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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|Two-dimensional electron gas in a magnetic field

A two-dimensional electron gas (2-DEG) is produced at semiconductor interfaces and in metal-oxide-semiconductor (MOS) structures. Electrons move freely in the $x-y$ plane but are localized in the $z$-direction. Absent a magnetic field, the most convenient single-particle states are plane waves $|\mathbf{k} \sigma\rangle$, which are characterized by three quantum numbers: $k_x, k_y$, and $\sigma$. In the presence of a magnetic field, however, these states are not very convenient.

Consider an electron gas, confined to a rectangular sheet of length $L_x$ and width $L_y$, i.e., in the presence of a static, uniform magnetic field $\mathbf{B}$ in the $z$-direction. What is the Hamiltonian for a charged particle in a magnetic field $\mathbf{B}$ ? To answer this question we go back to classical mechanics. The force on a particle of charge $q$ and velocity $\mathbf{v}$ is, in cgs units, $\mathbf{F}=(q / c) \mathbf{v} \times \mathbf{B}$; in SI units, $q / c \rightarrow q$. Defining the vector potential $\mathbf{A}$ by $\mathbf{B}=\nabla \times \mathbf{A}$, it is not difficult to check that the Lagrangian $L$, given by
$$L=\frac{1}{2} m v^2+\frac{q}{c} \mathbf{v} \cdot \mathbf{A} .$$
does indeed produce the correct equation for the force. The proof consists in using the Euler-Lagrange equation of motion
$$\frac{d}{d t} \frac{\partial L}{\partial \dot{x}}=\frac{\partial L}{\partial x}$$

with similar equations for $y$ and $z$, along with the following two equations:
$$\begin{gathered} \mathbf{v} \times \mathbf{B}=\mathbf{v} \times(\boldsymbol{\nabla} \times \mathbf{A})=\nabla(\mathbf{v} \cdot \mathbf{A})-(\mathbf{v} . \nabla) \mathbf{A} \ \frac{d \mathbf{A}}{d t}=\frac{\partial \mathbf{A}}{\partial t}+\frac{\partial \mathbf{A}}{\partial x} \frac{d x}{d t}+\frac{\partial \mathbf{A}}{\partial y} \frac{d y}{d t}+\frac{\partial \mathbf{A}}{\partial z} \frac{d z}{d t}=(\mathbf{v} \cdot \nabla) \mathbf{A} . \end{gathered}$$

## 物理代写|费曼图代写Feynman diagram代考|N -particle wave function

Suppose that we have a complete, onthononnal set of single-particle states $\left|\phi_1\right\rangle$. where $v$ is an index that represents all the quantum numbers that characterize the state. Orthonormality and completeness mean that
$$\left\langle\phi_v \mid \phi_{v^{\prime}}\right\rangle=\delta_{r v^{\prime}} \text { (orthonormality) } \sum_v\left|\phi_v\right\rangle\left\langle\phi_v\right|=\mathrm{I} \text { (completeness). }$$
We will show that the $N$-particle wave function $\Psi(1.2 \ldots N)$ can be expanded in terms of products of the single-particle states. We may proceed as follows. Suppose that we fix the spatial and spin coordinates of particles $2,3 \ldots . . N$. Then $\Psi(1,2 \ldots . N)$ is a function of the coordinates of particle I alone: hence, we can expand it in a complete set of states $\phi_1(1)$.

If we now allow the coordinates of particle 2 to vary, $A_{1_1}(2,3 \ldots, N)$ becomes a function of these coordinates, and we may expand it as
$$A_{v_1}(2,3, \ldots . . N)=\sum_{v_2} B_{v_1 v_2}(3,4 \ldots N) \phi_{v_2}(2) .$$
Continuing in this fashion, we end up with
$$\Psi(1,2 \ldots . . N)=\sum_{r_1 v_2 \ldots w_N} C_{v_1 v_2 \ldots 1_j} \phi_{v_1}(1) \phi_{v_2}(2) \ldots \phi_{1, N}(N)$$

# 费曼图代考

## 物理代写|费曼图代写Feynman diagram代考|Two-dimensional electron gas in a magnetic field

$$L=\frac{1}{2} m v^2+\frac{q}{c} \mathbf{v} \cdot \mathbf{A} .$$

$$\frac{d}{d t} \frac{\partial L}{\partial \dot{x}}=\frac{\partial L}{\partial x}$$

$$\mathbf{v} \times \mathbf{B}=\mathbf{v} \times(\boldsymbol{\nabla} \times \mathbf{A})=\nabla(\mathbf{v} \cdot \mathbf{A})-(\mathbf{v} . \nabla) \mathbf{A} \frac{d \mathbf{A}}{d t}=\frac{\partial \mathbf{A}}{\partial t}+\frac{\partial \mathbf{A}}{\partial x} \frac{d x}{d t}+\frac{\partial \mathbf{A}}{\partial y} \frac{d y}{d t}+\frac{\partial \mathbf{A}}{\partial z} \frac{d z}{d t}=(\mathbf{v} \cdot \nabla) \mathbf{A}$$

## 物理代写|费曼图代写Feynman diagram代考|N -particle wave function

$$\left\langle\phi_v \mid \phi_{v^{\prime}}\right\rangle=\delta_{r v^{\prime}} \text { (orthonormality) } \sum_v\left|\phi_v\right\rangle\left\langle\phi_v\right|=\mathrm{I} \text { (completeness). }$$

$$A_{v_1}(2,3, \ldots . . N)=\sum_{v_2} B_{v_{1 v_2}}(3,4 \ldots N) \phi_{v_2}(2) .$$

$$\Psi(1,2 \ldots N)=\sum_{r_1 v_2 \ldots w_N} C_{v_1 v_2 \ldots 1_j} \phi_{v_1}(1) \phi_{v_2}(2) \ldots \phi_{1, N}(N)$$

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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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