### 物理代写|原子物理代写Atomic and Molecular Physics代考|SL221463

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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|原子物理代写Atomic and Molecular Physics代考|Numerical Solutions at Large Times

In mathematically rigorous procedures the point at $t \rightarrow \infty$ is approached by defining a new variable $\tau=1 / \mathrm{t}$ and going to the point $\tau=0$. Use of the $\tau$ variable gives a version of the Schrödinger equation that simply moves the singularity at infinite time to another location. Fundamental obstacles still remain; in particular the divergent “explosion” factor is still present. This factor can be removed from the wave function and the solution, thus reduced, no longer oscillates rapidly at large $r$. The function, however, still expands to fill a large volume therefore to preserve normalization its magnitude must decrease accordingly. This means, for example, that at distances of the order of 1000 au the magnitude of the wave function is of the order of $10^{-9}$ compared with starting values of the order of unity, thus making it difficult to integrate to distances where Eq. (5.5) applies.

To circumvent the dimension problem one may scale the coordinates so that space expands with time. This does not lead to new singularities if the explosion factor has been removed. The corresponding theory closely follows the earlier hidden crossing theory of Solov’ev mentioned in Sec. IV. To make the transformations somewhat more general, scaling by a factor $R_{s}=\sqrt{b_{x}^{2}+v_{x}^{2} x^{2}}$ rather than the physical internuclear distance $R=\sqrt{b^{2}+v^{2} t^{2}}$ is employed. This has the advantage that scale of the coordinates in the region near $t=0$ is selected by the parameter $b_{s}$, which can be chosen to obtain optimal precision in that crucial region while still maintaining the linear scaling with $t$ for large times. The parameter $v_{s}$ is included only for dimensional consistency since changing $v_{\mathrm{s}}$ is equivalent to changing the mesh of the time step $\Delta t$.

## 物理代写|原子物理代写Atomic and Molecular Physics代考|Hydrodynamic Representation of the Schrödinger Equation

Fluid flow is governed by the Navier-Stokes equations. For our purposes the most general form of these equations is not needed, rather a special form where both the viscosity and the bulk viscosity of the fluid vanish, are used. This special case is one of Euler’s equations [29] for fluid flow for an ideal fluid with density $\rho$
$$\frac{\partial v}{\partial t}+\boldsymbol{v} \cdot \nabla \boldsymbol{v}+\frac{\nabla \boldsymbol{p}}{\rho}-\frac{\boldsymbol{f}}{\rho}=0$$
where $f$ is the force per unit volume.
Eq. (6.6) is essentially Newton’s equation for the acceleration of a fluid element under the influence of a force due to the pressure $p$ and force $f$ per unit volume due to an external field.

Note that $f / p$ has the dimensions of force per unit mass. In addition there is the equation of continuity expressing the conservation of mass:
$$\frac{\partial}{\partial t}+\nabla \cdot(\rho v)=0$$

## 物理代写|原子物理代写Atomic and Molecular Physics代考|Hydrodynamic Representation of the Schrödinger Equation

$$\frac{\partial v}{\partial t}+\boldsymbol{v} \cdot \nabla \boldsymbol{v}+\frac{\nabla \boldsymbol{p}}{\rho}-\frac{\boldsymbol{f}}{\rho}=0$$

$$\frac{\partial}{\partial t}+\nabla \cdot(\rho v)=0$$

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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。