# 物理代写|电动力学代写electromagnetism代考|PHYS2001

#### Doug I. Jones

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

The light scattered by a molecule in a uniform static field (either electric or magnetic), when this field makes an angle to the propagation direction of the light beam, is in general elliptically polarised. The Kerr and Cotton-Mouton effects correspond to the electric and magnetic field cases, respectively [60]. The rotation of the plane of polarisation in the absence of any external fields (‘optical activity’) is the characteristic property of chiral substances; the same effect can be induced in any fluid substance by an external magnetic field applied along the direction of the light (the Faraday effect). Analogous birefringence phenomena in the absence of applied external fields may be induced by the intense optical fields of powerful lasers.

Certain kinds of processes cannot be described using the simple interaction (10.176); for example, it cannot describe chirality (a change in the polarisation state of the beam) in isotropic media since the optical rotation angle far from resonance depends purely on the imaginary part of the T-matrix. Since $d$ is a real operator, (10.176) leads to a real T-matrix element; the generalised diamagnetic susceptibility $(9.136)$ is also purely real and so cannot contribute to optical activity. For such a case one must introduce the magnetic dipole interaction involving the magnetic induction vector $\mathbf{B}$; the magnetic dipole operator is pure imaginary. This means giving up the assumption that the electric field is approximately uniform, and for consistency one must also include the electric quadrupole term that couples to the electric field gradient; thus, in the next multipolar approximation one has $$V^1=-\mathbf{d} \cdot \mathbf{E}^{\perp}-\mathbf{m} \cdot \mathbf{B}-\mathbf{Q}: \nabla \mathbf{E}^{\perp}$$
It should also be mentioned that recent work has shown that (laser) light can be engineered to possess a twisting or helical phase structure that can be characterised by assigning orbital angular momentum to photons. The plane wave description of the field variables $\left(\mathbf{A}, \mathbf{B}, \mathbf{E}^{\perp}\right.$ ) cannot describe such properties, and a different formulation is required. Given the requisite field variable expansions (e.g. (7.246)), the perturbation theory of light scattering summarised here, based on the Kramers-Heisenberg dispersion formula, can be reworked and novel phenomena identified. A detailed study can be found in [61]; a striking prediction is of novel chiroptical birefringence effects in which the molecular quadrupole operator plays an essential role.

The quantum mechanical approach to the optical birefringence of a rarefied medium considers a beam of photons being scattered by a molecule. For such a system, the initial state can be represented by a molecule in a given initial state and photons linearly polarised along one direction of polarisation and in a single specified mode $\mathbf{k}$ of the field. In the distant future, the final state of the system has the molecule in its original state but recognises that there is a non-zero probability that photons have transferred from one polarisation direction $\lambda$ to the other $\lambda^{\prime}$, without a change of momentum. Thus, although this is a case of forward scattering, a transition (the ‘polarisation flip’) has occurred and the scattering theory based on the T-matrix is still appropriate for the calculation of this probability. The observations that one makes on the incident and emergent light beams in a birefringence experiment are their intensities and the characteristics of the polarisation ellipse of each expressed through the azimuth and ellipticity angles; these observables are summarised elegantly by the Stokes parameter formalism (Chapter 7).

## 物理代写|电动力学代写electromagnetism代考|Level Shifts and Self-Interaction

The diagram expansion is equally applicable to the perturbation theory approach to energy level shifts; recall from Chapter 6 that a discrete energy level of the reference system can be related formally to an energy level of the full problem by solving for the roots of the equation,
\begin{aligned} E_n & =E_n^0+\Delta_n(E) \ & \approx E_n^0+\Delta_n\left(E_n^0\right) \end{aligned}
Where
$$\Delta_n\left(E_n^0\right)=\left\langle\Phi_n^0\left|\left(\mathrm{~V}+\mathrm{VG}^0\left(E_n^0\right) \mathrm{V}+\ldots\right)^{\prime}\right| \Phi_n^0\right\rangle$$
This is essentially the same expansion as for the T-matrix with the supplementary condition that $\Phi_n^0$ must be excluded from sums over complete sets of states.

A diagram in Figure 10.2 can also be repurposed as the building block of the diagrammatic expansion for the energy shift $\Delta E$ due to Van der Waals interactions of pairs of neutral atoms/molecules/chromophores, supposed electronically distinct, via the exchange of virtual photons. One can imagine making a copy of diagram (10.2iia) and, taking the two copies together, joining the external lines 1 to $1^{\prime}, 2$ to $2^{\prime}$; the composite diagram is then relabelled so that the initial and final states are the same, $(1,2)$, and the virtual intermediate states are $\left(1^{\prime}, 2^{\prime}\right)$. There are six topologically distinct diagrams that can be formed in this way when all time orderings of the vertices are allowed for (excluding interchange of the two particles); the diagrams have four vertices and so, if $\Phi_n^0$ in (10.206) is taken to be the tensor product of the ground state of the atomic system and the photon vacuum, the diagrams describe the van der Waals interaction to order $e^4$ (i.e. $\alpha^2$ ); within the electric dipole approximation they lead to [11], [72],
$$\Delta E \approx \frac{C}{R^7}, \quad R \gg \lambda ; \quad \Delta E \approx \frac{C^{\prime}}{R^6}, \quad R \ll \lambda$$

# 电动力学代考

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

$$V^1=-\mathbf{d} \cdot \mathbf{E}^{\perp}-\mathbf{m} \cdot \mathbf{B}-\mathbf{Q}: \nabla \mathbf{E}^{\perp}$$

## 物理代写|电动力学代写electromagnetism代考|Level Shifts and Self-Interaction

$$E_n=E_n^0+\Delta_n(E) \quad \approx E_n^0+\Delta_n\left(E_n^0\right)$$

$$\Delta_n\left(E_n^0\right)=\left\langle\Phi_n^0\left|\left(\mathrm{~V}+\mathrm{VG}^0\left(E_n^0\right) \mathrm{V}+\ldots\right)^{\prime}\right| \Phi_n^0\right\rangle$$

$$\Delta E \approx \frac{C}{R^7}, \quad R \gg \lambda ; \quad \Delta E \approx \frac{C^{\prime}}{R^6}, \quad R \ll \lambda$$

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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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