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• Statistical Inference 统计推断
• Statistical Computing 统计计算
• Advanced Probability Theory 高等概率论
• Advanced Mathematical Statistics 高等数理统计学
• (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|量子光学代写Quantum Optics代考|Maxwell’s Equations

In electrodynamics we deal with time-dependent electric $\boldsymbol{E}(\boldsymbol{r}, t)$ and magnetic $\boldsymbol{B}(\boldsymbol{r}, t)$ fields. The force acting on a particle with charge $q$ at position $\boldsymbol{r}$ and moving with velocity $v$ can be computed from the Lorentz force
$$\boldsymbol{F}=q[\boldsymbol{E}(\boldsymbol{r}, t)+\boldsymbol{v} \times \boldsymbol{B}(\boldsymbol{r}, t)] .$$
The electromagnetic fields themselves are created by charge and current distributions $\rho(\boldsymbol{r}, t)$ and $\boldsymbol{J}(\boldsymbol{r}, t)$, respectively. The dynamics of these fields is determined by Maxwell’s equations which form the basis of the theory of electrodynamics,
Maxwell’s Equations
\begin{aligned} \text { Gauss’ law } & \nabla \cdot \boldsymbol{E}=\frac{\rho}{\varepsilon_0} \ \text { no name } & \nabla \cdot \boldsymbol{B}=0 \ \text { araday’s law } & \nabla \times \boldsymbol{E}=-\frac{\partial \boldsymbol{B}}{\partial t} \end{aligned}
Ampere’s law $\nabla \times \boldsymbol{B}=\mu_0 \boldsymbol{J}+\mu_0 \varepsilon_0 \frac{\partial \boldsymbol{E}}{\partial t}$.
$\mu_0$ is the vacuum permeability. There are various ways of “reading” Maxwell’s equations. The first one is in terms of the Helmholtz theorem stating that a vector function is determined once its divergence and curl together with the boundary conditions are known. In this way, Gauss’ and Faradays’s law determine the electric field $\boldsymbol{E}(\boldsymbol{r}, t)$, whereas the second equation (no name) and Ampere’s law determine the magnetic field $\boldsymbol{B}(\boldsymbol{r}, t)$.

## 物理代写|量子光学代写Quantum Optics代考|Electromagnetic Potentials

Electromagnetic potentials play an important role in classical electrodynamics $[1$, 2], but are of significantly less importance in the field of nano optics. As we will discuss in Chap. 5 , in nano optics one usually introduces different objects, the so-called Green’s functions, which take over many of the advantages of electromagnetic potentials. Nevertheless, at several places, most noteworthy certainly in quantum optics, we will rely on the concept of electromagnetic potentials.

We start by “reading” Maxwell’s equations in a way that will become even more important in the next section, namely in terms of homogeneous and inhomogeneous equations. The inhomogeneities in Maxwell’s equations are the external charge and current distributions. The idea behind electrodynamic potentials is to introduce new quantities, the scalar and vector potentials $V(\boldsymbol{r}, t)$ and $\boldsymbol{A}(\boldsymbol{r}, t)$, which are chosen such that the homogeneous Maxwell’s equations are automatically fulfilled. The equations that determine these potentials are then provided by the inhomogeneous Maxwell equations.

Let us start with $\nabla \cdot \boldsymbol{B}=0$. We may now relate $\boldsymbol{B}$ to the vector potential $\boldsymbol{A}$ through
$$\boldsymbol{B}=\nabla \times \boldsymbol{A}$$
With this choice the magnetic field is guaranteed to have no sources or sinks, because the divergence of a curl field $\nabla \cdot \nabla \times A$ is always zero. As for the other homogeneous equation, Faraday’s law, we start with
$$\nabla \times\left(\boldsymbol{E}+\frac{\partial \boldsymbol{A}}{\partial t}\right)=-\nabla \times \nabla V=0$$
Here we have replaced the expression in parentheses by $-\nabla V$ because the curl of a gradient field is automatically zero. $V$ is the scalar potential. The negative sign is a convention adopted from electrostatics where the potential can be related to the work done by a charge against the electric field [2]. Thus, we can express the electric field in terms of the scalar and vector potentials via
$$\boldsymbol{E}=-\nabla V-\frac{\partial \boldsymbol{A}}{\partial t}$$
With these relations between $\boldsymbol{E}, \boldsymbol{B}$ and the electromagnetic potentials $V, \boldsymbol{A}$ the homogeneous Maxwell equations are automatically fulfilled.

# 量子光学代考

## 物理代写|量子光学代写Quantum Optics代考|麦克斯韦方程组

$$\boldsymbol{F}=q[\boldsymbol{E}(\boldsymbol{r}, t)+\boldsymbol{v} \times \boldsymbol{B}(\boldsymbol{r}, t)] .$$

\begin{aligned} \text { Gauss’ law } & \nabla \cdot \boldsymbol{E}=\frac{\rho}{\varepsilon_0} \ \text { no name } & \nabla \cdot \boldsymbol{B}=0 \ \text { araday’s law } & \nabla \times \boldsymbol{E}=-\frac{\partial \boldsymbol{B}}{\partial t} \end{aligned}

$\mu_0$是真空导率。“解读”麦克斯韦方程有很多种方法。第一个是关于亥姆霍兹定理的，它说明了一个向量函数的散度和旋度以及边界条件都是已知的。这样，高斯和法拉第定律决定了电场$\boldsymbol{E}(\boldsymbol{r}, t)$，而第二个方程(没有名字)和安培定律决定了磁场$\boldsymbol{B}(\boldsymbol{r}, t)$。

## 物理代写|量子光学代写Quantum Optics代考|电磁势

$$\boldsymbol{B}=\nabla \times \boldsymbol{A}$$

$$\nabla \times\left(\boldsymbol{E}+\frac{\partial \boldsymbol{A}}{\partial t}\right)=-\nabla \times \nabla V=0$$

$$\boldsymbol{E}=-\nabla V-\frac{\partial \boldsymbol{A}}{\partial t}$$

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