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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|量子光学代写Quantum Optics代考|Maxwell’s Equations in Matter

We continue with our “reading” of Maxwell’s equations in terms of homogeneous and inhomogeneous equations. Apparently, the inhomogeneities of charge and current distributions $\rho, \boldsymbol{J}$ are the means of how the material world “communicates” with the electromagnetic fields. In fact, one of the main reasons why optics and nano optics have seen such a tremendous boost in recent years is the progress in nano and material science. This has brought up numerous novel charge and current sources that allow for an unprecedented control of light-matter interaction. Was it just for the electromagnetic part of Maxwell’s equations, the field of electrodynamics would have probably turned into a completely boring discipline by now.

When dealing with Maxwell’s equations in matter, it is convenient to separate the charge and current distributions into external parts, which can be controlled from the outside, and induced contributions associated with polarizations and magnetizations. The latter can usually not be easily controlled, yet, in presence of matter microscopic polarizations and magnetizations will he induced and will inevitably act back on the fields. Figure $2.5$ gives a brief sketch of the principle underlying this separation. The separation into external and induced contributions is not always completely clear and there is sometimes some freedom of choice what is “external” and what is “induced.”

We next introduce, in close analogy to the charge and current distributions, the polarization $\boldsymbol{P}(\boldsymbol{r}, t)$ as an electric dipole moment per unit volume, and the magnetization $\boldsymbol{M}(\boldsymbol{r}, t)$ as a magnetic dipole moment per unit volume. These quantities account for the material response in presence of electromagnetic fields, and we have to provide a prescription of how they are related to the electromagnetic fields. With $\boldsymbol{P}$ and $\boldsymbol{M}$ we can separate the charge and current distributions into free and bound contributions according to $[1,2]$

## 物理代写|量子光学代写Quantum Optics代考|Linear Materials

For a wide class of materials we can assume a linear relation between the material response and the external fields. More specifically, we get
Linear Materials
$$\boldsymbol{P}=\varepsilon_0 \chi_e \boldsymbol{E}, \quad \boldsymbol{M}=\chi_m \boldsymbol{H} .$$
Here $\chi_e$ and $\chi_h$ are the electric and magnetic susceptibilities, respectively.
Polarization. Let me first discuss the polarization expression. As I would like to argue, there is a strong physical motivation for relating $\boldsymbol{P}$ to the electric field $\boldsymbol{E}$. We first recall that $\boldsymbol{D}$ is an auxiliary field that is solely created by the external charge distribution $\rho_{\text {ext. }}$ If we had erroneously assumed $\boldsymbol{P}=\chi_e \boldsymbol{D}$ (wrong relation), the polarization at a given position would be only due to the external fields. In reality, however, the true field $\boldsymbol{E}$ is the sum of the external field and of the polarization field, which is produced by the entire polarized body under investigation, in agreement to the choice made in Eq. (2.27).
Magnetization. Things are different for the magnetization, which is only induced by the free currents governing $\boldsymbol{H}$. This directly points to the previously mentioned confusion regarding the proper role of $\boldsymbol{B}$ and $\boldsymbol{H}$. Fortunately, things are not as bad as they seem. For practically all materials under study the magnetization is very small, and for this reason the error made through $\boldsymbol{M}=$ $\chi_m \boldsymbol{H}$ is usually negligible in comparison to the arguably more correct choice $\boldsymbol{M}=\chi_m \boldsymbol{B}$ (wrong relation).
We can now continue to establish a relation between $\boldsymbol{D}$ and $\boldsymbol{E}$,
$$\boldsymbol{D}=\varepsilon_0\left(1+\chi_e\right) \boldsymbol{E}=\varepsilon \boldsymbol{E},$$
where we have introduced the permittivity $\varepsilon=\varepsilon_0\left(1+\chi_e\right)$. Similarly, we get
$$\boldsymbol{B}=\mu_0\left(1+\chi_m\right) \boldsymbol{H}=\mu \boldsymbol{H},$$
where we have introduced the permeability $\mu=\mu_0\left(1+\chi_m\right)$. In case of anisotropic materials both $\varepsilon$ and $\mu$ become tensorial quantities, but we will not consider such materials unless stated differently.

# 量子光学代考

## 物理代写|量子光学代写Quantum Optics代考|线性材料

$$\boldsymbol{P}=\varepsilon_0 \chi_e \boldsymbol{E}, \quad \boldsymbol{M}=\chi_m \boldsymbol{H} .$$

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## MATLAB代写

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