## 澳洲代写｜PHYC30018｜Quantum Physics量子计算 墨尔本大学

statistics-labTM为您提供墨尔本大学The University of Melbourne，简称UniMelb，中文简称“墨大”）Quantum Physics量子计算澳洲代写代考辅导服务！

Quantum mechanics plays a central role in our understanding of fundamental phenomena, primarily in the microscopic domain. It lays the foundation for an understanding of atomic, molecular, condensed matter, nuclear and particle physics.

## Quantum Physics量子计算问题集

(a) Suppose that the resistivity matrix is given by the classical result
$$\rho=\left(\begin{array}{cc} \rho_0 & -\rho_H \ \rho_H & \rho_0 \end{array}\right)$$
where $\rho_H=B /$ nec is the Hall resistivity and $\rho_0$ is the usual Ohmic resistivity. Find the conductivity matrix, $\sigma=\rho^{-1}$. Write it in the form:
$$\sigma=\left(\begin{array}{cc} \sigma_0 & \sigma_H \ -\sigma_H & \sigma_0 \end{array}\right) .$$
What are $\sigma_0$ and $\sigma_H$ ?
(b) Suppose $B=0$, so the Hall resistivity is zero. Notice that the Ohmic conductivity, $\sigma_0$, is just $1 / \rho_0$. In particular, note that $\sigma_0 \rightarrow \infty$ as $\rho_0 \rightarrow 0$. Now suppose $\rho_H \neq 0$. Show that $\sigma_0 \rightarrow 0$ as $\rho_0 \rightarrow 0$, so it is possible to have both $\sigma_0$ and $\rho_0$ equal to zero

This problem asks you to give a complete presentation of a calculation that is almost the same as one you saw in lecture.

Consider a constant electric field, $\vec{E}=\left(0, E_0, 0\right)$ and a constant magnetic field, $\vec{B}=\left(0,0, B_0\right)$.
(a) Choose an electrostatic potential $\phi$ and a vector potential $\vec{A}$ which describe the $\vec{E}$ and $\vec{B}$ fields, and write the Hamiltonian for a charged particle of mass $m$ and charge $q$ in these fields. Assume that the particle is restricted to move in the $x y$-plane.
(b) What are the allowed energies as a function of $B_0$ and $E_0$ ? Draw a figure to show how the Landau levels (energy levels when $E_0=0$ ) change as $E_0$ increases.

You will see the “standard presentation” of the Aharonov-Bohm effect in lecture, on the day that this problem set is due. The standard presentation has its advantages, and in particular is more general than the presentation you will work through in this problem. However, students often come away from the standard presentation of the Aharonov-Bohm effect thinking that the only way to detect this effect is to do an interference experiment. This is not true, and the purpose of this problem is to disabuse you of this misimpression before you form it.

As Griffiths explains on pages 385-387 (344-345 in 1st Ed.), the Aharonov-Bohm effect modifies the energy eigenvalues of suitably chosen quantum mechanical systems. In this problem, I ask you to work through the same physical example that Griffiths uses, but in a different fashion which makes more use of gauge invariance.

Imagine a particle constrained to move on a circle of radius $b$ (a bead on a wire ring, if you like.) Along the axis of the circle runs a solenoid of radius $a<b$, carrying a magnetic field $\vec{B}=\left(0,0, B_0\right)$. The field inside the solenoid is uniform

and the field outside the solenoid is zero. The setup is depicted in Griffiths’ Fig. 10.10. (10.12 in 1st Ed.)
(a) Construct a vector potential $\vec{A}$ which describes the magnetic field (both inside and outside the solenoid) and which has the form $A_r=A_z=0$ and $A_\phi=\alpha(r)$ for some function $\alpha(r)$. I am using cylindrical coordinates $z, r$, $\phi$.
(b) Since $\vec{\nabla} \times \vec{A}=0$ for $r>a$, it must be possible to write $\vec{A}=\vec{\nabla} f$ in any simply connected region in $r>a$. [This is a theorem in vector calculus.] Show that if we find such an $f$ in the region
$$r>a \text { and }-\pi+\epsilon<\phi<\pi-\epsilon,$$
then
$$f(r, \pi-\epsilon)-f(r,-\pi+\epsilon) \rightarrow \Phi \text { as } \epsilon \rightarrow 0 .$$
Here, the total magnetic flux is $\Phi=\pi a^2 B_0$. Now find an explicit form for $f$, which is a function only of $\phi$.
(c) Now consider the motion of a “bead on a ring”: write the Schrödinger equation for the particle constrained to move on the circle $r=b$, using the $\vec{A}$ you found in (a). Hint: the answer is given in Griffiths.
(d) Use the $f(\phi)$ found in (b) to gauge transform the Schrödinger equation for $\psi(\phi)$ within the angular region $-\pi+\epsilon<\phi<\pi-\epsilon$ to a Schrödinger equation for a free particle within this angular region. Call the original wave function $\psi(\phi)$ and the gauge-transformed wave function $\psi^{\prime}(\phi)$.
(e) The original wave function $\psi$ must be single-valued for all $\phi$, in particular at $\phi=\pi$. That is, $\psi(\pi-\epsilon)-\psi(-\pi+\epsilon) \rightarrow 0$ and $\frac{\partial \psi}{\partial \phi}(\pi-\epsilon)-\frac{\partial \psi}{\partial \phi}(-\pi+\epsilon) \rightarrow 0$ as $\epsilon \rightarrow 0$. What does this say about the gauge-transformed wave function? I.e., how must $\psi^{\prime}(\pi-\epsilon)$ and $\psi^{\prime}(-\pi+\epsilon)$ be related as $\epsilon \rightarrow 0$ ?
[Hint: because the $f(\phi)$ is not single valued at $\phi=\pi$, the gauge transformed wave function $\psi^{\prime}(\phi)$ is not single valued there either.]
(f) Solve the Schrödinger equation for $\psi^{\prime}$, and find energy eigenstates which satisfy the boundary conditions you derived in (e). What are the allowed energy eigenvalues?
(g) Undo the gauge transformation, and find the energy eigenstates $\psi(\phi)$ in the original gauge. Do the energy eigenvalues in the two gauges differ?
(h) Plot the energy eigenvalues as a function of the enclosed flux, $\Phi$. Show that the energy eigenvalues are periodic functions of $\Phi$ with period $\Phi_0$, where you must determine $\Phi_0$. For what values of $\Phi$ does the enclosed magnetic field have no effect on the spectrum of a particle on a ring? Show that the

Aharonov-Bohm effect can only be used to determine the fractional part of $\Phi / \Phi_0$.
[Note: you have shown that even though the bead on a ring is everywhere in a region in which $\vec{B}=0$, the presence of a nonzero $\vec{A}$ affects the energy eigenvalue spectrum. However, the effect on the energy eigenvalues is determined by $\Phi$, and is therefore gauge invariant. To confirm the gauge invariance of your result, you can compare your answer for the energy eigenvalues to Griffiths’ result, obtained using a different gauge.]

## 有限元方法代写

tatistics-lab作为专业的留学生服务机构，多年来已为美国、英国、加拿大、澳洲等留学热门地的学生提供专业的学术服务，包括但不限于Essay代写，Assignment代写，Dissertation代写，Report代写，小组作业代写，Proposal代写，Paper代写，Presentation代写，计算机作业代写，论文修改和润色，网课代做，exam代考等等。写作范围涵盖高中，本科，研究生等海外留学全阶段，辐射金融，经济学，会计学，审计学，管理学等全球99%专业科目。写作团队既有专业英语母语作者，也有海外名校硕博留学生，每位写作老师都拥有过硬的语言能力，专业的学科背景和学术写作经验。我们承诺100%原创，100%专业，100%准时，100%满意。

## 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。

## PHS2062｜Electromagnetism and optics电磁学和光学 蒙纳士大学

statistics-labTM为您提供蒙纳士大学（Monash University）Electromagnetism and optics电磁学和光学澳洲代写代考辅导服务！

Electromagnetism and optics fundamentally underpin such modern communication technologies as radio, cellular phones, GPS, Wi-Fi, laser and optical fibres.

1. Electromagnetism: classical electromagnetic theory; Maxwell’s equations; Gauss’s law; Faraday’s law; Ampere-Maxwell law; electric and magnetic fields in vacuum; electrodynamics.

2. Optics: geometric ray tracing; optical cavities; electromagnetic waves; Gaussian beam propagation; multiple-beam interference; polarisation; birefringence.

## Mathematical Optimisation for Data Science 数据科学的数学优化案例

(b) Which of the following vector fields could describe an electric field? Say yes or no for each, and give a very brief reason.
(i) $\vec{E}(\vec{r})=x \hat{e}_x-y \hat{e}_y$.
(ii) $\vec{E}(\vec{r})=y \hat{e}_x+x \hat{e}_y$.
(iii) $\vec{E}(\vec{r})=y \hat{e}_x-x \hat{e}_y$.

The curl of an electrostatic field must be zero, but otherwise there is no restriction. So the answer follows as
(i) $\vec{\nabla} \times \vec{E}(\vec{r})=\left(\frac{\partial E_y}{\partial x}-\frac{\partial E_x}{\partial y}\right) \hat{e}_z+\ldots=\overrightarrow{0}$. YES, it describes an electric field.
(ii) $\vec{\nabla} \times \vec{E}(\vec{r})=(1-1) \hat{e}_z=0$. YES, it describes an electric field.
(iii) $\vec{\nabla} \times \vec{E}(\vec{r})=(-1-1) \hat{e}_z=-2 \hat{e}_z$. NO, it does not describe an electric field.

A very long cylindrical object consists of an inner cylinder of radius $a$, which has a uniform charge density $\rho$, and a concentric thin cylinder, of radius $b$, which has an equal but opposite total charge, uniformly distributed on the surface.
(a) Calculate the electric field everywhere.

(a) This problem has enough symmetry to allow a solution by Gauss’s law. In particular, symmetry considerations imply that the electric field will point radially outward, and will have a magnitude that depends only on the distance from the axis. Following Griffiths, we use $s$ for the distance from the $z$-axis, and $\hat{s}$ for a unit vector pointing radially outward from the axis, and of course we choose the $z$-axis to be the axis of the cylindrical object. Then
$$\vec{E}=E(s) \hat{s} .$$
To evaluate $E(s)$, we apply Gauss’s law to a Gaussian cylinder of length $\ell$, concentric with the $z$-axis. Then
$$\oint \vec{E} \cdot \mathrm{d} \vec{a}=\frac{Q_{\text {enc }}}{\epsilon_0}=2 \pi s \ell E(s) .$$
For $sb$ the enclosed charge is zero, so $E(s)=0$. Putting this together,
$$\vec{E}=\frac{\rho}{2 \epsilon_0} \begin{cases}s \hat{s} & \text { if } sa .\end{cases}$$

## 有限元方法代写

tatistics-lab作为专业的留学生服务机构，多年来已为美国、英国、加拿大、澳洲等留学热门地的学生提供专业的学术服务，包括但不限于Essay代写，Assignment代写，Dissertation代写，Report代写，小组作业代写，Proposal代写，Paper代写，Presentation代写，计算机作业代写，论文修改和润色，网课代做，exam代考等等。写作范围涵盖高中，本科，研究生等海外留学全阶段，辐射金融，经济学，会计学，审计学，管理学等全球99%专业科目。写作团队既有专业英语母语作者，也有海外名校硕博留学生，每位写作老师都拥有过硬的语言能力，专业的学科背景和学术写作经验。我们承诺100%原创，100%专业，100%准时，100%满意。

## 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。