# 物理代写|电动力学代写electromagnetism代考|ELEC2300

#### Doug I. Jones

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## 物理代写|电动力学代写electromagnetism代考|Diagrammatic Perturbation Theory

The evaluation of the terms in the expansion (10.36) for the transition matrix for a given physical process often involves cumbersome algebraic computations which may be simplified using graphical or diagrammatic techniques [46]. This is especially so for multiphoton processes if only the lowest-order multipolar interactions are retained in the PZW-transformed Hamiltonian, since in this approximation the perturbation operators are proportional to the charge parameter $e$. If the full Hamiltonian is used, $\mathrm{V}$ also contains terms proportional to $e^2$, and so there are more types of diagram to include. The application of such methods in non-relativistic QED became well known in the 1960s [7]-[11], [47]-[49].

Every term in the perturbation series corresponds to a particular diagram, and to recover the complete perturbation formulae of a given order in the interaction, all topologically distinct diagrams must be considered. There are three basic components to the diagrams, open lines, vertices and propagators. At a vertex a photon is absorbed or emitted by a ‘particle’, which may be a single charge or a collection of charges, for example, an atom or molecule, or distinct parts of molecules (e.g. chromophores) if they can be assumed to be electronically distinct, and so is translated into a matrix element of the interaction $\mathrm{V}$. Open lines correspond to real particles, and in the absence of external fields the conservation laws for energy and momentum apply at the vertices where open lines terminate or start. The diagrams are to be read with time increasing from left to right. The basic absorption and emission vertex diagrams are shown in Figure 10.1. Thus, diagram (i) in Figure 10.1 represents absorption of a photon in the mode $\mathbf{Q}, \mu$ with the particle gaining energy $\hbar Q c$ and momentum $\hbar \mathbf{Q}$; similarly, diagram (ii) represents emission of a photon and loss of the same energy and momentum by the particle to the field. ${ }^9$

Diagrams are built up by glueing lines together; a closed line between two vertices represents the propagation of a virtual particle, and energy conservation does not apply to such vertices. A closed line is translated into a matrix element of a Green’s function or propagator, for example,
$$\left\langle n\left|G^0\left(E_i\right)\right| m\right\rangle=\frac{1}{E_n^0-E_m^0+i \varepsilon}$$ and so closed lines give the characteristic energy denominators in the perturbation expansion. For a free system conservation of momentum is maintained at vertices between closed lines. Static fields act instantaneously, and so external static electric and magnetic fields may be included using diagrams that have a vertical line ending at a vertex (with incoming and outgoing lines for the particles). The static Coulomb interaction between charges ${ }^{10}$ may be represented by a vertical line joining two vertices (see Figure 10.2). Conservation of momentum does not apply at external field vertices.

## 物理代写|电动力学代写electromagnetism代考|Absorption, Emission and Scattering – the Basic Processes

This section is devoted to an outline of the evaluation of the basic diagrams, Figures 10.1-10.3 using the Coulomb gauge Hamiltonian for a charged particle interacting with the quantised electromagnetic field; after that, the extension to the physically interesting cases involving many charges (atoms, molecules, condensed matter, plasmas etc.) will be seen to be quite straightforward. The matrix elements required are given in Appendix $\mathrm{F}$.

Figure 10.1 shows the primitive absorption and emission vertices that correspond to first-order perturbation theory; there is no denominator to evaluate. Consider the absorption vertex; according to Eq. (F.1.5), the perturbation operator is
$$\mathrm{K}a^1=-\sum{\mathbf{q}, \sigma} \mathbf{f}e(\mathbf{q}) \cdot \hat{\boldsymbol{\varepsilon}}(\mathbf{q})\sigma \mathrm{c}{\mathbf{q}, \sigma}$$ If the initial and final states for the absorption of a photon by a free charge are $\Phi_n^0=$ $\left|\varphi{\mathbf{P}}, \mu\left[n_{\mathbf{Q}}\right]\right\rangle$ and $\Phi_k^0=\left|\varphi_{\mathbf{P}^{\prime}}, \mu\left[n_{\mathbf{Q}}-1\right]\right\rangle$, respectively, the matrix element is (cf. (F.1.8))
$$\left\langle\Phi_k^0\left|\mathrm{~K}a^1\right| \Phi_n^0\right\rangle=-\frac{e}{m} \sqrt{\frac{\hbar^2}{2 \varepsilon_0 \Omega \mathcal{E}{\mathbf{Q}}}} \mathbf{P} \cdot \hat{\boldsymbol{\varepsilon}}(\mathbf{Q})_\mu \delta^3\left(\mathbf{P}+\hbar \mathbf{Q}-\mathbf{P}^{\prime}\right) .$$
The emission vertex has the same form, with $\mathbf{Q} \rightarrow-\mathbf{Q}$.
We have to recognise that a free charge cannot absorb or emit a (real) photon because energy cannot be conserved in such a transition. To see this, consider a charge initially at rest and an incident photon with wave vector $\mathbf{Q}$. After absorbing the photon, the particle must have momentum $\hbar \mathbf{Q}$. Thus, we have
\begin{aligned} E_n^0=\hbar Q c, & \mathbf{P}_n^0=\hbar \mathbf{Q}, \ E_k^0=\frac{\hbar^2 Q^2}{2 m}, & \mathbf{P}_k^0=\hbar \mathbf{Q}, \end{aligned}
for the initial and final energy and momentum of the (charge + photon) system. But since the final speed of the particle is $v_k=\hbar Q / m$, conservation of energy would require $v_k=2 c$ which is impossible. ${ }^{12}$ Photons are absorbed and emitted by free charges in virtual transitions to which energy conservation does not apply.

It is easy to see that collections of bound charges have a different behaviour as regards absorption and emission of photons because the requirement for momentum conservation can be met by the centre-of-mass motion of the composite (‘recoil’), and photon absorption/emission can be associated with transitions involving the internal states. It is then possible to satisfy the requirements for energy and momentum conservation simultaneously; thus, atoms and molecules undergo real energy-conserving transitions involving the absorption and emission of photons. The first-order amplitude $V_{f i}^{(1)}$ in $(10.35)$ is non-zero and yields the Einstein coefficients for absorption and emission of radiation [41].

# 电动力学代考

## 物理代写|电动力学代写electromagnetism代考|Diagrammatic Perturbation Theory

$$\left\langle n\left|G^0\left(E_i\right)\right| m\right\rangle=\frac{1}{E_n^0-E_m^0+i \varepsilon}$$

## 物理代写|电动力学代写electromagnetism代考|Absorption, Emission and Scattering – the Basic Processes

$$\mathrm{K} a^1=-\sum \mathbf{q}, \sigma \mathbf{f} e(\mathbf{q}) \cdot \hat{\boldsymbol{\varepsilon}}(\mathbf{q}) \sigma \mathrm{cq}, \sigma$$

$$\left\langle\Phi_k^0\left|K a^1\right| \Phi_n^0\right\rangle=-\frac{e}{m} \sqrt{\frac{\hbar^2}{2 \varepsilon_0 \Omega \mathcal{E} \mathbf{Q}}} \mathbf{P} \cdot \hat{\varepsilon}(\mathbf{Q})\mu \delta^3(\mathbf{P}$$ 发射顶点具有相同的形式， $\mathbf{Q} \rightarrow-\mathbf{Q}$. 我们必须认识到，自由电荷不能吸收或发射 (真实的) 光子，因为在这种转变中能量不能守恒。要看到这一 点，请考虑最初处于静止状态的电荷和具有波矢的入射 光子 $\mathbf{Q}$. 吸收光子后，粒子必须具有动量 $\hbar \mathbf{Q}$. 因此，我 们有 $$E_n^0=\hbar Q c, \mathbf{P}_n^0=\hbar \mathbf{Q}, E_k^0=\frac{\hbar^2 Q^2}{2 m}, \quad \mathbf{P}_k^0=\hbar \mathbf{Q}$$ 对于 (电荷 + 光子) 系统的初始和最终能量和动量。但 是由于粒子的最终速度是 $v_k=\hbar Q / m$, 能量守恒需要 $v_k=2 c$ 这是不可能的。 ${ }^{12}$ 光子在能量守恒不适用的虚 跃迁中被自由电荷吸收和发射。 很容易看出，束缚电荷的集合在光子的吸收和发射方面 具有不同的行为， 因为动量守恒的要求可以通过复合材 料的质心运动 (“反冲”) 和光子吸收来满足/emission 可 以与涉及内部状态的转换相关联。这样就可以同时满足 能量守恒和动量守恒的要求; 因此，原子和分子经历真 正的能量守恒跃迁，涉及光子的吸收和发射。一阶振幅 $V{f i}^{(1)}$ 在 $(10.35)$ 是非零的，并产生辐䁈吸收和发射的爱 因斯坦系数 [41]。

## 有限元方法代写

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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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