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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
$$
\begin{array}{ll}
E_n^0=\hbar Q c, & \mathbf{P}_n^0=\hbar \mathbf{Q}, \
E_k^0=\frac{\hbar^2 Q^2}{2 m}, & \mathbf{P}_k^0=\hbar \mathbf{Q},
\end{array}
$$
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.

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 ).

电动力学代考

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

本节专门介绍基本图的评估概要,图 10.1-10.3 使用库仑规范哈密顿量来计算与量子化 电磁场相互作用的带电粒子;之后,扩展到涉及许多电荷(原子、分子、凝聚态物质、 等离子体等) 的有趣物理案例将被视为非常简单。所需的矩阵元素在附录中给出F.
图 10.1 显示了对应于一阶微扰理论的原始吸收和发射顶点;没有分母可以评估。考虑 吸收顶点;根据等式。(F.1.5),扰动算子是
$$
\mathrm{K} a^1=-\sum \mathbf{q}, \sigma \mathbf{f} e(\mathbf{q}) \cdot \hat{\boldsymbol{\varepsilon}}(\mathbf{q}) \sigma \mathrm{C} \mathbf{q}, \sigma
$$
如果自由电荷吸收光子的初始和最终状态是 $\Phi_n^0=\left|\varphi \mathbf{P}, \mu\left[n_{\mathbf{Q}}\right]\right\rangle$ 和
$\Phi_k^0=\left|\varphi_{\mathbf{P}^{\prime}}, \mu\left[n_{\mathbf{Q}}-1\right]\right\rangle$, 矩阵元素分别为 (cf. (F.1.8))
$$
\left\langle\Phi_k^0\left|\mathrm{~K} a^1\right| \Phi_n^0\right\rangle=-\frac{e}{m} \sqrt{\frac{\hbar^2}{2 \varepsilon_0 \Omega \mathcal{E} \mathbf{Q}}} \mathbf{P} \cdot \hat{\varepsilon}(\mathbf{Q})_\mu \delta^3\left(\mathbf{P}+\hbar \mathbf{Q}-\mathbf{P}^{\prime}\right) .
$$
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
$$
E_n^0=\hbar Q c, \quad \mathbf{P}_n^0=\hbar \mathbf{Q}, \quad E_k^0=\frac{\hbar^2 Q^2}{2 m}, \quad \mathbf{P}_k^0=\hbar \mathbf{Q}
$$
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。

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

当该场与光束的传播方向成一定角度时,在均匀静态场 (电场或磁场) 中被分子散射的 光通常是椭圆偏振的。Kerr 和 Cotton-Mouton 效应分别对应于电场和磁场情况 [60]。 在没有任何外部场 (“光学活动”) 的情况下偏振平面的旋转是手性物质的特征; 通过沿 光的方向施加的外部磁场 (法拉第效应),可以在任何流体物质中引起相同的效应。在 没有施加外部场的情况下,类似的双折射现象可能是由强大激光的强光场引起的。
某些类型的过程无法使用简单交互作用 (10.176) 来描述;例如,它不能描述各向同性介 质中的手性 (光束偏振态的变化),因为远离共振的旋光角完全取决于 $T$ 矩阵的虚部。 自从 $d$ 是实算子,(10.176) 导致实 $\mathrm{T}$ 矩阵元素;广义抗磁磁化率 (9.136) 也是纯实数, 因此不会影响旋光性。对于这种情况,必须引入涉及磁感应矢量的磁偶极子相互作用 $\mathbf{B}$; 磁偶极子算符是纯虚数。这意味着放弃电场近似均匀的假设,并且为了保持一致性,还 必须包括与电场梯度耦合的电四极子项;因此,在下一个多极近似中有
$$
V^1=-\mathbf{d} \cdot \mathbf{E}^{\perp}-\mathbf{m} \cdot \mathbf{B}-\mathbf{Q}: \nabla \mathbf{E}^{\perp}
$$
还应该提到的是,最近的研究表明,(激光)光可以设计成具有扭曲或螺旋相结构,其 特征在于将轨道角动量分配给光子。场变量的平面波描述 $\left(\mathbf{A}, \mathbf{B}, \mathbf{E}^{\perp}\right)$ 无法描述此类属 性,因此需要不同的公式。给定必要的场变量展开 (例如 (7.246)),此处总结的基于 Kramers-Heisenberg 色散公式的光散射微尤理论可以被修改并识别新现象。详细的研 究可以在 [61] 中找到; 一个引人注目的预测是新型手性光学双折射效应,其中分子四 极算子起着至关重要的作用。
稀薄介质光学双折射的量子力学方法考虑了被分子散射的光子束。对于这样的系统,初 始状态可以由处于给定初始状态的分子和沿一个偏振方向并以单一指定模式线性偏振的 光子表示 $\mathbf{k}$ 领域的。在遥远的末来,系统的最终状态是分子处于原始状态,但认识到光 子从一个偏振方向转移的概率非零 $\lambda$ 对另一个 $\lambda^{\prime}$ ,动量没有变化。因此,尽管这是前向散 射的情况,但发生了转变 (“偏振翻转”),并且基于 $\mathrm{T}$ 矩阵的散射理论仍然适用于计算 此概率。在双折射实验中,人们对入射和出射光束的观察是它们的强度和通过方位角和 椭圆率角表示的每个偏振椭圆的特征; 斯托克斯参数形式主义 (第 7 章) 很好地总结了 这些可观测值。

物理代写|电动力学代写electromagnetism代考

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