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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|理论力学作业代写Theoretical Mechanics代考|The existence of a classical world

We come back now to the ambiguous nature attributed by Bohr to the measuring apparatus. Does it belong to the classical or to the quantum world? In order to answer to this question we must preliminarly discuss the issue of the classical limit of Quantum Mechanics. We know that in the standard formulation of QM a system’s state is represented by a wave function in the cohordinate’s space (or a state vector in Hilbert space) which contains all the statistical properties of the system’s variables. The wave function allows to calculate the probability of finding a given value of any variable of the system as a result of a measurement by means of a suitable instrument. More precisely, if the wave function is given by
$$\psi=\mathrm{c}{1} \psi{1}+\mathrm{c}{2} \psi{2}$$
(where $\psi_{1}\left(\psi_{2}\right)$ represents a state in which the variable $G$ has with certainty the value $g_{1}\left(g_{2}\right)$ ), the probability of finding $g_{1}\left(g_{2}\right)$ is $\left|c_{1}\right|^{2}\left(\left|c_{2}\right|^{2}\right)$. In Bohr’s interpretation this means that the variable $G$ does not have one of these values before its measurement but assumes one or the other value with the corresponding probability during the act of measurement. Now comes the question: is this interpretation always valid, even when $g_{1}$ and $g_{2}$ are macroscopically different?

The answer poses a serious problem. One can in effect prove that in the limit when Planck’s constant $h$ tends to zero the probability distribution of the quantum state represented by $\Psi$ tends to the probability distribution in phase space of the corresponding classical statistical ensemble labeled by the same values of the system’s quantum variables. More precisely, in this limit $\left|c_{1}\right|^{2}$ and $\left|c_{2}\right|^{2}$ represent the probabilities of finding the values $g_{1}\left(g_{2}\right)$ of the classical variable corresponding to the quantum variable $G$. In this case, however, the interpretation of these probabilities is completely different. In classical statistical mechanics we assume that they express an incomplete knowlwdge of the values of $G$ actually possessed by the different systems of the ensemble. We assume in fact that, if the ensemble is made of $N$ systems, there are $N\left|c_{1}\right|^{2}$ systems wuth the value $g_{1}$ of $G$ and $N\left|c_{2}\right|^{2}$ systems with the value $g_{2}$ of $G$ to start with. Each system has a given value of $G$ from the beginning, even if we don’t know it.

## 物理代写|理论力学作业代写Theoretical Mechanics代考|Quantum and classical uncertainties

A way out of this dilemma, however, exists. We have shown, with Maurizio Serva (Cini M. Serva M. 1990, 1992), that, without changing the basic principlees and the predictions of Quantum Mechanics, one can save at the same time both Bohr’s interpretation of the phenomena of the quantum domain, and Einstein’s belief in the objective relity of the classical world in which we live. We have shown in fact that the uncertainty product between $x$ and $p$ can be written for any state of a quanton in the form
$$(\Delta x \Delta p)^{2}=(\Delta x \Delta p){\mathrm{cl}^{2}}+(\Delta x \Delta p){\mathrm{q}^{2}}{ }^{2}$$
where $(\Delta x \Delta p){\mathrm{q}}$ is of the order of the minimum value $h / 4 \pi$ of the Heisenberg uncertainty relation and $(\Delta x \Delta p){\mathrm{c}}$ is the classical expression of the product of the indeterminations $\Delta x$ and $\Delta p$ predicted by the probability distribution of the classical statistical mechanics distribution corresponding to the quantum state when $h \rightarrow 0$. It is therefore reasonable to attribute to each of these two terms the meaning relevant to its physical domain.

In the typical quantum domain the clssical term vanishes and the indeterminacy is ontological, namely the variables $x$ and $p$ do not have a definite value before the system’s interaction with a measuring instrument. When the accuracy of the act of measurement reduces the indeterminacy of one variable, the the indeterminacy of the other one increases. Their product cannot become smaller than $\mathrm{h} / 4 \pi$.

As soon as the uncertainty product calculated from the state y acquires a classical term (which survives in the limit $h \rightarrow 0$ ) the total indeterminacy becomes epistemic, namely it represents an incomplete knowledge of the value that the measured variable really had before being measured. In this case it is possible to measure the variables $x$ and $p$ in such a way as to reduce at the same time both $\Delta x$ et $\Delta p$ without violating any quantum principle. These measurements reduce simply our ignorance. There is no instantaneous localization of the quanton in coordinate or momentum space as a consequence of the interaction between system and instrument, because position and momentum (within the intrinsic quantum uncertainty) were already localized.

This solution solves therefore the contradiction between the different interpretations of the total uncertainty product, and allows a reconciliation of the two alternative conceptions of physical reality proposed by Einstein and Bohr. It saves a realistic conception of the world as a whole by recognizing that macroscopic objects have objective properties independently of their being observed by any “observer”, and, at the same time, that at the microscopic objects have properties dependent of the macroscopic objects with which they interact.

## 物理代写|理论力学作业代写Theoretical Mechanics代考|The existence of a classical world

$$\psi=\mathrm{c} 1 \psi 1+\mathrm{c} 2 \psi 2$$
(在哪里 $\psi_{1}\left(\psi_{2}\right)$ 表示变量的状态 $G$ 有确定的价值 $g_{1}\left(g_{2}\right)$ )，找到的概率 $g_{1}\left(g_{2}\right)$ 是 $\left|c_{1}\right|^{2}\left(\left|c_{2}\right|^{2}\right)$. 在玻尔的解释中，这意味着变量 $G$ 在测量之 前不具有这些值之一，但在测量行为期间假定具有相应概率的一个或另一个值。现在问题来了：这种解释是否总是有效的，即使 $g_{1}$ 和 $g_{2}$ 宏 观上有区别吗?

## 物理代写|理论力学作业代写Theoretical Mechanics代考|Quantum and classical uncertainties

$$(\Delta x \Delta p)^{2}=(\Delta x \Delta p) \mathrm{cl}^{2}+(\Delta x \Delta p) \mathrm{q}^{22}$$

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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。

assignmentutor™您的专属作业导师
assignmentutor™您的专属作业导师