## 物理代写|光学代写Optics代考|CSCl031

statistics-lab™ 为您的留学生涯保驾护航 在代写光学Optics方面已经树立了自己的口碑, 保证靠谱, 高质且原创的统计Statistics代写服务。我们的专家在代写光学Optics代写方面经验极为丰富，各种代写光学Optics相关的作业也就用不着说。

• Statistical Inference 统计推断
• Statistical Computing 统计计算
• Advanced Probability Theory 高等概率论
• Advanced Mathematical Statistics 高等数理统计学
• (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|Entangled States

In Chap. 6 (Sect. 6.7), we saw that the beam splitter output was an entangled state. Let us examine a two-photon state, which can be written in general as a product of two superpositions:
$$|\psi\rangle=|\psi\rangle_1|\psi\rangle_2=\left(\alpha_1|0\rangle_1+\beta_1|1\rangle_1\right)\left(\alpha_2|0\rangle_2+\beta_2|1\rangle_2\right)$$
where the subscript on each ket indicates the photon number (photon 1 or photon 2 ), and $|0\rangle$ and $|1\rangle$ represent the two possible orthogonal states of each photon-for example, horizontal and vertical polarization, or two different paths in a beam splitter or interferometer. Expanding this two-photon state gives
$$|\psi\rangle=\alpha_1 \alpha_2|0\rangle_1|0\rangle_2+\alpha_1 \beta_2|0\rangle_1|1\rangle_2+\beta_1 \alpha_2|1\rangle_1|0\rangle_2+\beta_1 \beta_2|1\rangle_1|1\rangle_2$$
Suppose you are given a composite state of two photons (e.g., Eq. (8.2)) and asked for the state of the individual photons. To answer this question, you would work backward to factor the state, obtaining Eq. (8.1). This is known as a separable state.

There are some states for which this factoring is impossible; that is, you cannot write the composite state as a product of the individual states:
$$|\psi\rangle \neq|\psi\rangle_1|\psi\rangle_2$$
These are known as entangled states. In entangled states, you cannot talk about the state of the photons individually – they are somehow intertwined. Note that a general two-photon state, $|\psi\rangle=\alpha_{00}|0\rangle_1|0\rangle_2+\alpha_{01}|0\rangle_1|1\rangle_2+\alpha_{10}|1\rangle_1|0\rangle_2+\alpha_{11}|1\rangle_1|1\rangle_2$, is usually entangled rather than separable – entanglement is normal in quantum mechanics!

There are four entangled two-photon states that are commonly encountered, known as the Bell states (we have dropped the particle subscripts):
\begin{aligned} \left|\Phi^{+}\right\rangle & =\frac{1}{\sqrt{2}}(|00\rangle+|11\rangle) \ \left|\Psi^{+}\right\rangle & =\frac{1}{\sqrt{2}}(|01\rangle+|10\rangle) \ \left|\Phi^{-}\right\rangle & =\frac{1}{\sqrt{2}}(|00\rangle-|11\rangle) \ \left|\Psi^{-}\right\rangle & =\frac{1}{\sqrt{2}}(|01\rangle-|10\rangle) \end{aligned}

## 物理代写|光学代写Optics代考|EPR Paradox and Hidden Variables

In a famous 1935 paper [1], Albert Einstein, Boris Podolsky, and Nathan Rosen (known as EPR) sought to demonstrate by the “EPR paradox” that quantum mechanics was incomplete. EPR were concerned by the instantaneous action at a distance, or “nonlocality”, implied by entanglement. Quantum mechanics also seems to violate “realism”. “Realism” means that particles have definite properties that are independent of any measurement.

Suppose we toss a coin. In principle, it is possible to know whether it will land heads or tails if we keep track of a lot of information about the system (called “degrees of freedom”), such as the forces applied during the toss, the air currents, the height of the toss, etc. However, all these physical properties are impossible to calculate in practice, so the most we can do is ascribe a probability distribution for the toss outcome resulting in $P_{\text {heads }}=\frac{1}{2}$ and $P_{\text {tails }}=\frac{1}{2}$. This outcome occurs from averaging the many degrees of freedom that we do not have access to. This principle also forms the basis for statistical thermodynamics.

Einstein and many others believed that quantum mechanics was like this; that is, they proposed that the probabilities in quantum mechanics are deterministic (versus probabilistic) and have some underlying causes that are “hidden”; that is, that we cannot access (analogous to the unknown variables during the coin toss). These underlying causes were called “hidden variables”. If we knew the hidden variables, we would be able to calculate a definite measurement outcome, rather than just probabilities.

Many quantum pioneers, exemplified by Einstein, believed in “local realism” where the state of particles is defined when they are created. However, the “hidden variables” only allow us to determine the probahility of these states. Finstein famously said: “God does not play dice with the universe”. Also, with regards to realism, Einstein said “Do you believe the moon exists only when you look at it?”
Others, exemplified by Bohr, believed in the possibility of superpositions and entanglement. They believed that no definitive statements about a physical system may be made until a measurement is made. Particle properties do not exist until we measure them. It turns out that Bohr was correct; but how do we prove it?

# 光学代考

## 物理代写|光学代写Optics代考|Entangled States

$$|\psi\rangle=|\psi\rangle_1|\psi\rangle_2=\left(\alpha_1|0\rangle_1+\beta_1|1\rangle_1\right)\left(\alpha_2|0\rangle_2+\beta_2|1\rangle_2\right)$$

$$|\psi\rangle=\alpha_1 \alpha_2|0\rangle_1|0\rangle_2+\alpha_1 \beta_2|0\rangle_1|1\rangle_2+\beta_1 \alpha_2|1\rangle_1|0\rangle_2+\beta_1 \beta_2|1\rangle_1|1\rangle_2$$

$$|\psi\rangle \neq|\psi\rangle_1|\psi\rangle_2$$

$$\left|\Phi^{+}\right\rangle=\frac{1}{\sqrt{2}}(|00\rangle+|11\rangle)\left|\Psi^{+}\right\rangle \quad=\frac{1}{\sqrt{2}}(|01\rangle+|10\rangle)\left|\Phi^{-}\right\rangle=\frac{1}{\sqrt{2}}(|00\rangle-|11\rangle)\left|\Psi^{-}\right\rangle$$

## 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。

## 物理代写|光学代写Optics代考|PHS2062

statistics-lab™ 为您的留学生涯保驾护航 在代写光学Optics方面已经树立了自己的口碑, 保证靠谱, 高质且原创的统计Statistics代写服务。我们的专家在代写光学Optics代写方面经验极为丰富，各种代写光学Optics相关的作业也就用不着说。

• Statistical Inference 统计推断
• Statistical Computing 统计计算
• Advanced Probability Theory 高等概率论
• Advanced Mathematical Statistics 高等数理统计学
• (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|Entangled State

If we have no input to the beam splitter, then we expect the action of the beam splitter to give
\begin{aligned} & \text { beam } \ & \underbrace{|0\rangle_1|0\rangle_2}{\text {input }} \stackrel{\text { splitter }}{\longrightarrow} \underbrace{|0\rangle_3|0\rangle_4}{\text {output }} \ & \end{aligned}
The single photon input state in Fig. $6.4$ can be expressed as
$$\left|\psi_{\text {in }}\right\rangle=|1\rangle_1|0\rangle_2=\widehat{a}_1^{\dagger}|0\rangle_1|0\rangle_2$$
According to Eqs. (6.34) and (6.71), Eq. (6.72) can be expressed in terms of the output space using Eq. (6.27):
\begin{aligned} & \text { beam } \ & \widehat{a}_1^{\dagger}|0\rangle_1|0\rangle_2 \stackrel{\text { splitter }}{\longrightarrow}\left(r \widehat{a}_3^{\dagger}+t \widehat{a}_4^{\dagger}\right)|0\rangle_3|0\rangle_4 \ & =r|1\rangle_3|0\rangle_4+t|0\rangle_3|1\rangle_4 \ & \end{aligned}
The output in Eq. (6.74) is called an entangled state of a photon in the $D_3$ path and the $D_4$ path. An entangled state is a state, which cannot be separated or factored into individual product states; that is, $|\psi\rangle \neq\left|\psi_3\right\rangle_3\left|\psi_4\right\rangle_4$. Equation (6.74) cannot be factored into the product of two individual states (try it). If you are not familiar with entanglement, do not worry. We will cover this topic in more detail in Chap. 8. Equation (6.74) tells us that the single photon input on port 1 results in a superposition of the single photon in mode 3 with zero photons in mode 4, and vice versa. The probability amplitude for a single photon along the $D_3$ path is given by $r$, that is, the coefficient of $|1\rangle_3|0\rangle_4$ in Eq. (6.74). The corresponding probability is the modulus squared of the probability amplitude, $|r|^2=R$, the same as Eq. (6.40). The probability amplitude for a single photon along the $D_4$ path is given by $t$, that is, the coefficient of $|0\rangle_3|1\rangle_4$ in Eq. (6.74). The corresponding probability is the modulus squared of the probability amplitude, $|t|^2=T$, the same as Eq. (6.45). As seen in Eq. (6.74), the probability of joint detection at $D_3$ and $D_4$, represented by the state $|1\rangle_3|1\rangle_4$, is zero.

## 物理代写|光学代写Optics代考|Quantum Light Interference

Let us now derive the output for the case of a single photon input, as shown in Fig. 7.3. Here, we clearly need a quantum description. The annihilation operator associated with photodetection at $D_3$ is
$$\widehat{a}3=\left(t^2 e^{i k z_1}-r^2 e^{i k z_2}\right) \widehat{a}_1+\left(-t r e^{i k z_1}-r t e^{i k z_2}\right) \widehat{a}_2$$ The first term follows from Eq. (7.1). The second term derives from the input on port 2, which is vacuum. Similar to Eq. (6.35), the probability of single photon detection at $D_3$ is $$P_3=\left\langle\psi{\text {out }}\left|\widehat{N}3\right| \psi{\text {out }}\right\rangle$$
Using Eq. (7.17), we can express Eq. (7.18) in terms of the input space. Assuming $r$ and $t$ are real, we get
$$P_3={ }_2\left\langle\left. 0\right|_1\left\langle 1\left|\left[\left(t^2 e^{-i k z_1}-r^2 e^{-i k z_2}\right)\left(t^2 e^{i k z_1}-r^2 e^{i k z_2}\right) \widehat{a}_1^{\dagger} \widehat{a}_1\right]\right| 1\right\rangle_1 \mid 0\right\rangle_2$$
where all terms related to $\widehat{a}_2$ are omitted, because they result in zero when applied to the vacuum input, $|0\rangle_2$, on port 2. Evaluating Eq. (7.19) gives
$$P_3=\left[R^2+T^2-2 R T \cos (k \Delta z)\right]$$
which is the same as the classical result.
Similarly, the annihilation operator associated with photodetection at $D_4$ is
$$\widehat{a}_4=\left(r t e^{i k z_1}+\operatorname{tr} e^{i k z_2}\right) \widehat{a}_1+\left(-r^2 e^{i k z_1}+t^2 e^{i k z_2}\right) \widehat{a}_2$$
The probability of single photon detection at $D_4$ is $$P_4=\left\langle\psi_{\text {out }}\left|\widehat{N}4\right| \psi{\text {out }}\right\rangle$$
or, in terms of the input space:
\begin{aligned} P_4 & ={ }_2\left\langle\left. 0\right|_1\left\langle 1\left|\left[\left(r t e^{-i k z_1}+t r e^{-i k z_2}\right)\left(r t e^{i k z_1}+t r e^{i k z_2}\right) \widehat{a}_1^{\dagger} \widehat{a}_1\right]\right| 1\right\rangle_1 \mid 0\right\rangle_2 \ & =2 R T+2 R T \cos (k \Delta z) \end{aligned}
which is the same as the classical result.

# 光学代考

## 物理代写|光学代写Optics代考|Entangled State

$$\text { beam } \underbrace{|0\rangle_1|0\rangle_2} \text { input } \stackrel{\text { splitter }}{\longrightarrow} \underbrace{|0\rangle_3|0\rangle_4} \text { output }$$

$$\left|\psi_{\text {in }}\right\rangle=|1\rangle_1|0\rangle_2=\widehat{a}_1^{\dagger}|0\rangle_1|0\rangle_2$$

$$\text { beam } \quad \widehat{a}_1^{\dagger}|0\rangle_1|0\rangle_2 \stackrel{\text { splitter }}{\longrightarrow}\left(r \widehat{a}_3^{\dagger}+t \widehat{a}_4^{\dagger}\right)|0\rangle_3|0\rangle_4=r|1\rangle_3|0\rangle_4+t|0\rangle_3|1\rangle_4$$

## 物理代写|光学代写Optics代考|Quantum Light Interference

$$\widehat{a} 3=\left(t^2 e^{i k z_1}-r^2 e^{i k z_2}\right) \widehat{a}1+\left(-t r e^{i k z_1}-r t e^{i k z_2}\right) \widehat{a}_2$$ 第一项来自方程式。(7.1)。第二项来自端口 2 上的输入，即真空。类似于方程式。(6.35)，单光子探测概 率 $D_3$ 是 使用方程式。 (7.17)，我们可以表达Eq。(7.18) 在输入空间方面。假设 $r$ 和 $t$ 是真实的，我们得到 $$P_3={ }_2\left\langle\left. 0\right|_1\left\langle 1\left|\left[\left(t^2 e^{-i k z_1}-r^2 e^{-i k z_2}\right)\left(t^2 e^{i k z_1}-r^2 e^{i k z_2}\right) \widehat{a}_1^{\dagger} \widehat{a}_1\right]\right| 1\right\rangle_1 \mid 0\right\rangle_2$$ 所有与相关的条款 $\widehat{a}_2$ 被省略，因为当应用于真空输入时它们会导致零， $|0\rangle_2$ ，在端口 2 上。评估 Eq。 (7.19) 给出 $$P_3=\left[R^2+T^2-2 R T \cos (k \Delta z)\right]$$ 这与经典结果相同。 同样，与光电检测相关的湮灭算子 $D_4$ 是 $$\widehat{a}_4=\left(r t e^{i k z_1}+\operatorname{tr} e^{i k z_2}\right) \widehat{a}_1+\left(-r^2 e^{i k z_1}+t^2 e^{i k z_2}\right) \widehat{a}_2$$ 单光子探测概率 $D_4$ 是 $$P_4=\left\langle\psi{\text {out }}|\widehat{N} 4| \psi \text { out }\right\rangle$$

$$P_4={ }_2\left\langle\left. 0\right|_1\left\langle 1\left|\left[\left(r t e^{-i k z_1}+t r e^{-i k z_2}\right)\left(r t e^{i k z_1}+t r e^{i k z_2}\right) \widehat{a}_1^{\dagger} \widehat{a}_1\right]\right| 1\right\rangle_1 \mid 0\right\rangle_2=2 R T+2 R T$$

## 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。

## 物理代写|光学代写Optics代考|UNITS24

statistics-lab™ 为您的留学生涯保驾护航 在代写光学Optics方面已经树立了自己的口碑, 保证靠谱, 高质且原创的统计Statistics代写服务。我们的专家在代写光学Optics代写方面经验极为丰富，各种代写光学Optics相关的作业也就用不着说。

• Statistical Inference 统计推断
• Statistical Computing 统计计算
• Advanced Probability Theory 高等概率论
• Advanced Mathematical Statistics 高等数理统计学
• (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|Coincident Measurements

Let us calculate the probability, $P_{34}$, of simultaneous detection at $D_3$ and $D_4$. Simultaneous detection events are called coincidence or correlation measurements. Classically, we would expect a fraction $R$ of a classical light intensity reflected to $D_3$ and a fraction $T$ transmitted to $D_4$, allowing simultaneous detection. Classically, we expect the intensity to be proportional to $\left(E_3\right)^2=R\left(E_1\right)^2$ at detector $D_3$ and $\left(E_4\right)^2=T\left(E_1\right)^2$ at detector $D_4$, giving the probability of double detection:
$$\text { Classical : } P_{34}=\frac{R\left(E_1\right)^2 I\left(E_1\right)^2}{\left(E_1\right)^2\left(E_1\right)^2}=R T$$
Let us calculate the probability of a coincident detection, $P_{34}$, for a single photon input on the beam splitter, as illustrated in Fig. 6.5. The simultaneous measurement is described by the operator $\widehat{a}4 \widehat{a}_3$ : $$P{34}=\left\langle\psi_{\text {out }}\left|\left(\widehat{a}4 \widehat{a}_3\right)^{\dagger}\left(\widehat{a}_4 \widehat{a}_3\right)\right| \psi{\text {out }}\right\rangle=\left\langle\psi_{\text {out }}\left|\widehat{a}3^{\dagger} \widehat{a}_4^{\dagger} \widehat{a}_4 \widehat{a}_3\right| \psi{\text {out }}\right\rangle$$
Using Eqs. (6.15) to (6.18) gives

Evaluating all the terms of Eq. (6.48) gives
$$P_{34}=0$$
Quantum mechanically, double detections are not possible for a single photon, which is very different than the classical result of Eq. (6.46). The single photon is detected at $D_3$ with probability $R$, or at $D_4$ with probability $T$, but never both simultaneously. Here, we have a nonclassical correlation. The absence of double detections must be the case if the concept of “single photon” is to make any sense at all. You can only detect a single photon once, either at $D_3$ or $D_4$.
Exercise 6.4 Evaluate Eq. (6.48), verifying that $P_{34}=0$.
The single photon beam splitter could be used as a random number generator. With a 50:50 beam splitter $(R=T=0.5)$, we have a probability $P_3=P_4=0.5$ that a single photon is detected at either $D_3$ or $D_4$. A single photon is launched into the heam splitter, and a 0 hit is assigned for detection at $D_3$, while a 1 hit is assigned for detection at $D_4$. After launching many single photons, one at a time, into the beam splitter, a random sequence of bits is generated, 00110101110… The random sequence of bits can be used to generate a random number.

## 物理代写|光学代写Optics代考|Second-Order Correlation Function

The correlations described in the previous section are usually described by a secondorder correlation function, $g^{(2)}(\tau)$, introduced in 1963 by Roy Glauber (Fig. 6.6), a pioneer of quantum optics [7]. The 2005 Nobel Prize in Physics was divided, one half awarded to Roy J. Glauber “for his contribution to the quantum theory of optical coherence,” the other half jointly to John L. Hall and Theodor W. Hänsch “for their contributions to the development of laser-based precision spectroscopy, including the optical frequency comb technique.”

First, we look at the classical definition of the second-order correlation function, which is given by
$$g_{\text {classical }}^{(2)}(\tau)=\frac{\langle I(t) I(t+\tau)\rangle}{\langle I(t)\rangle^2}=\frac{\left\langle E^(t) E(t) E^(t+\tau) E(t+\tau)\right\rangle}{\left\langle E^(t) E(t)\right\rangle^2}$$ where $I \propto|E|^2=E^ E$. The brackets, \langle\rangle , indicate an average to account for intensity fluctuations during the measurement time. $g_{\text {classical }}^{(2)}(\tau)$ describes the correlation between two temporally separated intensity signals with time difference $\tau$ from one source. If $\tau=0, g_{\text {classical }}^{(2)}(0)$ is especially interesting, because it gives the probability of simultaneous detection events at two detectors, normalized to the probability of individual detection events at either detector. The ” 0 ” means no time delay between the two simultaneous detections.

Suppose the input to the beam splitter is treated as a classical source of light. For classical light, we have
$$g_{\text {classical }}^{(2)}(0)=\frac{\left\langle R E_1^2 T E_1^2\right\rangle}{\left(R\left\langle E_1\right\rangle^2\right)\left(T\left\langle E_1\right\rangle^2\right)}$$
$R$ and $T$ cancel out, and since $E_1^2$ is proportional to the light intensity, we get
$$g_{\text {classical }}^{(2)}(0)=\frac{\left\langle I^2\right\rangle}{\langle I\rangle^2}$$
Next, we can use the Cauchy-Schwarz inequality, which states $$\left\langle I^2\right\rangle \geq\langle I\rangle^2$$
for any positive random variable.

# 光学代考

## 物理代写|光学代写Optics代考|Coincident Measurements

$$\text { Classical : } P_{34}=\frac{R\left(E_1\right)^2 I\left(E_1\right)^2}{\left(E_1\right)^2\left(E_1\right)^2}=R T$$

## 物理代写|光学代写Optics代考|Second-Order Correlation Function

$$g_{\text {classical }}^{(2)}(\tau)=\frac{\langle I(t) I(t+\tau)\rangle}{\langle I(t)\rangle^2}=\frac{\left.\left.\left\langle E^{(} t\right) E(t) E^{(} t+\tau\right) E(t+\tau)\right\rangle}{\left.\left\langle E^{(} t\right) E(t)\right\rangle^2}$$

$$g_{\text {classical }}^{(2)}(0)=\frac{\left\langle R E_1^2 T E_1^2\right\rangle}{\left(R\left\langle E_1\right\rangle^2\right)\left(T\left\langle E_1\right\rangle^2\right)}$$
$R$ 和 $T$ 抵消，因为 $E_1^2$ 与光强度成正比，我们得到
$$g_{\text {classical }}^{(2)}(0)=\frac{\left\langle I^2\right\rangle}{\langle I\rangle^2}$$

$$\left\langle I^2\right\rangle \geq\langle I\rangle^2$$

## 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。

## 物理代写|光学代写Optics代考|CSCI031

statistics-lab™ 为您的留学生涯保驾护航 在代写光学Optics方面已经树立了自己的口碑, 保证靠谱, 高质且原创的统计Statistics代写服务。我们的专家在代写光学Optics代写方面经验极为丰富，各种代写光学Optics相关的作业也就用不着说。

• Statistical Inference 统计推断
• Statistical Computing 统计计算
• Advanced Probability Theory 高等概率论
• Advanced Mathematical Statistics 高等数理统计学
• (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|Quantum Harmonic Oscillator

As described in Chap. 1, the Hamiltonian for the quantum harmonic oscillator (QHO) is obtained by canonnical quantization where the coordinătès $(x, \not p)$ aree replaced by their quantum operators:

\begin{aligned} & \text { canonical } \ & H=\frac{p^2}{2 m}+\frac{1}{2} m \omega^2 x^2 \stackrel{\text { quantization }}{\longrightarrow} \widehat{H}=\frac{\widehat{p}^2}{2 m}+\frac{1}{2} m \omega^2 \widehat{x}^2 \ & \end{aligned}
where $\widehat{x}$ and $\widehat{p}$ obey the commutation relation:
$$[\widehat{x}, \widehat{p}]=i \hbar$$
Equation (2.12) can be used to find the momentum operator $\hat{p}$ in terms of the $x$ coordinate. Starting from Eq. (2.12) and according to the definition of the commutation relation:
$$(\widehat{x} \hat{p}-\widehat{p} \hat{x})|\psi\rangle=i \hbar|\psi\rangle$$
Expanding the left side of Eq. (2.13) gives
$$\widehat{x} \hat{p}|\psi\rangle-\widehat{p} \hat{x}|\psi\rangle=i \hbar|\psi\rangle$$
If $|\psi\rangle$ is in the position representation (i.e., $|\psi\rangle$ represents the familiar wavefunction, $\psi(\mathrm{x})$ ), then the operator $\hat{x}$ is simply the position, $x$; that is, $\hat{x}|\psi\rangle=x|\psi\rangle$. Thus, Eq. (2.14) becomes
$$x \hat{p}|\psi\rangle-\widehat{p} x|\psi\rangle=i \hbar|\psi\rangle$$
In the second term on the left, $\widehat{p} x|\psi\rangle$, we apply the rules of partial differentiation, that is, the operator $\widehat{p}$ operates on $x$ while keeping $|\psi\rangle$ constant, and then $\widehat{p}$ operates on $|\psi\rangle$ while keeping $x$ constant. This gives
$$x \widehat{p}|\psi\rangle-(\widehat{p} x)|\psi\rangle-x(\widehat{p}|\psi\rangle)=i \hbar|\psi\rangle$$
In the second term on the left, $\widehat{p}$ operates on $x$ only.

## 物理代写|光学代写Optics代考|Dirac Formalism

Paul Dirac formulated an alternative approach to solve the QHO. Suppose $\hat{H}$ can be factorized as follows:
$$\widehat{H}=\widehat{O}^{\dagger} \widehat{O}+E_0$$
where $\widehat{O}$ is some operator and $\widehat{O}^{\dagger}$ is the Hermitian conjugate. If $\left|\psi_n\right\rangle$ is an eigenstate of $\widehat{H}$, then the eigenenergies are
$$E_n=\left\langle\psi_n|\widehat{H}| \psi_n\right\rangle$$
Substituting Eq. (2.30) for $\widehat{H}$ gives
\begin{aligned} E_n & =\left\langle\psi_n\left|\left(\widehat{O}^{\dagger} \widehat{O}+E_0\right)\right| \psi_n\right\rangle \ & =\left\langle\psi_n\left|\widehat{O}^{\dagger} \widehat{O}\right| \psi_n\right\rangle+E_0 \end{aligned}
This means:
$$E_n \geq E_0$$
If $\widehat{O}\left|\psi_0\right\rangle=0$, then the minimum energy (ground state energy, $E_0$ ) is found. At this point, it is helpful to define dimensionless operators, $\widehat{Q}$ and $\widehat{P}$ :
$$\widehat{Q}=\sqrt{\frac{m \omega}{\hbar}} \widehat{x}$$ $$\widehat{P}=\sqrt{\frac{1}{m \hbar \omega}} \widehat{p}$$
It is easily shown that Eqs. (2.11) and (2.12) become
$$\begin{gathered} \widehat{H}=\frac{\hbar \omega}{2}\left(\widehat{Q}^2+\widehat{P}^2\right) \ {[\widehat{Q}, \widehat{P}]=i} \end{gathered}$$

# 光学代考

## 物理代写|光学代写Optics代考|Quantum Harmonic Oscillator

$$\text { canonical } \quad H=\frac{p^2}{2 m}+\frac{1}{2} m \omega^2 x^2 \stackrel{\text { quantization }}{\longrightarrow} \widehat{H}=\frac{\hat{p}^2}{2 m}+\frac{1}{2} m \omega^2 \widehat{x}^2$$

$$[\widehat{x}, \hat{p}]=i \hbar$$

$$(\widehat{x} \hat{p}-\hat{p} \hat{x})|\psi\rangle=i \hbar|\psi\rangle$$

$$\widehat{x} \hat{p}|\psi\rangle-\hat{p} \hat{x}|\psi\rangle=i \hbar|\psi\rangle$$

$$x \hat{p}|\psi\rangle-\hat{p} x|\psi\rangle=i \hbar|\psi\rangle$$

$$x \hat{p}|\psi\rangle-(\hat{p} x)|\psi\rangle-x(\hat{p}|\psi\rangle)=i \hbar|\psi\rangle$$

## 物理代写|光学代写Optics代考|Dirac Formalism

Paul Dirac 制定了另一种解决 $\mathrm{QHO}$ 的方法。认为 $\hat{H}$ 可以分解如下:
$$\widehat{H}=\widehat{O}^{\dagger} \widehat{O}+E_0$$

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

$$E_n=\left\langle\psi_n\left|\left(\widehat{O}^{\dagger} \widehat{O}+E_0\right)\right| \psi_n\right\rangle=\left\langle\psi_n\left|\widehat{O}^{\dagger} \widehat{O}\right| \psi_n\right\rangle+E_0$$

$$E_n \geq E_0$$

\begin{aligned} \widehat{Q} & =\sqrt{\frac{m \omega}{\hbar}} \widehat{x} \ \widehat{P} & =\sqrt{\frac{1}{m \hbar \omega}} \hat{p} \end{aligned}

$$\widehat{H}=\frac{\hbar \omega}{2}\left(\widehat{Q}^2+\widehat{P}^2\right)[\widehat{Q}, \widehat{P}]=i$$

## 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。

## 物理代写|光学代写Optics代考|UNITS24

statistics-lab™ 为您的留学生涯保驾护航 在代写光学Optics方面已经树立了自己的口碑, 保证靠谱, 高质且原创的统计Statistics代写服务。我们的专家在代写光学Optics代写方面经验极为丰富，各种代写光学Optics相关的作业也就用不着说。

• Statistical Inference 统计推断
• Statistical Computing 统计计算
• Advanced Probability Theory 高等概率论
• Advanced Mathematical Statistics 高等数理统计学
• (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|Commutation Relations

Dirac showed that the canonically conjugate variables $\left(\widehat{q}i, \widehat{p}_j\right)$ of the quantum system satisfy the commutation relation: $$\left[\widehat{q}_i, \widehat{p}_j\right]=i \hbar \delta{i j}$$
where $\delta_{i j}$ is the Kronecker function and, by definition,
$$\left[\widehat{q}_i, \widehat{p}_j\right]=\widehat{q}_i \widehat{p}_j-\widehat{p}_j \widehat{q}_i$$
Thus, when $i=j$, we say that $\widehat{q}_i$ and $\widehat{p}_i$ “do not commute”; that is, $\left[\widehat{q}_i, \widehat{p}_i\right]=$ $\widehat{q}_i \widehat{p}_i-\widehat{p}_i \widehat{q}_i=i \hbar$. Otherwise, the operators commute. For example, when the generalized coordinates $\left(\widehat{q}_i, \widehat{p}_i\right)$ are the position and momentum, we have
\begin{aligned} & {\left[\widehat{x}, \widehat{p}_x\right]=i \hbar} \ & {\left[\widehat{y}, \widehat{p}_y\right]=i \hbar} \ & {\left[\widehat{z}, \widehat{p}_z\right]=i \hbar} \end{aligned}
Thus, position and momentum along the same direction do not commute (e.g., $\hat{x}$ and $\widehat{p}_x$ do not commute), while position and momentum along different directions do commute (e.g., $\widehat{x}$ and $\widehat{p}_y$ commute). Eq. (1.20) leads to the well-known Heisenberg uncertainty relation:
$$\Delta x \Delta p_x \geq \frac{\hbar}{2}$$
with the same relation for the $y$ and $z$ directions arising from Eq. (1.21) and (1.22), respectively. In Eq. (1.23), $\Delta x$ is the uncertainty in position $x$ and $\Delta p_x$ is the uncertainty in momentum along $x$. Uncertainty is defined as the standard deviation or root mean square (rms) error:
$$\begin{gathered} \Delta x=\sqrt{\left\langle(x-\langle x\rangle)^2\right\rangle} \ =\sqrt{\left\langle x^2+\langle x\rangle^2-2 x\langle x\rangle\right\rangle} \ =\sqrt{\left\langle x^2\right\rangle-\langle x\rangle^2} \end{gathered}$$
where the brackets \langle\rangle denote an average (in quantum mechanics, this is called the “expectation value” of $x$ ).

## 物理代写|光学代写Optics代考|Classical Harmonic Oscillator

Consider a classical system comprised of a particle of mass, $m$, and position, $x$, moving in a one-dimensional parabolic potential:
$$U(x)=\frac{1}{2} k x^2=\frac{1}{2} m \omega^2 x^2$$
where $k$ is a force constant and $\omega=\sqrt{k / m}$ is the angular frequency. The harmonic oscillator arises in a wide variety of classical systems, but most often as a mass on a spring described by Hooke’s law $(F=-k x)$. The generalized coordinates for this system are simply the position and momentum:
$$\begin{gathered} q \rightarrow x \ p \rightarrow m \frac{d x}{d t} \end{gathered}$$
and the Hamiltonian becomes

$$H=\frac{p^2}{2 m}+\frac{1}{2} m \omega^2 x^2$$
The Hamilton equations become
$$\begin{gathered} \frac{d x}{d t}=\frac{\partial H}{\partial p}=\frac{p}{m}=v \ \frac{d p}{d t}=-\frac{\partial H}{\partial x}=-m \omega^2 x=-\frac{\partial U}{\partial x}=F \end{gathered}$$
The first equation is the definition of momentum $(p=m v)$, while the second equation reproduces Newton’s equation $\left(F=\frac{d p}{d t}\right)$. Thus, $x$ and $p$ satisfy the Hamilton equations (they give the correct dynamical behavior) and are therefore canonically conjugate variables.

Equations (2.5) and (2.6) are easily solved. Combining the two equations gives
$$\frac{d^2 x}{d t^2}=-\omega^2 x$$
with the solution
$$x=a \cos (\omega t+\varphi)$$
where the amplitude, $a$, and phase, $\varphi$, are determined by initial conditions. Equivalently, the solution may be written as
$$x=A e^{-i \omega t}+\text { c.c. }$$

# 光学代考

## 物理代写|光学代写Optics代考|Commutation Relations

$$\left[\hat{q}_i, \hat{p}_j\right]=\hat{q}_i \hat{p}_j-\hat{p}_j \hat{q}_i$$

$$\left[\widehat{x}, \hat{p}_x\right]=i \hbar \quad\left[\hat{y}, \hat{p}_y\right]=i \hbar\left[\hat{z}, \hat{p}_z\right]=i \hbar$$

$$\Delta x \Delta p_x \geq \frac{\hbar}{2}$$

$$\Delta x=\sqrt{\left\langle(x-\langle x\rangle)^2\right\rangle}=\sqrt{\left\langle x^2+\langle x\rangle^2-2 x\langle x\rangle\right\rangle}=\sqrt{\left\langle x^2\right\rangle-\langle x\rangle^2}$$

## 物理代写|光学代写Optics代考|Classical Harmonic Oscillator

$$U(x)=\frac{1}{2} k x^2=\frac{1}{2} m \omega^2 x^2$$

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

$$H=\frac{p^2}{2 m}+\frac{1}{2} m \omega^2 x^2$$

$$\frac{d x}{d t}=\frac{\partial H}{\partial p}=\frac{p}{m}=v \frac{d p}{d t}=-\frac{\partial H}{\partial x}=-m \omega^2 x=-\frac{\partial U}{\partial x}=F$$

$$\frac{d^2 x}{d t^2}=-\omega^2 x$$

$$x=a \cos (\omega t+\varphi)$$

$$x=A e^{-i \omega t}+\text { c.c. }$$

## 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。

## 物理代写|光学代写Optics代考|PHS2062

statistics-lab™ 为您的留学生涯保驾护航 在代写光学Optics方面已经树立了自己的口碑, 保证靠谱, 高质且原创的统计Statistics代写服务。我们的专家在代写光学Optics代写方面经验极为丰富，各种代写光学Optics相关的作业也就用不着说。

• Statistical Inference 统计推断
• Statistical Computing 统计计算
• Advanced Probability Theory 高等概率论
• Advanced Mathematical Statistics 高等数理统计学
• (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:
$$H\left(q_1, \ldots, q_i, \ldots ; p_1, \ldots, p_i, \ldots\right) \stackrel{\text { canonical quantization }}{\longrightarrow} \widehat{H}\left(\hat{q}_1, \ldots, \hat{q}_i, \ldots ; \hat{p}_1, \ldots, \hat{p}_i\right.$$

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

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

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

## 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。

## 物理代写|光学代写Optics代考|CSCl031

statistics-lab™ 为您的留学生涯保驾护航 在代写光学Optics方面已经树立了自己的口碑, 保证靠谱, 高质且原创的统计Statistics代写服务。我们的专家在代写光学Optics代写方面经验极为丰富，各种代写光学Optics相关的作业也就用不着说。

• Statistical Inference 统计推断
• Statistical Computing 统计计算
• Advanced Probability Theory 高等概率论
• Advanced Mathematical Statistics 高等数理统计学
• (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|MIXTURES, POLYMER-DISPERSED

In general, temperature ranges for the various mesophases of single constituent liquid crystals are quite limited. Therefore, while many fundamental studies are still conducted on such liquid crystalline materials, industrial applications employ mostly mixtures, composites, or specially doped liquid crystals with large operating temperature range and tailor-made physical and optical properties.

There are many techniques for modifying the physical properties of a liquid crystal. At the most fundamental level, various chemical groups such as bonds or atoms can be introduced to modify the LC molecule. A good example is the cyanobiphenyl homologous series $n \mathrm{CB}(n=1,2,3 \ldots)$. As $n$ is increased through synthesis, the viscosities, anisotropies, molecular sizes, and many other parameters are greatly modified. Some of these physical properties can also be modified by substitution. For example, the hydrogen in the 2,3 , and 4 positions of the phenyl ring may be substituted by some fluoro $(\mathrm{F})$ or chloro $(\mathrm{Cl})$ group [14].

Besides these molecular synthesis techniques, there are other ways to dramatically improve the performance characteristics of liquid crystals. In the following sections, we describe three well-developed methods, focusing our discussion on nematic liquid crystals as they exemplify the unique characteristics of liquid crystals widely used in optical and photonic applications.

## 物理代写|光学代写Optics代考|Dye-doped Liquid Crystals

An obvious effect of introducing dye molecule to liquid crystals is to increase the absorption of a particular liquid crystal at some specified wavelength region. In particular, dye molecules with absorption anisotropy, or those that undergo conformation changes such as trans-cis isomorphism or produce photo-charges, are often used for photonic applications [15-17]. For example, dichroic dye molecules that are more absorptive for optical field polarization parallel than perpendicular to its long axis are often used for the guest-host effect as their oblong shape makes them compatible for dispersing in the host nematic liquid crystals without disturbing the order. These dichroic molecules can then be oriented and reoriented by an external field applied to the host NLC to switch the transmission of the cell (cf. Figure 1.17); such dichroic dye-doped liquid crystals have been utilized to demonstrate optical diode action [15] in the transmission of polarized light.

If the dye molecules undergo some physical changes such as trans-cis isomorphism or produce space charges following photon absorption, they could give rise to nonlinear optical effects [16]; others [17] have shown that dye molecules deposited on the cell windows can be optically aligned as an effective means of surface alignment mechanism for I.C. cell fabrication. These and other effects due to the presence of dye molecules or other photosensitive agents in liquid crystals are discussed in more detail in Chapter 8.

## 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。

## 物理代写|光学代写Optics代考|PHS2062

statistics-lab™ 为您的留学生涯保驾护航 在代写光学Optics方面已经树立了自己的口碑, 保证靠谱, 高质且原创的统计Statistics代写服务。我们的专家在代写光学Optics代写方面经验极为丰富，各种代写光学Optics相关的作业也就用不着说。

• Statistical Inference 统计推断
• Statistical Computing 统计计算
• Advanced Probability Theory 高等概率论
• Advanced Mathematical Statistics 高等数理统计学
• (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|Lyotropic Liquid Crystals

Lyotropic liquid crystals are obtained when an appropriate concentration of material is dissolved in some solvent. The most common systems are those formed by water and amphiphilic molecules (molecules that possess a hydrophilic part that interacts strongly with water and a hydrophobic part that is water insoluble) such as soaps, detergents, and lipids. Here the most important variable controlling the existence of the liquid crystalline phase is the amount of solvent (or concentration). There are quite a number of phases observed in such water-amphiphilic systems, as the composition and temperature are varied; some appear as spherical micelles, and others possess ordered structures with 1-, 2-, or 3-D positional order.

Examples of these kinds of molecules are soaps (Figure 1.8) and various phospholipids like those present in cell membranes. Lyotropic liquid crystals are of interest in biological studies [7].

Polymeric liquid crystals are basically the polymer versions of the monomers discussed in Section 1.1. A good account of polymeric liquid crystals may be found in [9]. There are three common types of polymers, as shown in Figure 1.9a-c, which are characterized by the degree of flexibility. The vinyl type (Figure 1.9a) is the most flexible, the Dupont Kevlar polymer (Figure 1.9b) is semirigid, and the polypeptide chain (Figure 1.9c) is the most rigid. Mesogenic (or liquid crystalline) polymers are classified in accordance with the molecular architectural arrangement of the mesogenic monomer. Main-chain polymers are built by linking rigid mesogenic groups in a manner depicted schematically in Figure 1.10a; the link may be a direct bond or some flexible spacer. Liquid crystal side-chain polymers are formed by pendant side attachment of mesogenic monomers to a conventional polymeric chain, as depicted in Figure 1.10b.

## 物理代写|光学代写Optics代考|Thermotropic Liquid Crystals

Although the molecular structures of thermotropic liquid crystals are quite complicated, they are often represented as “rigid rods” that interact with one another to form distinctive ordered structures (or phases) as a function of ascending temperature: crystals, smectic, nematic, cholesteric (including blue-phase), and the isotropic liquid phase. In smectic liquid crystals, there are several subclassifications in accordance with the positional and directional arrangement of the molecules.

As explained in greater detail in the following chapters, these mesophases are defined and characterized by many physical parameters such as long- and shortrange order, orientational distribution functions, and so on. Here we continue to use the rigid-rod model and pictorially describe these phases in terms of their molecular arrangement.

Figure $1.11$ depicts the collective arrangement of the rodlike molecules in the nematic phase schematically. These molecules are, however, directionally correlated; they are aligned in a general direction defined by a unit vector $\tilde{n}$, the so-called director axis, which may be regarded as the crystal axis. Nevertheless, the molecules are positionally random and exhibit flow very much like liquids; X-ray diffraction from nematics does not exhibit any diffraction peak.

Although individual molecules of nematic liquid crystal (NLC), cholesteric liquid crystal (CLC), and blue-phase liquid crystal (BPLC) may be polar, i.e. carry a permanent dipole, they tend to self-assemble themselves in such a manner that bulk liquid crystals are centrosymmetric, cf. Figure 1.12; their physical properties are the same in the $+\hat{n}$ and the optically uniaxial $-\hat{n}$ directions.

Cholesteric liquid crystals, often also called chiral nematic liquid crystals, resemble nematic liquid crystals except that the molecules assembled in a helical manner, as depicted in Figure 1.11. This property results from the addition of chiral agents to nematic constituents in the starting mixture. Owing to the spatially (helical) varying refractive index, CLCs possess special optical properties such as photonic bandgaps for transmission of circularly polarized lights. More details on CLC as well as cholesteric BPLCs obtained by increasing the concentration of the chiral constituent $[10]$ in the starting mixture are presented in Chapter 4.

## 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。

## 物理代写|光学代写Optics代考|UNITS24

statistics-lab™ 为您的留学生涯保驾护航 在代写光学Optics方面已经树立了自己的口碑, 保证靠谱, 高质且原创的统计Statistics代写服务。我们的专家在代写光学Optics代写方面经验极为丰富，各种代写光学Optics相关的作业也就用不着说。

• Statistical Inference 统计推断
• Statistical Computing 统计计算
• Advanced Probability Theory 高等概率论
• Advanced Mathematical Statistics 高等数理统计学
• (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|MOLECULAR STRUCTURES AND CHEMICAL COMPOSITIONS

With few exceptions, liquid crystals are composed of organic substances with a typical structure, as depicted in Figure 1.1. They are aromatic and, if they contain benzene rings, they are often referred to as benzene derivatives. In general, aromatic liquid crystal molecules such as those shown in Figure $1.1$ comprise a side chain $\mathrm{R}$, two or more aromatic rings $\mathrm{A}$ and $\mathrm{A}^{\prime}$, connected by linkage groups $\mathrm{X}$ and $\mathrm{Y}$, and at the other end connected to a terminal group $\mathrm{R}^{\prime}$.

Examples of side chain and terminal groups are alkyl $\left(\mathrm{C}n \mathrm{H}{2 n+1}\right)$, alkoxy $\left(\mathrm{C}n \mathrm{H}{2 n+1} \mathrm{O}\right)$, and others such as acyloxyl, alkyl carbonate, alkoxy carbonyl, and the nitro and cyano groups. The Xs of the linkage groups are simple bonds or groups such as stilbene $(-\mathrm{CH}==\mathrm{CH}-)$, ester $(-\underset{\mathrm{C}}{\mathrm{O}} \mathrm{O}-)$, tolane $(-\mathrm{C} \equiv \equiv \mathrm{C}-)$, azoxy $(-\mathrm{N}==\mathrm{N}-)$, Schiff base $(-\mathrm{CH}==\mathrm{N}-)$, acetylene $(-\mathrm{C} \equiv \equiv \mathrm{C}-)$, and diacetylene $(-\mathrm{C} \equiv \equiv \mathrm{C}-\mathrm{C} \equiv \equiv \mathrm{C}-)$. The names of liquid crystals are often fashioned after the linkage group (e.g. Schiff-base liquid crystal). There are quite a number of aromatic rings. These include saturated cyclohexane or unsaturated phenyl, biphenyl, and terphenyl in various combinations.

The majority of liquid crystals are benzene derivatives; the rest include heterocyclics, organometallics, sterols, and some organic salts or fatty acids. Their typical structures are shown in Figures 1.2-1.4. Heterocyclic liquid crystals are similar in

structure to benzene derivatives, with one or more of the benzene rings replaced by a pyridine, pyrimidine, or another similar group. Cholesterol derivatives are the most common chemical compounds that exhibit the cholesteric (or chiral nematic) phase of liquid crystals. Organometallic compounds are special in that they contain metal-
lic atoms and possess interesting dynamical and magneto-optical properties.
All the physical and optical properties of liquid crystals such as dielectric. constants, elastic constants, viscosities, absorption spectra, transition temperatures, the existence of mesophases, anisotropies, and optical nonlinearities are governed by the properties of these constituent groups and how they are chemically synthesized together. Since these molecules are quite large and anisotropic, and therefore very complex, it would take a treatise to discuss all the possible variations in the molecular architecture and the resulting changes in their physical properties. Nevertheless, there are some general observations one can make on the dependence of the physical properties on the molecular constituents [2]. For example, the chemical stability of liquid crystals depends very much on the central linkage group. Schiff-base liquid crystals are usually quite unstable. Ester, azo, and azoxy compounds are more stable but are also quite susceptible to moisture, temperature change, and ultraviolet (UV) radiation. Compounds without a central linkage group are among the most stable liquid crystals ever synthesized. The most widely studied one is $5 \mathrm{CB}$ (pentyl cyanobiphenyl), whose structure is shown in Figure 1.5. Other compounds such as pyrimide and phenyl cyclohexane are also quite stable.

## 物理代写|光学代写Optics代考|Electronic Optical Transitions and UV Absorption

Since liquid crystal constituent molecules are quite large, their energy level structures are rather complex. In essence, the basic quantum mechanical theory is similar to the one described in Chapter 10 for a multilevel molecule. Generally, the energy levels are referred to as orbitals: $\pi, n$, and $\sigma$ orbitals for the ground and low-lying levels and $\pi^{+}, n^{+}$, and $\sigma^{+}$for their excited counterparts. Since most liquid crystals are aromatic compounds, containing one or more aromatic rings, the energy levels or orbitals of aromatic rings play a major role. In particular, the $\pi \rightarrow \pi^$ transitions in a benzene molecule have been extensively studied. Figure $1.6$ shows three possible $\pi \rightarrow \pi^$ transitions in a benzene molecule.

In general, these transitions correspond to the absorption of light in the near-UV spectral region $(\leq 200 \mathrm{~nm})$ [2]. These results for a benzene molecule can also be used for interpreting the absorption of liquid crystals containing phenyl rings. On the other hand, in a saturated cyclohexane ring or band, usually only $\sigma$ electrons are involved. The $\sigma \rightarrow \sigma^$ transitions correspond to absorption of light of shorter wavelength $(\leq 180 \mathrm{~nm})$ in comparison to the $\pi \rightarrow \pi^$ transition mentioned previously.
These optical properties are also related to the presence or absence of conjugation (i.e. alternations of single and double bonds, as in the case of a benzene ring). In such conjugated molecules, the $\pi$ electron’s wave function is delocalized along the conjugation length, resulting in absorption of light in a longer wavelength region compared to, for example, that associated with the $\sigma$ electron in compounds that do not possess conjugation. Absorption data and spectral dependence for a variety of molecular constituents, including phenyl rings, biphenyls, terphenyls, tolanes, and diphenyl-diacetylenes, may be found in [2].

## 物理代写|光学代写Optics代考|Electronic Optical Transitions and UV Absorption

\pi \rightarrow \pi^苯分子中的跃迁已被广泛研究。数字1.6显示三种可能\ipi \rightarrow \pi^苯分子中的跃迁。

## 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。

## 物理代写|光学代写Optics代考|CSCl031

statistics-lab™ 为您的留学生涯保驾护航 在代写光学Optics方面已经树立了自己的口碑, 保证靠谱, 高质且原创的统计Statistics代写服务。我们的专家在代写光学Optics代写方面经验极为丰富，各种代写光学Optics相关的作业也就用不着说。

• Statistical Inference 统计推断
• Statistical Computing 统计计算
• Advanced Probability Theory 高等概率论
• Advanced Mathematical Statistics 高等数理统计学
• (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|General Stress Tensor for Nematic Liquid Crystals

The general theoretical framework for describing the hydrodynamics of liquid crystals has been developed principally by Leslie [16] and Ericksen [17]. Their approaches account for the fact that the stress tensor depends not only on the velocity gradients but also on the orientation and rotation of the director. Accordingly, the stress tensor is given by
$$\sigma_{\alpha \beta}=\alpha_1 n_\gamma n_\delta A_{\gamma \delta} n_\alpha n_\beta+\alpha_2 n_{\alpha \alpha} n_\beta+\alpha_3 n_\beta n_\alpha+\alpha_4 A_{\alpha \beta}+\alpha_5 n_\gamma A_{\gamma \beta}+\alpha_6 n_\beta n_\gamma A_{\gamma \alpha}$$
where the $A_{\alpha \beta}$ s are defined by
$$A_{\alpha \beta}=\frac{1}{2}\left[\frac{\partial v_\beta}{\partial x_\alpha}+\frac{\partial v_\alpha}{\partial x_\beta}\right]$$
Note that all the other terms on the right-hand side of Eq. (3.59) involve the director orientation, except the fourth term, $\alpha_4 A_{\alpha \beta}$.This is the same term as that for an isotropic fluid (cf. Eq. [3.57]), that is, $\alpha_4=2 \eta$.

Therefore, in this formalism, we have six so-called Leslie coefficients, $\alpha_1, \alpha_2, \ldots$, $\alpha_6$, which have the dimension of viscosity coefficients. It was shown by Parodi [18] that
$$\alpha_2+\alpha_3=\alpha_6-\alpha_5$$
and so there are really five independent coefficients.
In the next few sections, we will study exemplary cases of director axis orientation and deformation, and we will show how these Leslie coefficients are related to other commonly used viscosity coefficients.

## 物理代写|光学代写Optics代考|Flows with Fixed Director Axis Orientation

Consider here the simplest case of flows in which the director axis orientation is held fixed. This may be achieved by a strong externally applied magnetic field (see Figure 3.11), where the magnetic field is along the direction $\hat{n}$. Consider the case of shear flow, where the velocity is in the $z$-direction, and the velocity gradient is along the $x$-direction. This process could occur, for example, in liquid crystals confined by two parallel plates in the $y$-z plane.

In terms of the orientation of the director axis, there are three distinct possibilities involving three corresponding viscosity coefficients:

1. $\eta_1: \hat{n}$ is parallel to the velocity gradient, that is, along the $x$-axis $\left(\theta=90^{\circ}, \phi=0^{\circ}\right)$.
2. $\eta_2: \hat{n}$ is parallel to the flow velocity, that is, along the $z$-axis and lies in the shear plane $x-z\left(\theta=0^{\circ}, \phi=0^{\circ}\right)$.
3. $\eta_3: \hat{n}$ is perpendicular to the shear plane, that is, along the $y$-axis $\left(\theta=0^{\circ}, \phi\right.$ $\left.=90^{\circ}\right)$.

These three configurations have been investigated by Miesowicz [19], and the $\eta$ s are known as Miesowicz coefficients. In the original paper, as well as in the treatment by deGennes [3], the definitions of $\eta_1$ and $\eta_3$ are interchanged. In deGennes notation, in terms of $\eta_a, \eta_b$, and $\eta_c$, we have $\eta_a=\eta_1, \eta_b=\eta_2$, and $\eta_c=\eta_3$. The notation used here is attributed to Helfrich [6], which is now the conventional one.

To obtain the relations between $\eta_{1,2,3}$ and the Leslie coefficients $\alpha_{1,2, \ldots, 6}$, one could evaluate the stress tensor $\sigma_{\alpha \beta}$ and the shear rate $A_{\alpha \beta}$ for various director orientations and flow and velocity gradient directions. From these considerations, the following relationships are obtained [3]:

\begin{aligned} &\eta_1=\frac{1}{2}\left(\alpha_4+\alpha_5-\alpha_2\right) \ &\eta_2=\frac{1}{2}\left(\alpha_3+\alpha_4+\alpha_6\right) \ &\eta_3=\frac{1}{2} \alpha_4 \end{aligned}
In the shear plane $x-z$, the general effective viscosity coefficient is actually more correctly expressed in the form [20]
$$\eta_{\mathrm{eff}}=\eta_1+\eta_2 \cos ^2 \theta+\eta_2$$
in order to account for angular velocity gradients. The coefficient $\eta_{1,2}$ is related to the Leslie coefficient $\alpha_1$ by
$$\eta_{1,2}=\alpha_1 .$$

## 物理代写|光学代写Optics代考|General Stress Tensor for Nematic Liquid Crystals

$$\sigma_{\alpha \beta}=\alpha_1 n_\gamma n_\delta A_{\gamma \delta} n_\alpha n_\beta+\alpha_2 n_{\alpha \alpha} n_\beta+\alpha_3 n_\beta n_\alpha+\alpha_4 A_{\alpha \beta}+\alpha_5 n_\gamma A_{\gamma \beta}+\alpha_6 n_\beta n_\gamma A_{\gamma \alpha}$$

$$A_{\alpha \beta}=\frac{1}{2}\left[\frac{\partial v_\beta}{\partial x_\alpha}+\frac{\partial v_\alpha}{\partial x_\beta}\right]$$

$$\alpha_2+\alpha_3=\alpha_6-\alpha_5$$

## 物理代写|光学代写Optics代考|Flows with Fixed Director Axis Orientation

1. $\eta_1: \hat{n}$ 平行于速度梯度，即沿 $x$-轴 $\left(\theta=90^{\circ}, \phi=0^{\circ}\right)$.
2. $\eta_2: \hat{n}$ 平行于流速，即沿 $z$-轴并且位于剪切平面内 $x-z\left(\theta=0^{\circ}, \phi=0^{\circ}\right)$.
3. $\eta_3: \hat{n}$ 垂直于剪切面，即沿 $y$-轴 $\left(\theta=0^{\circ}, \phi=90^{\circ}\right)$.
Miesowicz [19] 研究了这三种配置，并且 $\eta$ s 称为 Miesowicz 系数。在原始论文以及 deGennes [3] 的处理中，定 义 $\eta_1$ 和 $\eta_3$ 被互换。在 deGennes 表示法中，根据 $\eta_a, \eta_b$ ，和 $\eta_c$ ，我们有 $\eta_a=\eta_1, \eta_b=\eta_2$ ，和 $\eta_c=\eta_3$. 这里 使用的符号归功于 Helfrich [6]，它现在是传统的符号。
获得之间的关系 $\eta_{1,2,3}$ 和莱斯利系数 $\alpha_{1,2, \ldots, 6}$ ，可以评估应力张量 $\sigma_{\alpha \beta}$ 和剪切速率 $A_{\alpha \beta}$ 适用于各种导向器方向以及 流动和速度梯度方向。从这些考虑，得到以下关系[3]:
$$\eta_1=\frac{1}{2}\left(\alpha_4+\alpha_5-\alpha_2\right) \quad \eta_2=\frac{1}{2}\left(\alpha_3+\alpha_4+\alpha_6\right) \eta_3=\frac{1}{2} \alpha_4$$
在剪切平面 $x-z ， 一$ 般有效粘度系数实际上更正确地表示为 [20]
$$\eta_{\text {eff }}=\eta_1+\eta_2 \cos ^2 \theta+\eta_2$$
为了考虑角速度梯度。系数 $\eta_{1,2}$ 与莱斯利系数有关 $\alpha_1$ 经过
$$\eta_{1,2}=\alpha_1$$

## 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。