# 物理代写|电动力学代写electromagnetism代考|PHYC20014

#### Doug I. Jones

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A classical electromagnetic field is conveniently described in terms of its amplitude and phase; it is of interest, therefore, to establish the limitations imposed on such a description by the quantum theory. In $\$ 3.4$we noted that there exists a classical transformation that reduces the Hamiltonian for an oscillator to cyclic form; this involves a canonical transformation from position$(x)$and momentum$(p)$variables to action-angle variables,$(J, \theta)$. In quantum theory the analogous transformation is to the number operator,$\mathrm{n}$, and a conjugate phase angle operator$\theta$, and it is not surprising that classical electrodynamics was initially quantised in terms of this pair. The approach can still be seen (1930) in the first edition of Dirac’s famous book where there is only the briefest mention of annihilation and creation operators which were adopted universally soon after. Dirac noted that the idea of conjugate number and phase operators was deeply problematic in quantum theory. By analogy with the classical discussion of a mode a ‘phase operator’$\theta$is introduced such that the operator$\mathrm{E}(\theta)=\exp (i \theta)$derived from it appears in the polar decomposition n$^4$the annihilation operator $$\mathrm{c}=e^{i \theta} \frac{1}{\sqrt{\mathrm{n}}}$$ There is then a formal commutation relation, $$[\mathrm{n} \hbar, \theta]=i \hbar 1$$ Equation (7.96) does not yield the usual uncertainty relation for conjugate variables since the eigenvalues of$\theta$are assumed to lie in the interval$[0,2 \pi]$and$n$is bounded from below with discrete spectrum. Dirac was aware that no such$\thetacan be defined, and that its exponential is not unitary [10]. Much later it was shown that no unitary operator is available for the decomposition (7.95), and modified definitions were proposed for an adjoint pair of operators [11] $$\mathrm{E}(\theta) \equiv e^{i \theta}=\frac{1}{\sqrt{\mathrm{n}+1}} \mathrm{c}, \quad \mathrm{E}(\theta)^{+} \equiv e^{-i \theta^{+}}=\mathrm{c}^{+} \frac{1}{\sqrt{\mathrm{n}+1}},$$ together with their ‘trigonometric’ companions, usually, and loosely, called sin and cos, \begin{aligned} & \cos (\theta)=\frac{1}{2}\left(e^{i \theta}+e^{-i \theta^{+}}\right) \ & \sin (\theta)=\frac{1}{2 i}\left(e^{i \theta}-e^{-i \theta^{+}}\right) \end{aligned} ## 物理代写|电动力学代写electromagnetism代考|Quantisation of the Field Operators If no gauge condition is imposed, the vector potential operator has a longitudinal degree of freedom in addition to the two transverse degrees of freedom that describe polarised photons; similarly its conjugate\pi$also has three degrees of freedom. Their mutual commutation relation is canonical, $$\left[a(\mathbf{x})^r, \pi\left(\mathbf{x}^{\prime}\right)s\right]=i \hbar \delta{r s} \delta^3\left(\mathbf{x}-\mathbf{x}^{\prime}\right)$$ which implies that the vector potential operator’s canonical conjugate,$\pi$, may be realised as a functional derivative, $$\pi_s=-i \hbar \frac{\delta}{\delta \mathrm{a}_s} .$$ Since the commutation relations fix the Hilbert of states, it will be ‘too large’, and at the outset the calculations will involve the additional degrees of freedom; an extra condition on the state space is thus required to pick out the physically significant states. If we choose a representation in which the vector potential operator is diagonal, $$\mathbf{a} \mid \mathbf{a})=\mathbf{a}|\mathbf{a}\rangle$$ the ‘eigenvalue’ a is a classical vector potential and the ‘eigenfunctions’ are wave functionals$\Phi[\mathbf{a}]=\langle\mathbf{a} \mid \Phi\rangle$. The classical primary constraint for the free-field problem $$\boldsymbol{\nabla} \cdot \boldsymbol{\pi} \approx 0$$ after quantisation must be expressed as a condition on the states$\Phi$to pick out the physical Hilbert space. In close correspondence with$\$3.6$ we define the Gauss’s law operator for the free field as $\mathrm{G}_0=\nabla \cdot \pi$ and require that the physical states for the field be annihilated by $\mathrm{G}_0$ :
$$\mathrm{G}_0 \Phi[\mathbf{a}]=0$$
We may interpret the Gauss’s law operator by considering the effect of a continuous linear superposition of it with a smooth scalar field $\sigma$,
$$\mathcal{G}_0^\sigma=\int \mathrm{G}_0 \sigma \mathrm{d}^3 \mathbf{x}$$

# 电动力学代考

## 物理代写|电动力学代写electromagnetism代考|The Number and Phase Representation

$$\mathrm{G}_0 \Phi[\mathbf{a}]=0$$

$$\mathcal{G}_0^\sigma=\int \mathrm{G}_0 \sigma \mathrm{d}^3 \mathbf{x}$$

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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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