### 物理代写|光学代写Optics代考|PHS2062

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• Statistical Inference 统计推断
• Statistical Computing 统计计算
• (Generalized) Linear Models 广义线性模型
• Statistical Machine Learning 统计机器学习
• Longitudinal Data Analysis 纵向数据分析
• Foundations of Data Science 数据科学基础

## 物理代写|光学代写Optics代考|Hamiltonian Mechanic

Hamiltonian mechanics was formulated by William Rowan Hamilton in 1833. Hamiltonian mechanics is equivalent to Newton’s laws of motion but provides a simplification of the analysis for many dynamical systems. Another approach is Lagrangian mechanics, which we leave to the reader as a topic for independent study. In Hamiltonian mechanics, a system is described by canonically conjugate variables denoted by $q_i$ and $p_i$ :
$$q_1, q_2, \ldots, q_i, \ldots ; p_1, p_2, \ldots, p_i, \ldots$$
$q_i$ and $p_i$ are also called the generalized position and momentum coordinates, respectively. For example, $q_1, q_2$ and $q_3$ may refer to the actual position coordinates $(x, y, z)$ of a particle and $p_1, p_2$ and $p_3$ correspond to its linear momentum $\left(p_x, p_y\right.$ and $\left.p_2\right)$. If there is more than one particle, then $q_4, q_5, q_6, p_4, p_5$ and $p_6$ are the corresponding variables for the second particle, and so on. In general, $q_i$ and $p_i$ may represent dynamic variables other than position and momentum, depending on the system. For example, to describe a pendulum (Exercise 1.1), it is easier to assign $q_i$ as the angle of the pendulum and $p_i$ as the angular momentum. The $q_i$ and $p_i$ variables, if they are canonically conjugate variables, satisfy the Hamilton equations:

$$\begin{gathered} \frac{d q_i}{d t}=\frac{\partial H}{\partial p_i} \ \frac{d p_i}{d t}=-\frac{\partial H}{\partial q_i} \end{gathered}$$
where $H$ is the Hamiltonian and $t$ is the time. The Hamiltonian is the total energy of the system (kinetic energy plus potential energy) expressed in terms of the generalized coordinates.

To illustrate Hamilton’s approach, let us find the equations of motion for a particle of mass $m$ in a one-dimensional potential, $U(x)$, shown in Fig. 1.1. Although $U(x)$ is actually the potential energy, physicists often abbreviate this simply as “the potential”. In this example, suppose the generalized coordinates $\left(q_i, p_i\right)$ are the position $(x)$ and momentum $(p)$ of the particle:
$$\begin{gathered} q \rightarrow x \ p \rightarrow m \frac{d x}{d t} \end{gathered}$$
The Hamiltonian is the total energy (kinetic energy plus potential energy) expressed in terms of the generalized coordinates from Eqs. (1.4) and (1.5):
$$H-\frac{p^2}{2 m}+U(x)$$

## 物理代写|光学代写Optics代考|Canonical Quantization

Canonical quantization is a prescribed method of finding the Hamiltonian of a quantum system. The procedure was developed by Paul Dirac in 1925 (Fig. 1.2). Dirac proposed that any system for which we have a classical description can be quantized according to the procedure of canonical quantization. In canonical quantization, the generalized coordinates of the classical description, found by Hamilton’s approach (described in the previous section), are replaced by the corresponding quantum operators (denoted by a “hat”, )):
$$H\left(q_1, \ldots, q_i, \ldots ; p_1, \ldots, p_i, \ldots\right) \stackrel{\substack{\text { canonical } \ \text { quantization }}}{\longrightarrow} \widehat{H}\left(\widehat{q}_1, \ldots, \widehat{q}_i, \ldots ; \widehat{p}_1, \ldots, \widehat{p}_i, \ldots\right)$$ where the classical description is on the left and the quantum description is on the right. The Hamiltonian of the quantum system, $\widehat{H}$, is expressed in terms of the generalized coordinates (now operators) on the right-hand side of Eq. (1.14). For example, according to Sect. 1.1, the generalized coordinates for a particle of mass $m$ in a potential, $U(x)$, are $x$ and $p$. The Hamiltonian for the corresponding quantum system becomes
\begin{aligned} & \text { canonical } \ & H=\frac{p^2}{2 m}+U(x) \stackrel{\text { quantization }}{\longrightarrow} \widehat{H}=\frac{\widehat{p}^2}{2 m}+U(\hat{x}) \ & \end{aligned}
Once you know $\widehat{H}$ of the quantum system, you can determine its quantum properties from the time-dependent Schrodinger equation:
$$i \hbar \frac{\partial|\psi\rangle}{\partial t}=\widehat{H}|\psi\rangle$$
where $|\psi\rangle$ is the state of the system and $\hbar$ is the reduced Planck constant $(\hbar=h / 2 \pi)$. You may remember from introductory quantum mechanics that Eq. (1.16) reduces to the time-independent Schrodinger equation for stationary states:
$$\widehat{H}\left|\psi_n\right\rangle=E_n\left|\psi_n\right\rangle$$
where $E_n$ are the eigenenergies and $\left|\psi_n\right\rangle$ are the eigenstates (basis states) of the system.

# 光学代考

## 物理代写|光学代写Optics代考|Hamiltonian Mechanic

$$q_1, q_2, \ldots, q_i, \ldots ; p_1, p_2, \ldots, p_i, \ldots$$
$q_i$ 和 $p_i$ 也分别称为广义位置和动量坐标。例如， $q_1, q_2$ 和 $q_3$ 可参考实际位置坐标 $(x, y, z)$ 一个 粒子和 $p_1, p_2$ 和 $p_3$ 对应于它的线性动量 $\left(p_x, p_y\right.$ 和 $\left.p_2\right)$. 如果有一个以上的粒子，则 $q_4, q_5, q_6, p_4, p_5$ 和 $p_6$ 是第二个粒子的相应变量，依此类推。一般来说， $q_i$ 和 $p_i$ 可能表示位置 和动量以外的动态变量，具体取决于系统。例如，要描述一个钟摆（练习 1.1），更容易分配 $q_i$ 作为摆的角度和 $p_i$ 作为角动量。这 $q_i$ 和 $p_i$ 变量，如果它们是典型共轭变量，则满足 Hamilton 方程:
$$\frac{d q_i}{d t}=\frac{\partial H}{\partial p_i} \frac{d p_i}{d t}=-\frac{\partial H}{\partial q_i}$$

$$q \rightarrow x p \rightarrow m \frac{d x}{d t}$$

$$H-\frac{p^2}{2 m}+U(x)$$

## 物理代写|光学代写Optics代考|Canonical Quantization

1.2）。狄拉克提出，任何具有经典描述的系统都可以根据规范量化过程进行量化。在规范量化 中，通过哈密顿方法 (在上一节中描述) 找到的经典描述的广义坐标被相应的量子算子 (用“帽 子”表示) 代替:
$$H\left(q_1, \ldots, q_i, \ldots ; p_1, \ldots, p_i, \ldots\right) \stackrel{\text { canonical quantization }}{\longrightarrow} \widehat{H}\left(\hat{q}_1, \ldots, \hat{q}_i, \ldots ; \hat{p}_1, \ldots, \hat{p}_i\right.$$

$$\text { canonical } \quad H=\frac{p^2}{2 m}+U(x) \stackrel{\text { quantization }}{\longrightarrow} \widehat{H}=\frac{\hat{p}^2}{2 m}+U(\hat{x})$$

$$i \hbar \frac{\partial|\psi\rangle}{\partial t}=\widehat{H}|\psi\rangle$$

$$\widehat{H}\left|\psi_n\right\rangle=E_n\left|\psi_n\right\rangle$$

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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。