最高质量的固体物理Solid Physics作业代写，服务覆盖北美、欧洲、澳洲等 国家。 在代写价格方面，考虑到同学们的经济条件，在保障代写质量的前提下，我们为客户提供最合理的价格。 由于统计Statistics作业种类很多，同时其中的大部分作业在字数上都没有具体要求，因此固体物理Solid Physics作业代写的价格不固定。通常在经济学专家查看完作业要求之后会给出报价。作业难度和截止日期对价格也有很大的影响。

avatest™ 为您的留学生涯保驾护航 在物理Physics代写方面已经树立了自己的口碑, 保证靠谱, 高质且原创的物理Physics代写服务。我们的专家在固体物理Solid Physics代写方面经验极为丰富，各种固体物理Solid Physics相关的作业也就用不着 说。

## 物理代写|固体物理代写SOLID PHYSICS代考|Bloch’sTheorem

In Section 1.2, we used the periodicity of the system to guess a solution that was the same in every cell except for a phase factor that is constant within each cell. It turns out that this is a general property of all periodic systems, for any number of dimensions. This is known as Bloch’s theorem For any potential that is periodic such that $U(\vec{r}+\vec{R})=$ $U(\vec{r})$ for all $\vec{R}=N \vec{a}$, where $\vec{a}$ is some vector, the eigenstates of the Hamitonian have the property
$$\psi_{n \vec{k}}(\vec{r}+\vec{R})=\psi_{n \vec{k}}(\vec{r}) e^{i \vec{k} \cdot \vec{R}}$$
where $n$ is a band index that we add because, as we have seen with the Kronig-Penney model, there can be more than one eigenstate with the same $k$.

This can be restated in another way. Multiplying through by a phase factor $e^{-i \vec{k} \cdot \vec{r}}$, we get
\begin{aligned} & \psi_{n \vec{k}}(\vec{r}+\vec{R}) e^{-i \vec{k} \cdot \vec{r}}=\psi_{n \vec{k}}(\vec{r}) e^{i \vec{k} \cdot \vec{R}} e^{-i \vec{k} \cdot \vec{r}} \ & \psi_{n \vec{k}}(\vec{r}+\vec{R}) e^{-i \vec{k} \cdot(\vec{R}+\vec{r})}=\psi_{n \vec{k}}(\vec{r}) e^{-i \vec{k} \cdot \vec{r}} . \end{aligned}
This implies that $\psi_{n \vec{k}}(\vec{r}) e^{-i \vec{k} \cdot \vec{r}}$ is a periodic function. We can therefore write the eigenstates as
$$\psi_{n \vec{k}}(\vec{r})=\frac{1}{\sqrt{V}} u_{n \vec{k}}(\vec{r}) e^{i \vec{k} \cdot \vec{r}},$$
where $V$ is the volume of the crystal (introduced for normalization of the wave function) and $u_{n \vec{k}}(\vec{r})$ has the same periodicity as the potential. Note that here the phase factor depends on the continuous variable $\vec{r}$ instead of the discrete vector $\vec{R}$. The function $\psi_{n \vec{k}}(\vec{r})$ is called a Bloch function and the function $u_{n \vec{k}}(\vec{r})$ can be called the cell function.

## 物理代写|固体物理代写SOLID PHYSICS代考|BravaisLattices and ReciprocalSpace

As discussed in Section 1.1, solid state physics does not only deal with periodic structures. Nevertheless, the theory of periodic structures is extremely important because many solids do have periodicity. Solids that have periodic arrays of atoms are called crystals. Most metals and most semiconductors are crystals.

Crystals are common in nature because an ordered structure has lower entropy than a disordered structure, and lower entropy states are favored at low temperatures. Whether or not a system forms an ordered crystal at room temperature depends on the ratio of the thermal energy $k_B T$ to the binding energy of two atoms. If $k_B T$ is small compared to the binding energy, then the system is essentially in a zero-temperature state, even if it is quite hot compared to room temperature. We will return to discuss solid phase transitions in Section 5.4.

Bravais lattices In order to fill all of space with a periodic structure, we take a finite volume of space, which we call the primitive cell or unit cell and make copies of it adjacent to each other by translating it without rotation through integer multiples of three vectors, $\vec{a}_1, \vec{a}_2$, and $\vec{a}_3$. These vectors, known as the primitive vectors must be linearly independent, but need not be orthogonal. Some examples of lattices generated from primitive vectors are shown in Figure 1.13. The set of all locations of the unit cells is given by
$$\vec{R}=N_1 \vec{a}_1+N_2 \vec{a}_2+N_3 \vec{a}_3$$
where $N_1, N_2$, and $N_3$ are three integers. This set of all the vectors $\vec{R}$ makes up the Bravais lattice of the crystal. These vectors point to a set of points which define the origin of each primitive cell.

The primitive cell that is copied throughout space does not need to be cubic or rectangular; it can be any shape that will fill all space when copied periodically – it can be as complicated as the repeated elements of an Escher print. The most natural choice, however, is a parallelepiped with three edges equal to the primitive vectors.

A crystal can have more than one atom per primitive cell. Within each primitive cell, we can specify a basis, which is a set of vectors giving the location of the atoms relative to the origin of each cell. Figure 1.14 shows two examples of lattices with a basis. Table 1.1 gives the standard primitive vectors and basis vectors of some of the more common types of crystals.

## 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。