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# 物理代写|粒子物理代写Particle Physics代考|PHYS696 The Quantum Theory of Radiation

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## 物理代写|粒子物理代写Particle Physics代考|Maxwell’s theory as a classical field theory

The simplest version of Maxwell’s equations takes the form ${ }^{3}$
$$\boldsymbol{\nabla} \cdot \boldsymbol{E}=\rho, \quad \boldsymbol{\nabla} \cdot \boldsymbol{B}=0$$
$$\boldsymbol{\nabla} \wedge \boldsymbol{B}-\frac{\partial \boldsymbol{E}}{\partial t}=\boldsymbol{j}, \quad \boldsymbol{\nabla} \wedge \boldsymbol{E}+\frac{\partial \boldsymbol{B}}{\partial t}=0$$
where $\boldsymbol{E}(t, \boldsymbol{x})$ and $\boldsymbol{B}(t, \boldsymbol{x})$ are the electric and magnetic fields, respectively, while $\rho(t, \boldsymbol{x})$ and $\boldsymbol{j}(t, \boldsymbol{x})$ are the external electric charge and current densities. Consistency of (2.5), (2.6), requires that $\rho$ and $j$ satisfy the continuity condition: $\partial \rho / \partial t+\nabla \cdot \boldsymbol{j}=0$. According to our recipe, if we want to describe this system as a classical field theory, we must first identify a set of independent variables $q_{\alpha}(t, x)$. These cannot be the six components of $\boldsymbol{E}(t, \boldsymbol{x})$ and $\boldsymbol{B}(t, \boldsymbol{x})$ because they are constrained by the first two equations (2.5). We should solve these constraints and eliminate the redundant variables.

It seems that a first step in this direction was taken by Gauss in 1835, long before Maxwell wrote his equations. It consists of introducing the vector and scalar potentials $\boldsymbol{A}(t, x)$ and $\phi(t, x)$. It will be convenient to use a compact relativistic notation in which $x=(t, x)$ and introduce the four-vectors $j^{\mu}(x)=(\rho, \boldsymbol{j})$ and $A^{\mu}(x)=(\phi, \boldsymbol{A})$. We then construct the two-index antisymmetric tensor
$$F_{\mu \nu}(x)=\frac{\partial A_{\nu}}{\partial x^{\mu}}-\frac{\partial A_{\mu}}{\partial x^{\nu}}$$
Since the derivative operator $\partial / \partial x_{\mu}$ will appear very often, we introduce a short-hand notation for it: $\partial / \partial x_{\mu}=\partial^{\mu}$. Similarly, $\partial / \partial x^{\mu}=\partial_{\mu}$. The electric and magnetic fields are given in terms of $F_{\mu \nu}$ by
$$F_{0 i}=-\partial_{0} A^{i}-\partial_{i} A^{0}=\left(-\frac{\partial \boldsymbol{A}}{\partial t}-\nabla A^{0}\right)^{i}=E^{i}, \quad B^{i}=\frac{1}{2} \epsilon_{i j k} F_{j k}$$
${ }^{3}$ We use the symbol $\wedge$ to denote the vector product of two three-dimensional vectors : $(a \wedge b){i}=$ $\epsilon{i j k} a^{j} b^{k}$.

## 物理代写|粒子物理代写Particle Physics代考|Quantum theory of the free electromagnetic field-photons

We are now ready to apply our general quantisation prescription and obtain the corresponding quantum theory. The canonical variables $(2.17)$ are promoted to operators satisfying the canonical commutation relations $(2.4) .^{7}$
$$\begin{gathered} {\left[\widetilde{A}^{(\lambda)}(t, k), \dot{\widetilde{A}}^{\left(\lambda^{\prime}\right) \dagger}\left(t, \boldsymbol{k}^{\prime}\right)\right]=\mathrm{i} \delta^{\lambda \lambda^{\prime}}(2 \pi)^{3} \delta^{3}\left(\boldsymbol{k}-\boldsymbol{k}^{\prime}\right)} \ {\left[\widetilde{A}^{(\lambda)}(t, \boldsymbol{k}), \widetilde{A}^{\left(\lambda^{\prime}\right)}\left(t, \boldsymbol{k}^{\prime}\right)\right]=0, \quad\left[\dot{\tilde{A}}^{(\lambda) \dagger}(t, \boldsymbol{k}), \dot{\tilde{A}}^{\left(\lambda^{\prime}\right) \dagger}\left(t, \boldsymbol{k}^{\prime}\right)\right]=0} \end{gathered}$$
where $\dagger$ denotes the Hermitian adjoint of the operator. The presence of the factor $(2 \pi)^{3}$ is due to our convention on the Fourier transform. Here, and throughout this book, we have adopted the system of units we introduce in Appendix A in which $c=\hbar=1$. Physical units will be restored only when it is necessary.

The relations (2.23) imply for the operators corresponding to the coefficients of the plane wave expansion $(2.20)$ the commutation relations
$$\begin{gathered} {\left[a^{(\lambda)}(\boldsymbol{k}), a^{\left(\lambda^{\prime}\right) \dagger}\left(\boldsymbol{k}^{\prime}\right)\right]=\delta^{\lambda \lambda^{\prime}} 2 E_{k}(2 \pi)^{3} \delta^{3}\left(\boldsymbol{k}-\boldsymbol{k}^{\prime}\right)} \ {\left[a^{(\lambda)}(\boldsymbol{k}), a^{\left(\lambda^{\prime}\right)}\left(\boldsymbol{k}^{\prime}\right)\right]=0, \quad\left[a^{(\lambda) \dagger}(\boldsymbol{k}), a^{\left(\lambda^{\prime}\right) \dagger}\left(\boldsymbol{k}^{\prime}\right)\right]=0} \end{gathered}$$

## 物理代写粒子物理代写Particle Physics代考|Maxwell’s theory as a classical field theory

$$\begin{gathered} \boldsymbol{\nabla} \cdot \boldsymbol{E}=\rho, \quad \boldsymbol{\nabla} \cdot \boldsymbol{B}=0 \ \boldsymbol{\nabla} \wedge \boldsymbol{B}-\frac{\partial \boldsymbol{E}}{\partial t}=\boldsymbol{j}, \quad \boldsymbol{\nabla} \wedge \boldsymbol{E}+\frac{\partial \boldsymbol{B}}{\partial t}=0 \end{gathered}$$ 连续性条件: $\partial \rho / \partial t+\nabla \cdot \boldsymbol{j}=0$. 根据㧴们的配方，如果㧴们要将这个系统描述为经典场论，我们必须首先确定一组自变量 $q_{\alpha}(t, x)$. 这些不可能是六个组成部分 $\boldsymbol{E}(t, \boldsymbol{x})$ 和 $\boldsymbol{B}(t, \boldsymbol{x})$ 因为它们受前两个方程 (2.5) 的约束。我们应该解隹这些约束并消除冗 余变量。

$$F_{\mu \nu}(x)=\frac{\partial A_{\nu}}{\partial x^{\mu}}-\frac{\partial A_{\mu}}{\partial x^{\nu}}$$

$$F_{0 i}=-\partial_{0} A^{i}-\partial_{i} A^{0}=\left(-\frac{\partial \boldsymbol{A}}{\partial t}-\nabla A^{0}\right)^{i}=E^{i}, \quad B^{i}=\frac{1}{2} \epsilon_{i j k} F_{j k}$$

## 物理代写粒子物理代写Particle Physics代考|Quantum theory of the free electromagnetic field-photons

$$\left[\widetilde{A}^{(\lambda)}(t, k), \dot{\tilde{A}}^{(\lambda) \dagger}\left(t, \boldsymbol{k}^{\prime}\right)\right]=\mathrm{i} \delta^{\lambda \lambda^{\prime}}(2 \pi)^{3} \delta^{3}\left(\boldsymbol{k}-\boldsymbol{k}^{\prime}\right)\left[\widetilde{A}^{(\lambda)}(t, \boldsymbol{k}), \widetilde{A}^{\left(\lambda^{\prime}\right)}\left(t, \boldsymbol{k}^{\prime}\right)\right]=0, \quad\left[\dot{\tilde{A}}^{(\lambda) \dagger}(t, \boldsymbol{k}), \dot{\bar{A}}^{\left(\lambda^{\prime}\right) \dagger}\left(t, \boldsymbol{k}^{\prime}\right)\right]=0$$

$$\left[a^{(\lambda)}(\boldsymbol{k}), a^{\left(\lambda^{\prime}\right) \dagger}\left(\boldsymbol{k}^{\prime}\right)\right]=\delta^{\lambda \lambda^{\prime}} 2 E_{k}(2 \pi)^{3} \delta^{3}\left(\boldsymbol{k}-\boldsymbol{k}^{\prime}\right)\left[a^{(\lambda)}(\boldsymbol{k}), a^{\left(\lambda^{\prime}\right)}\left(\boldsymbol{k}^{\prime}\right)\right]=0, \quad\left[a^{(\lambda) \dagger}(\boldsymbol{k}), a^{\left(\lambda^{\prime}\right) \dagger}\left(\boldsymbol{k}^{\prime}\right)\right]=0$$

## 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。