物理代写|电动力学代写electromagnetism代考|S-Matrix Theory and Linewidths

Doug I. Jones

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物理代写|电动力学代写electromagnetism代考|S-Matrix Theory and Linewidths

We now consider the modifications of the S-matrix theory required for the description of spectral linewidths. In the conventional presentation of the quantum theory of atoms and molecules, it is customary to regard the atomic system as a closed entity; it is recognised, of course, that an ‘isolated molecule’ is a fiction and that in real practical situations atoms and molecules are always coupled to their environment, to varying degrees, in what may be called ‘persistent interactions’ [21], [22]. As a consequence, infinitely long-lived discrete states with perfectly sharp energies in the sense of Bohr’s ‘stationary states’, other than ground states, do not exist; excited energy levels gain widths (equivalently, finite life times) in various ways which are revealed in spectroscopy as lineshapes. The description of this situation in quantum theory is based on coupling between the idealised discrete states and one or more continua that characterise the ‘environment’, which typically is taken as macroscopic. Macroscopic systems in quantum theory have purely continuous spectra and are describable by quantum field theories; examples include the electromagnetic field itself, a crystalline host lattice which supports phonons, solutions and liquids characterised by Brownian motion and so on, which may be described as ‘heat baths’ or ‘reservoirs’.

A formal solution to this problem can be achieved by the refinement of the scattering theory summarised at the end of $\$ 6.5$[23], [24]. The reaction matrix$\mathbf{D}$is closely related to the matrix$\mathbf{\Sigma}$in (6.32); only its on-energy-shell elements$\left{D_{k n}\right}$are required for the T-matrix, and these can be seen to be gauge invariant from the analysis of the gauge dependence of$\boldsymbol{\Sigma}$presented in what follows. In practice the Heitler equation (6.91) has not been used in atomic and molecular quantum electrodynamics. Instead the results of the non-resonant perturbation theory described in$\$10.2 .2$ (or other formulations that are equivalent to it) are modified by the incorporation of phenomenological damping factors that account for energy level shifts and lifetimes due to resonant interactions. In condensed media there are many other factors that give rise to linewidths that can be incorporated in this way; such factors are constructed as gauge-invariant quantities.

物理代写|电动力学代写electromagnetism代考|Friedrichs–Fano Models

An early treatment of linewidths in continuous absorption spectra, and one that is highly instructive, is the theory of atomic autoionisation due to Fano [28], [29]. Asymmetric lineshapes associated with so-called Fano resonances have been observed in diverse areas of physics [30]-[32]. In the simplest case considered, a single discrete state above the first ionisation threshold is coupled to a continuum. ${ }^8$ These states correspond to specified atomic ‘configurations’ constructed in the independent electron approximation, so that they do not diagonalise the atomic Hamiltonian $\mathrm{H}$, and the true states of the atom arise from ‘configuration interaction’.

Let the normalised discrete state be denoted by $\varphi_k$, and the continuum states by $\psi\left(E^{\prime}\right)$ with $E_1 \leq E^{\prime} \leq E_2$; the continuum states are normalised according to Dirac’s delta function prescription, $\S 5.2 .1$ and are orthogonal to the discrete state. Then we restrict attention to the sub-matrix block of the Hamiltonian matrix with elements
\begin{aligned} \left\langle\varphi_k|\mathrm{H}| \varphi_k\right\rangle & =E_k \ \left\langle\psi\left(E^{\prime}\right)|\mathrm{H}| \varphi_k\right\rangle & =V_k\left(E^{\prime}\right) \ \left\langle\psi\left(E^{\prime \prime}\right)|\mathrm{H}| \psi\left(E^{\prime}\right)\right\rangle & =E^{\prime} \delta\left(E^{\prime \prime}-E^{\prime}\right) \end{aligned}
with $E_k$ lying in the energy range of the considered continuum.
The solution of the Schrödinger equation
$$H \Psi(E)=E \Psi(E)$$
restricted to this subspace may be expressed as the superposition
$$\Psi(E)=a_k(E) \varphi_k+\int_{E_1}^{E_2} b\left(E: E^{\prime}\right) \psi\left(E^{\prime}\right) \mathrm{d} E^{\prime}$$
with coefficients $a, b$ that satisfy
\begin{aligned} E_k a_k(E)+\int_{E_1}^{E_2} V_k\left(E^{\prime}\right)^* b\left(E: E^{\prime}\right) \mathrm{d} E^{\prime} & =E a_k(E) \ V_k\left(E^{\prime}\right) a_k(E)+E^{\prime} b\left(E: E^{\prime}\right) & =E b\left(E: E^{\prime}\right) \end{aligned}
according to (10.77)-(10.79).

电动力学代考

$6.5 [23]、[24]末尾总结的散射理论来实现。反应矩阵\mathbf{D}与 (6.32) 中的矩阵\mathbf{\Sigma}D密切相关；T 矩阵只需要它的能量壳元素\left{D_{kn}\right} ，从对\boldsymbol{\Sigma}的规范依赖性的分析中可以看出这些元素是规范不变的在接下来的内容中。实际上，海特勒方程 (6.91) 尚未用于原子和分子量子电动力学。取而代之的是\$ 10.2 .2中描述的非共振微扰理论的结果Σ\left{D_{k n}\right}\left{D_{k n}\right}Σ$10.2.2（或与其等效的其他公式）通过合并现象学阻尼因子进行修改，这些因子解释了由于共振相互作用引起的能级变化和寿命。在浓缩介质中，还有许多其他因素会导致可以以这种方式合并的线宽；这些因素被构造为规范不变的数量。 物理代写|电动力学代写electromagnetism代考|Friedrichs–Fano Models Fano [28]、[29] 提出的原子自电离理论是连续吸收光谱 中线宽的早期处理方法，也是一种很有启发性的方法。 与所谓的 Fano 共振相关的不对称线形已在物理学的不 同领域被观察到 [30]-[32]。在考虑的最简单的情况下， 高于第一电离阈值的单个离散状态耦合到连续体。${ }^8$这些 状态对应于在独立电子近似中构造的指定原子”配置”，因 此它们不会对角化原子哈密顿量$\mathrm{H}$，并且原子的真实状 态来自“配置相互作用”。 让归一化离散状态用|varphi_k表示$\varphi_k$，连续状态用$\psi\left(E^{\prime}\right)$和$E_1 \leq E^{\prime} \leq E_2$；连续状态根据 Dirac 的 delta 函数规定$\S 5.2 .1$进行归一化，并且与离散状态正 交。然后我们将注意力限制在具有元素的哈密顿矩阵的 子矩阵块上 $$\left\langle\varphi_k|\mathrm{H}| \varphi_k\right\rangle=E_k\left\langle\psi\left(E^{\prime}\right)|\mathrm{H}| \varphi_k\right\rangle \quad=V_k\left(E^{\prime}\right)$$ 与$E_k$位于所考虑的连续体的能量范围内。 薛定谔方程 $$H \Psi(E)=E \Psi(E)$$ 限制在这个子空间的解可以表示为叠加 $$\Psi(E)=a_k(E) \varphi_k+\int_{E_1}^{E_2} b\left(E: E^{\prime}\right) \psi\left(E^{\prime}\right) \mathrm{d} E^{\prime}$$$, a, b\$ 满足
$$E_k a_k(E)+\int_{E_1}^{E_2} V_k\left(E^{\prime}\right)^* b\left(E: E^{\prime}\right) \mathrm{d} E^{\prime}=E a_k(E)$$

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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