## 物理代写|电磁学代写electromagnetism代考|PCS624

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$$\int_{-\infty}^{+\infty} e^{-x^2} f(x) d x=\sum_{i=1}^n w_i f\left(x_i\right)+R_n$$
On neglecting the remainder term, it can be written as
$$\int_{-\infty}^{+\infty} e^{-x^2} f(x) d x=\sum_{i=1}^n w_i f\left(x_i\right)$$
The alternative form of the above equation is
$$\int_{-\infty}^{+\infty} g(x) d x=\sum_{i=1}^n w_i e^{x_i^2} g\left(x_i\right)$$
In Equations 9.8a through 9.8c, $x_i$ is the $i$ th zero of $H_n(x), H_n(x)$ is the Hermite polynomials, $w_i$ is the weight and $R_n$ is the remainder
$$\begin{gathered} w_i=\frac{2^{n-1} n ! \sqrt{\pi}}{n^2\left[H_{n-1}\left(x_i\right)\right]^2} \ R_n=\frac{n ! \sqrt{\pi}}{2^n(2 n) !} f^{2 n}(\xi) \quad(-\infty<\xi<\infty) \end{gathered}$$
The weight factors $\left(w_i\right)$ and the product $w_i e^{x_i{ }^2}$ for the values of abscissas $\left(x_i\right)$ representing zeros of Hermite polynomials are available ${ }^{14}$ for $n=2,3,4,5,6$, $7,8,9,10,12,16$ and 20 . Table 9.1 gives these values for an arbitrarily selected $n(=9)$.

$$\int_0^{+\infty} e^{-x} f(x) d x=\sum_{i=1}^n w_i f\left(x_i\right)+R_n$$
On neglecting the remainder term, it can be written as
$$\int_0^{+\infty} e^{-x} f(x) d x=\sum_{i=1}^n w_i f\left(x_i\right)$$
The above equation can be written in the following alternative form:
$$\int_0^{+\infty} g(x) d x=\sum_{i=1}^n w_i e^{x_i} g\left(x_i\right)$$

In Equations 9.10 through $9.10 \mathrm{c}, x_i$ is the $i$ th zero of $L_n(x), L_n(x)$ is the Laguerre polynomials, $w_i$ is the weight and $R_n$ is the remainder
$$\begin{gathered} w_i=\frac{(n !)^2 x_i}{(n+1)^2\left[L_{n+1}\left(x_i\right)\right]^2} \ R_n=\frac{(n !)^2}{(2 n) !} f^{2 n}(\xi) \quad(0<\xi<\infty) \end{gathered}$$
Weight factors $\left(w_i\right)$ and the product $w_i e^{x_i}$ for some selected values of abscissas $\left(x_i\right)$ representing zeros of Laguerre polynomials are available ${ }^{14}$ for $n=2,3,4,5,6$, $7,8,9,10,12$ and 15 . Table 9.2 gives these values for an arbitrarily selected $n(=9)$.

## 物理代写|电磁学代写electromagnetism代考|Change of Variable for Infinite Intervals

If, in Equation 9.8a, $x$ is replaced by $t /\left(1-t^2\right)$, then $d x=\left(\left(1+t^2\right) /\left(1-t^2\right)^2\right)$. In view of this replacement the limits of ‘ $-\infty$ ‘ to ‘ $+\infty$ ‘ change to ‘ -1 ‘ to ‘ +1 ‘. Thus, the integral of infinite interval reduces to that of finite interval
$$\int_{-\infty}^{+\infty} f(x) d x=\int_{-1}^{+1} f\left(\frac{t}{1-t^2}\right) \frac{1+t^2}{\left(1-t^2\right)^2} d t$$

In this case, $x$ is replaced by $a+(t /(1-t))$ then $d x=d t /(1-t)^2$ and the limits ‘ $a$ ‘ to ‘ $\infty$ ‘ change from ‘ 0 ‘ to ‘ 1 ‘. Thus, the integral becomes
$$\int_a^{+\infty} f(x) d x=\int_0^1 f\left(a+\frac{t}{1-t}\right) \frac{d t}{(1-t)^2}$$

Here $x$ is replaced by $a-((1-t) / t)$ then $d x=d t / t^2$ and the limits ‘ $\infty$ ‘ to ‘ $a$ ‘ change to ‘ 0 ‘ to ‘ 1 ‘. The integral thus becomes
$$\int_{-\infty}^a f(x) d x=\int_0^1 f\left(a-\frac{1-t}{t}\right) \frac{d t}{t^2}$$

The quadrature rules as such are designed to compute one-dimensional integrals. The multi-dimensional integrals can, however, also be evaluated by repeating one-dimensional integrals. In this approach, the function evaluations exponentially grow with the number of dimensions and some methods to overcome this effect are to be used. Monte Carlo or quasi-Monte Carlo methods provide better alternatives. These methods are easy to apply to multi-dimensional integrals. Besides, these may yield greater accuracy for the same number of function evaluations than repeated integrations using one-dimensional methods. Markov chain Monte Carlo algorithms, which include Metropolis-Hestings algorithm and Gibbs sampling, belong to a large class of useful Monte Carlo methods. Besides, sparse grids are developed by Smolyak for the quadrature of high-dimensional functions. Although it is based on a one-dimensional quadrature rule, it performs more sophisticated combination of univariate results.

# 电磁学代考

$$\int_{-\infty}^{+\infty} e^{-x^2} f(x) d x=\sum_{i=1}^n w_i f\left(x_i\right)+R_n$$

$$\int_{-\infty}^{+\infty} e^{-x^2} f(x) d x=\sum_{i=1}^n w_i f\left(x_i\right)$$

$$\int_{-\infty}^{+\infty} g(x) d x=\sum_{i=1}^n w_i e^{x_i^2} g\left(x_i\right)$$

$$\begin{gathered} w_i=\frac{2^{n-1} n ! \sqrt{\pi}}{n^2\left[H_{n-1}\left(x_i\right)\right]^2} \ R_n=\frac{n ! \sqrt{\pi}}{2^n(2 n) !} f^{2 n}(\xi) \quad(-\infty<\xi<\infty) \end{gathered}$$

$$\int_0^{+\infty} e^{-x} f(x) d x=\sum_{i=1}^n w_i f\left(x_i\right)+R_n$$

$$\int_0^{+\infty} e^{-x} f(x) d x=\sum_{i=1}^n w_i f\left(x_i\right)$$

$$\int_0^{+\infty} g(x) d x=\sum_{i=1}^n w_i e^{x_i} g\left(x_i\right)$$

$$\begin{gathered} w_i=\frac{(n !)^2 x_i}{(n+1)^2\left[L_{n+1}\left(x_i\right)\right]^2} \ R_n=\frac{(n !)^2}{(2 n) !} f^{2 n}(\xi) \quad(0<\xi<\infty) \end{gathered}$$

## 物理代写|电磁学代写electromagnetism代考|Change of Variable for Infinite Intervals

$$\int_{-\infty}^{+\infty} f(x) d x=\int_{-1}^{+1} f\left(\frac{t}{1-t^2}\right) \frac{1+t^2}{\left(1-t^2\right)^2} d t$$

$$\int_a^{+\infty} f(x) d x=\int_0^1 f\left(a+\frac{t}{1-t}\right) \frac{d t}{(1-t)^2}$$

$$\int_{-\infty}^a f(x) d x=\int_0^1 f\left(a-\frac{1-t}{t}\right) \frac{d t}{t^2}$$

## 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|电磁学代写electromagnetism代考|PHYS355

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## 物理代写|电磁学代写electromagnetism代考|Numerical Analysis

Numerical analysis is the study of algorithms that use numerical approximation for the mathematical problems that evolve out of some physical systems or processes. Its overall goal is the design and analysis of techniques to give approximate but acceptable solutions to the complicated problems. These problems may be related to weather predictions, computation of the trajectories of spacecraft, the crash safety of cars, stresses developed in physical structures or the distribution of fields and so on. For estimating trajectories, the accurate numerical solution of a system of ordinary differential equations may be required, whereas car safety may require numerical solutions of partial differential equations. The problem of structure or that of fields may also involve ordinary or partial differential equations, integral equations and so on.

In numerical analysis, the process of interpolation, extrapolation and regression are quite frequently employed. In case of interpolation, the value of some unknown function can be evaluated in between the two given values of the function. In extrapolation, the value of some unknown function is to be evaluated, which falls outside the given points. This process first assesses the nature of variation of previous values and based on this trend estimates the new values. Regression is also a similar process, but it takes into account that the data are imprecise. Given some points, and a measurement of the value of some function at these points (with an error), it determines the unknown function. It mostly relies on the least square error to achieve the goal.
9.2.1 Computational Errors
No technique, which falls in the domain of numerical analysis, is error free. These errors creep in mainly due to the following reasons:

1. In general, all practical computers have a finite memory and it is impossible to exactly represent all the real numbers on such a computing machine. Thus, a class of error referred to as the round-off errors are bound to occur.
2. When an iterative method is terminated or a mathematical procedure is approximated, the error due to which the approximate solution differs from the exact solution is referred to as truncation errors.
3. Similarly, the discretisation induces a discretisation error because the solution of the discrete problem does not coincide with the solution of the continuous problem.
It may be noted that once an error is generated, it generally propagates through subsequent calculations.

## 物理代写|电磁学代写electromagnetism代考|Domain of Numerical Analysis

The field of numerical analysis includes many subdisciplines and encompasses problems of multi-facial nature. These may include the following.

The evaluation of a function at a given point is one of the simplest problems. In the case of polynomials, the Horner scheme is a better approach, since it requires a lesser number of multiplications and additions. In this case, the estimation and control of round-off errors due to the use of floating point arithmetic is of immense importance.

These can be further classified into linear and nonlinear forms. Linear equations are an important class of the numerical analysis. There are many methods for solving the systems of linear equations. Some standard methods employ matrix decomposition techniques. These include Gaussian elimination, LU (lower-upper) decomposition, Cholesky decomposition for symmetric (or Hermitian) and positive-definite matrix and QR decomposition for nonsquare matrices. For large systems preference is given to iterative methods, which include Jacobi method, Gauss-Seidel method, successive over relaxation method and conjugate gradient method. General iterative methods can be developed by using a matrix splitting.

Nonlinear equations are solved by using root-finding algorithms. In this case, if the function is differentiable and the derivative is known it can be solved by using Newton’s method. The technique referred to as linearisation can also be employed for solving nonlinear equations.

In context to the system of equations, it seems to be appropriate to describe the formation of matrices from linear algebraic equations. An algebraic equation in which each term is either a constant or the product of a constant and (the first power of) a single variable is referred to as a linear equation. A linear equation can involve a number of variables but does not include exponents. An equation involving $n$ variables can be written in the following form:
$$a_1 x_1+a_2 x_2+\cdots+a_n x_n=b$$
where $a_1, a_2, \ldots, a_n$ represent numbers and are called the coefficients. The parameters $x_1, x_2, \ldots, x_n$ are the unknowns and $b$ is called the constant term. The present analysis gives rise to a set of such equations, which can be written as
$$\begin{gathered} A_{11} x_1+a_{12} x_2+\cdots+a_{1 N} x_N=b_1 \ A_{21} x_1+a_{22} x_2+\cdots+a_{2 N} x_N=b_2 \ \cdots \ A_{M 1 x 1}+a_{M 2} x_2+\cdots+a_{M N} x_N=b_M \end{gathered}$$

# 电磁学代考

9.2.1计算误差

## 物理代写|电磁学代写electromagnetism代考|Domain of Numerical Analysis

$$a_1 x_1+a_2 x_2+\cdots+a_n x_n=b$$

$$\begin{gathered} A_{11} x_1+a_{12} x_2+\cdots+a_{1 N} x_N=b_1 \ A_{21} x_1+a_{22} x_2+\cdots+a_{2 N} x_N=b_2 \ \cdots \ A_{M 1 x 1}+a_{M 2} x_2+\cdots+a_{M N} x_N=b_M \end{gathered}$$

## 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|电磁学代写electromagnetism代考|PHYSICS7536

statistics-lab™ 为您的留学生涯保驾护航 在代写电磁学electromagnetism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。我们的专家在代写电磁学electromagnetism代写方面经验极为丰富，各种代写电磁学electromagnetism相关的作业也就用不着说。

## 物理代写|电磁学代写electromagnetism代考|Power Components

Equation 8.76 a contains the following four components:
$$\begin{gathered} \mathcal{P}{H L}=(s \cdot \omega) \cdot \frac{1}{2} \cdot \alpha \cdot \sin (\beta) \cdot\left(H{1 y} \cdot H_{1 y}^+H_{1 z} \cdot H_{1 z}^\right) \ \mathcal{P}{E L}=\frac{1}{2} \cdot \frac{1}{\sigma_1} J{1 x} \cdot J_{1 x}^* \ \mathcal{P}{E M}=u_y \cdot \frac{1}{2} \alpha \cdot \mathcal{R} e\left[-e^{-j \beta} J{1 x}^* \cdot H_{1 z}\right] \ \mathcal{P}{H M}=u_y \cdot \frac{1}{2} \ell \cdot \alpha \cdot \sin (\beta) \cdot\left(H{1 y} \cdot H_{1 y}^+H_{1 z} \cdot H_{1 z}^\right) \end{gathered}$$
These four terms bear the following meaning:

1. The first term $\left(\mathcal{P}_{H L}\right.$ ) given by Equation $8.76 \mathrm{~b}$ represents the power density proportional to the slip frequency. For hysteresis-free media, this term is zero. Therefore, it can be considered as hysteresis loss per unit volume of the rotor ring.
2. The second term $\left(\mathcal{P}_{E L}\right)$ given by Equation $8.76 \mathrm{c}$ represents the eddy current loss per unit rotor ring volume. This term vanishes for zero conductivity resulting in the absence of eddy currents.
3. The third term $\left(\mathcal{P}_{E M}\right)$ is due to eddy currents in the rotor ring. As it is proportional to the rotor speed, it indicates the mechanical power developed due to induction machine action.
4. The fourth term $\left(\mathcal{P}_{H M}\right)$ is also proportional to the rotor speed and thus indicates the mechanical power developed due to hysteresis machine action. This term vanishes for zero value of the hysteretic angle, $\beta$.

## 物理代写|电磁学代写electromagnetism代考|Slip-Power Relation

From Equations $8.76 \mathrm{~b}$ and $8.76 \mathrm{e}$, we get
$$\frac{\mathcal{P}{H M}}{\mathcal{P}{H L}}=\frac{1-s}{s}$$
The total hysteretic power is given as
$$\mathcal{P}H=\mathcal{P}{H L}+\mathcal{P}{H M}=\frac{1}{2} \cdot \omega \cdot \alpha \cdot \sin (\beta) \cdot\left(H{1 y} \cdot H_{1 y}^+H_{1 z} \cdot H_{1 z}^\right)$$
For a hysteresis machine with zero conductivity of the rotor, this term in view of Equations $8.57 \mathrm{c}, 8.58 \mathrm{a}, 8.58 \mathrm{~b}, 8.58 \mathrm{c}, 8.60 \mathrm{a}$ and $8.61 \mathrm{~b}$ becomes slipindependent for a given stator current, whereas the remaining two terms on the right-hand side (RHS) of Equation 8.76a disappear if eddy currents in the rotor ring are absent. Thus, for an ideal hysteresis machine with zero eddy currents, we have
$$\frac{P_{H M}}{P_{H L}}=\frac{1-s}{s}$$
where $P_{H M}$ indicates the total mechanical power developed due to hysteresis machine action, and $P_{H L}$ indicates total hysteresis loss in the rotor of the machine. Rotor power input, $P_R$, being the sum of power loss, $P_L$ and mechanical power developed, $P_M$, we have
$$\frac{P_R}{1}=\frac{P_L}{s}=\frac{P_M}{1-s}$$
It may be noted that Equations 8.69 and 8.79 indicate that induction machines and hysteresis machines belong to the same class of machines, both satisfying Equation 8.80.

# 电磁学代考

## 物理代写|电磁学代写electromagnetism代考|Power Components

$$\begin{gathered} \mathcal{P}{H L}=(s \cdot \omega) \cdot \frac{1}{2} \cdot \alpha \cdot \sin (\beta) \cdot\left(H{1 y} \cdot H_{1 y}^+H_{1 z} \cdot H_{1 z}^\right) \ \mathcal{P}{E L}=\frac{1}{2} \cdot \frac{1}{\sigma_1} J{1 x} \cdot J_{1 x}^* \ \mathcal{P}{E M}=u_y \cdot \frac{1}{2} \alpha \cdot \mathcal{R} e\left[-e^{-j \beta} J{1 x}^* \cdot H_{1 z}\right] \ \mathcal{P}{H M}=u_y \cdot \frac{1}{2} \ell \cdot \alpha \cdot \sin (\beta) \cdot\left(H{1 y} \cdot H_{1 y}^+H_{1 z} \cdot H_{1 z}^\right) \end{gathered}$$

## 物理代写|电磁学代写electromagnetism代考|Slip-Power Relation

$$\frac{\mathcal{P}{H M}}{\mathcal{P}{H L}}=\frac{1-s}{s}$$

$$\mathcal{P}H=\mathcal{P}{H L}+\mathcal{P}{H M}=\frac{1}{2} \cdot \omega \cdot \alpha \cdot \sin (\beta) \cdot\left(H{1 y} \cdot H_{1 y}^+H_{1 z} \cdot H_{1 z}^\right)$$

$$\frac{P_{H M}}{P_{H L}}=\frac{1-s}{s}$$

$$\frac{P_R}{1}=\frac{P_L}{s}=\frac{P_M}{1-s}$$

## 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|量子力学代写quantum mechanics代考|PHYS662

statistics-lab™ 为您的留学生涯保驾护航 在代写量子力学quantum mechanics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。我们的专家在代写量子力学quantum mechanics代写方面经验极为丰富，各种代写量子力学quantum mechanics相关的作业也就用不着说。

Notice that the angular part of the wave function, $Y(\theta, \phi)$, is the same for all spherically symmetric potentials; the actual shape of the potential, $V(r)$, affects only the radial part of the wave function, $R(r)$, which is determined by Equation 4.16:
$$\frac{d}{d r}\left(r^2 \frac{d R}{d r}\right)-\frac{2 m r^2}{\hbar^2}[V(r)-E] R=\ell(\ell+1) R .$$
This simplifies if we change variables: Let
$$u(r) \equiv r R(r),$$
so that $R=u / r, d R / d r=[r(d u / d r)-u] / r^2,(d / d r)\left[r^2(d R / d r)\right]=r d^2 u / d r^2$, and hence
$$-\frac{\hbar^2}{2 m} \frac{d^2 u}{d r^2}+\left[V+\frac{\hbar^2}{2 m} \frac{\ell(\ell+1)}{r^2}\right] u=E u .$$
This is called the radial equation; ${ }^9$ it is identical in form to the one-dimensional Schrödinger equation (Equation 2.5), except that the effective potential,
$$V_{\mathrm{eff}}=V+\frac{\hbar^2}{2 m} \frac{\ell(\ell+1)}{r^2},$$
contains an extra piece, the so-called centrifugal term, $\left(\hbar^2 / 2 m\right)\left[\ell(\ell+1) / r^2\right]$. It tends to throw the particle outward (away from the origin), just like the centrifugal (pseudo-)force in classical mechanics. Meanwhile, the normalization condition (Equation 4.31) becomes
$$\int_0^{\infty}|u|^2 d r=1$$

## 物理代写|量子力学代写quantum mechanics代考|The Hydrogen Atom

The hydrogen atom consists of a heavy, essentially motionless proton (we may as well put it at the origin), of charge $e$, together with a much lighter electron (mass $m_e$, charge $-e$ ) that orbits around it, bound by the mutual attraction of opposite charges (see Figure 4.4). From Coulomb’s law, the potential energy of the electron $\frac{13}{}$ (in SI units) is
$$V(r)=-\frac{e^2}{4 \pi \epsilon_0} \frac{1}{r},$$
and the radial equation (Equation $4.37$ ) says
$$-\frac{\hbar^2}{2 m_e} \frac{d^2 u}{d r^2}+\left[-\frac{e^2}{4 \pi \epsilon_0} \frac{1}{r}+\frac{\hbar^2}{2 m_e} \frac{\ell(\ell+1)}{r^2}\right] u=E u .$$
(The effective potential-the term in square brackets—is shown in Figure 4.5.) Our problem is to solve this equation for $u(r)$, and determine the allowed energies. The hydrogen atom is such an important case that $\mathrm{I} m$ not going to hand you the solutions this time-we’ll work them out in detail, by the method we used in the analytical solution to the harmonic oscillator. (If any step in this process is unclear, you may want to refer back to Section 2.3 .2 for a more complete explanation.) Incidentally, the Coulomb potential (Equation 4.52) admits continuum states (with $E>0$ ), describing electron-proton scattering, as well as discrete bound states, representing the hydrogen atom, but we shall confine our attention to the latter. ${ }^{14}$

# 量子力学代考

## 物理代写|量子力学代写quantum mechanics代考|The Qudit Bell States

$$|\Phi\rangle_{A B} \equiv \frac{1}{\sqrt{d}} \sum_{i=0}^{d-1}|i\rangle_A|i\rangle_B .$$

$$|\Gamma\rangle_{A B} \equiv \sum_{i=0}^{d-1}|i\rangle_A|i\rangle_B$$

$$\left|\Phi^{x, z}\right\rangle_{A B} \equiv\left(X_A(x) Z_A(z) \otimes I_B\right)|\Phi\rangle_{A B}$$

$d^2$状态$\left{\left|\Phi^{x, z}\right\rangle_{A B}\right}_{x, z=0}^{d-1}$被称为qudit Bell状态，在qudit量子协议和量子香农理论中很重要。练习3.7.11要求您验证这些状态是否构成一个完整的标准正交基。因此，可以在qudit Bell基中测量两个qudit。与量子位的情况类似，通过扩展3.6.1节中的参数，很容易看出量子位状态可以生成共享随机性的dit。

## 物理代写|量子力学代写quantum mechanics代考|Schmidt Decomposition

Schmidt分解是量子信息论中分析二部纯态最重要的工具之一，它表明任何纯二部态都可以分解为协调正交态的叠加。它是线性代数中著名的奇异值分解定理的一个结果。我们将这一结果形式化地表述为以下定理:

$$|\psi\rangle_{A B} \in \mathcal{H}A \otimes \mathcal{H}B$$其中$\mathcal{H}_A$和$\mathcal{H}_B$是有限维希尔伯特空间，不一定是相同的维数，还有$||\psi\rangle{A B} |_2=1$。那么，可以将这种状态表示为: $$|\psi\rangle{A B} \equiv \sum_{i=0}^{d-1} \lambda_i|i\rangle_A|i\rangle_B,$$

$$d \leq \min \left{\operatorname{dim}\left(\mathcal{H}_A\right), \operatorname{dim}\left(\mathcal{H}_B\right)\right}$$

## 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|量子力学代写quantum mechanics代考|Isometric Extension of a Quantum Channel

statistics-lab™ 为您的留学生涯保驾护航 在代写量子力学quantum mechanics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。我们的专家在代写量子力学quantum mechanics代写方面经验极为丰富，各种代写量子力学quantum mechanics相关的作业也就用不着说。

## 物理代写|量子力学代写quantum mechanics代考|Isometric Extension of a Quantum Channel

We now give a general definition for an isometric extension of a quantum channel:
DEfinition 5.2.1 (Isometric Extension) Let $\mathcal{H}A$ and $\mathcal{H}_B$ be Hilbert spaces, and let $\mathcal{N}: \mathcal{L}\left(\mathcal{H}_A\right) \rightarrow \mathcal{L}\left(\mathcal{H}_B\right)$ be a quantum channel. Let $\mathcal{H}_E$ be a Hilbert space with dimension no smaller than the Choi rank of the channel $\mathcal{N}$. An isometric extension or Stinespring dilation $U: \mathcal{H}_A \rightarrow \mathcal{H}_B \otimes \mathcal{H}_E$ of the channel $\mathcal{N}$ is a linear isometry such that $$\operatorname{Tr}_E\left{U X_A U^{\dagger}\right}=\mathcal{N}{A \rightarrow B}\left(X_A\right),$$
for $X_A \in \mathcal{L}\left(\mathcal{H}A\right)$. The fact that $U$ is an isometry is equivalent to the following conditions: $$U^{\dagger} U=I_A, \quad U U^{\dagger}=\Pi{B E},$$
where $\Pi_{B E}$ is a projection of the tensor-product Hilbert space $\mathcal{H}B \otimes \mathcal{H}_E$. NOtATION 5.2.1 We often write a channel $\mathcal{N}: \mathcal{L}\left(\mathcal{H}_A\right) \rightarrow \mathcal{L}\left(\mathcal{H}_B\right)$ as $\mathcal{N}{A \rightarrow B}$ in order to indicate the input and output systems explicitly. Similarly, we often write an isometric extension $U: \mathcal{H}A \rightarrow \mathcal{H}_B \otimes \mathcal{H}_E$ of $\mathcal{N}$ as $U{A \rightarrow B E}^{\mathcal{N}}$ in order to indicate its association with $\mathcal{N}$ explicitly, as well the fact that it accepts an inputsystem $A$ and has output systems $B$ and $E$. The system $E$ is often referred to as an “environment” system. Finally, there is a quantum channel $\mathcal{U}{A \rightarrow B E}^{\mathcal{N}}$ associated to an isometric extension $U{A \rightarrow B E}^{\mathcal{N}}$, which is defined by
$$\mathcal{U}{A \rightarrow B E}^{\mathcal{N}}\left(X_A\right)=U X_A U^{\dagger}$$ for $X_A \in \mathcal{L}\left(\mathcal{H}_A\right)$. Note that $\mathcal{U}{A \rightarrow B E}^{\mathcal{N}}$ is a quantum channel with a single Kraus operator $U$ given that $U^{\dagger} U=I_A$.

## 物理代写|量子力学代写quantum mechanics代考|Isometric Extension from Kraus Operators

It is possible to determine an isometric extension of a quantum channel directly from a set of Kraus operators. Consider a quantum channel $\mathcal{N}{A \rightarrow B}$ with the following Kraus representation: $$\mathcal{N}{A \rightarrow B}\left(\rho_A\right)=\sum_j N_j \rho_A N_j^{\dagger} .$$

An isometric extension of the channel $\mathcal{N}{A \rightarrow B}$ is the following linear map: $$U{A \rightarrow B E}^{\mathcal{N}} \equiv \sum_j N_j \otimes|j\rangle_E .$$
It is straightforward to verify that the above map is an isometry:
\begin{aligned} \left(U^{\mathcal{N}}\right)^{\dagger} U^{\mathcal{N}} & =\left(\sum_k N_k^{\dagger} \otimes\left\langle\left. k\right|E\right)\left(\sum_j N_j \otimes|j\rangle_E\right)\right. \ & =\sum{k, j} N_k^{\dagger} N_j\langle k \mid j\rangle \ & =\sum_k N_k^{\dagger} N_k \ & =I_A . \end{aligned}
The last equality follows from the completeness condition of the Kraus operators. As a consequence, we get that $U^{\mathcal{N}}\left(U^{\mathcal{N}}\right)^{\dagger}$ is a projector on the joint system $B E$, which follows by the same reasoning given in (4.259). Finally, we should verify that $U^{\mathcal{N}}$ is an extension of $\mathcal{N}$. Applying the channel $\mathcal{U}{A \rightarrow B E}^{\mathcal{N}}$ to an arbitrary density operator $\rho_A$ gives the following map: \begin{aligned} \mathcal{U}{A \rightarrow B E}^{\mathcal{N}}\left(\rho_A\right) & \equiv U^{\mathcal{N}} \rho_A\left(U^{\mathcal{N}}\right)^{\dagger} \ & =\left(\sum_j N_j \otimes|j\rangle_E\right) \rho_A\left(\sum_k N_k^{\dagger} \otimes\left\langle\left. k\right|E\right)\right. \ & =\sum{j, k} N_j \rho_A N_k^{\dagger} \otimes|j\rangle\left\langle\left. k\right|E,\right. \end{aligned} and tracing out the environment system gives back the original quantum channel $\mathcal{N}{A \rightarrow B}$ :
$$\operatorname{Tr}E\left{\mathcal{U}{A \rightarrow B E}^{\mathcal{N}}\left(\rho_A\right)\right}=\sum_j N_j \rho_A N_j^{\dagger}=\mathcal{N}_{A \rightarrow B}\left(\rho_A\right)$$

# 量子力学代考

## 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。