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

## 物理代写|量子力学代写quantum mechanics代考|Numerov method

The Numerov method is used to calculate numerically the ordinary second order differential equation such as the one-particle quantum harmonic oscillator (Chapter 14) and radial wave function of the hydrogen atom (Chapter 17):
\begin{aligned} &-\frac{\hbar^{2}}{2 m} \frac{d^{2} \psi}{d x^{2}}+V(x) \psi(x)=E \psi(x) \ &\frac{d^{2} \psi}{d x^{2}}=\frac{2 m}{\hbar^{2}}(V(x)-E) \psi(x) \ &K^{2}(x)=\frac{2 m}{\hbar^{2}}(E-V(x)) \ &\frac{d^{2} \psi}{d x^{2}}+K^{2}(x) \psi(x)=0 \end{aligned}
Any smooth function can be rewritten as a sum of infinite polynomial terms. Then, let us use the general Taylor series (see previous section):
$$f(x+a)-f(a)+\frac{x}{1 !} f^{\prime}(a)+\frac{x^{2}}{2 !} f^{\prime \prime}(a)+\frac{x^{3}}{3 !} f^{\prime \prime}(a)+\ldots+R_{n}(x)$$
to represent the wave function $\psi(\mathrm{x})$ :
$$\psi(x+h)=\psi(x)+h \psi^{\prime}(x)+\frac{h^{2}}{2 !} \psi^{\prime \prime}(x)+\frac{h^{3}}{3 !} \psi^{\prime \prime \prime}(x)+\frac{h^{4}}{4 !} \psi^{i v}(x)+\ldots$$
Where $\mathrm{h}$ is a small increment. Observe that $\mathrm{h}$ and $\mathrm{x}$ were interchanged in their positions in the equation.
The Taylor series can also be written in the following way (see previous section):
$$\psi(x-h)=\psi(x)-h \psi^{\prime}(x)+\frac{h^{2}}{2 !} \psi^{\prime \prime}(x)-\frac{h^{3}}{3 !} \psi^{\prime \prime \prime}(x)+\frac{h^{4}}{4 !} \psi^{i v}(x)+\ldots$$

## 物理代写|量子力学代写quantum mechanics代考|Matrix and matrix multiplication

A matrix is used to arrange real or complex scalars (or functions) in a rectangular array of $m, n$ dimensions where, $m$ is the dimension of the row (m-rows) and $n$ is the dimension of the column ( $\mathrm{n}$-column). The matrix is designated as $\mathrm{m} \mathrm{x} \mathrm{n}$ matrix or matrix of orders $m, n$. The order is always: number of rows $x$ number of columns. The elements of the matrix are designated by $\mathrm{i}$ and $\mathrm{j}$ indexes.
$$\mathbf{A}=\left[\begin{array}{cccc} a_{11} & a_{12} & \cdots & a_{1 n} \ a_{21} & a_{22} & \cdots & a_{2 n} \ \vdots & \vdots & \cdots & \vdots \ a_{n 1} & a_{2 n} & \cdots & a_{n n} \end{array}\right]$$
A very important property to the quantum matrix mechanics is that the commutative property is not applied to matrices (see section 39 -commutators). In chapter nine we see that the linear momentum and position matrices do not commute and this brings important consequences to the quantum angular momentum, uncertainty principle among other consequences.

When both matrices are not square matrices (whose rows and columns have the same number), probably the inverted order of the product provides no matrix at all. Even when both matrices are square matrices, the matrix $\mathbf{A B}$ is mostly different from matrix BA.

The matrix product $\mathbf{A B}$ (in this order) is defined only if: the number of columns of $\mathbf{A}$ is equal to the number of rows of $\mathbf{B}$. The dimensions of this matrix product is the number of rows of $\mathrm{A}$ and the number of columns of $\mathbf{B}$. Likewise, the product $\mathbf{B A}$ is defined only if: the number of columns of $\mathbf{B}$ is equal to the number of rows of $\mathbf{A}$.
$$\mathbf{A}{n \times m} \cdot \mathbf{B}{m \times p}=\mathbf{C}{n \times p}$$ However: $\mathbf{B}{m \times p} \cdot \mathbf{A}{n \times m}=$ impossible \begin{aligned} &\mathbf{A}{n \times n} \cdot \mathbf{B}{n \times n} \neq \mathbf{B}{n \times n} \cdot \mathbf{A}{n \times n} \ &\mathbf{C}=\mathbf{A} \cdot \mathbf{B} \therefore c{i j}=\sum_{k=1}^{n} a_{i k} b_{k j} \end{aligned}

# 量子力学代考

## 物理代写|量子力学代写quantum mechanics代考|Numerov method

Numerov 方法用于数值计算普通二阶微分方程，例如单粒子量子谐振子 (第 14 章) 和氢原子的径向波函数 (第 17 章)：
$$-\frac{\hbar^{2}}{2 m} \frac{d^{2} \psi}{d x^{2}}+V(x) \psi(x)=E \psi(x) \quad \frac{d^{2} \psi}{d x^{2}}=\frac{2 m}{\hbar^{2}}(V(x)-E) \psi(x) K^{2}(x)=\frac{2 m}{\hbar^{2}}(E-V(x)) \quad \frac{d^{2} \psi}{d x^{2}}+K^{2}(x) \psi(x)=0$$

$$f(x+a)-f(a)+\frac{x}{1 !} f^{\prime}(a)+\frac{x^{2}}{2 !} f^{\prime \prime}(a)+\frac{x^{3}}{3 !} f^{\prime \prime}(a)+\ldots+R_{n}(x)$$

$$\psi(x+h)=\psi(x)+h \psi^{\prime}(x)+\frac{h^{2}}{2 !} \psi^{\prime \prime}(x)+\frac{h^{3}}{3 !} \psi^{\prime \prime \prime}(x)+\frac{h^{4}}{4 !} \psi^{i v}(x)+\ldots$$

$$\psi(x-h)=\psi(x)-h \psi^{\prime}(x)+\frac{h^{2}}{2 !} \psi^{\prime \prime}(x)-\frac{h^{3}}{3 !} \psi^{\prime \prime \prime}(x)+\frac{h^{4}}{4 !} \psi^{i v}(x)+\ldots$$

## 物理代写|量子力学代写quantum mechanics代考|Matrix and matrix multiplication

$$\mathbf{A}=\left[\begin{array}{lllllllllllll} a_{11} & a_{12} & \cdots & a_{1 n} a_{21} & a_{22} & \cdots & a_{2 n} & \vdots & \cdots & \vdots a_{n 1} & a_{2 n} & \cdots & a_{n n} \end{array}\right]$$

$$\mathbf{A} n \times m \cdot \mathbf{B} m \times p=\mathbf{C} n \times p$$

$$\mathbf{A} n \times n \cdot \mathbf{B} n \times n \neq \mathbf{B} n \times n \cdot \mathbf{A} n \times n \quad \mathbf{C}=\mathbf{A} \cdot \mathbf{B} \therefore c i j=\sum_{k=1}^{n} a_{i k} b_{k j}$$

## 有限元方法代写

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## MATLAB代写

MATLAB 是一种用于技术计算的高性能语言。它将计算、可视化和编程集成在一个易于使用的环境中，其中问题和解决方案以熟悉的数学符号表示。典型用途包括：数学和计算算法开发建模、仿真和原型制作数据分析、探索和可视化科学和工程图形应用程序开发，包括图形用户界面构建MATLAB 是一个交互式系统，其基本数据元素是一个不需要维度的数组。这使您可以解决许多技术计算问题，尤其是那些具有矩阵和向量公式的问题，而只需用 C 或 Fortran 等标量非交互式语言编写程序所需的时间的一小部分。MATLAB 名称代表矩阵实验室。MATLAB 最初的编写目的是提供对由 LINPACK 和 EISPACK 项目开发的矩阵软件的轻松访问，这两个项目共同代表了矩阵计算软件的最新技术。MATLAB 经过多年的发展，得到了许多用户的投入。在大学环境中，它是数学、工程和科学入门和高级课程的标准教学工具。在工业领域，MATLAB 是高效研究、开发和分析的首选工具。MATLAB 具有一系列称为工具箱的特定于应用程序的解决方案。对于大多数 MATLAB 用户来说非常重要，工具箱允许您学习应用专业技术。工具箱是 MATLAB 函数（M 文件）的综合集合，可扩展 MATLAB 环境以解决特定类别的问题。可用工具箱的领域包括信号处理、控制系统、神经网络、模糊逻辑、小波、仿真等。

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