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## 物理代写|理论力学代写theoretical mechanics代考|Measurements and Errors

The search for laws prepares the ground on which the principles of nature are built. We generalize by relating comparable things. Of course, this has its limitations. When are two things equal to each other, and when are they only similar? The following is important for all measurements, but also for quantum theory and for thermodynamics and statistics.

We consider an arbitrary physical quantity which we assume does not change with time and can be measured repeatedly, e.g., the length of a rod or the oscillation period of a pendulum. Each measurement is carried out in terms of a “multiple of a scale unit”. It may be that a tenth of the unit can be estimated, but certainly not essentially finer divisions. An uncertainty is therefore attached to each of our measured values, and this uncertainty can be estimated rather simply.

It is more difficult to find a statement about how well an instrument is adjusted and whether there are further systematic errors. We will not deal with these questions here, but we do want to be able to estimate the bounds on the error from the statistical fluctuations of our measured data.

In particular, if we repeat our measurement in order to ensure against erroneous readings, then the values $x_{n}(n \in{1, \ldots, N})$ may not all be equal, e.g., we may find three times $10.1$ scale units (that is, three values with $10.05<x_{n}<10.15$ ), eight times $10.2$ (eight values with $10.15<x_{n}<10.25$ ), and one $10.3$ (with $10.25<x_{n}<$ $10.35)$ in an arbitrary order. Apparently, there are always “measurement errors”, the origin of which we do not know. (Systematic errors can be estimated separately.) Therefore, we have to assign a greater uncertainty than the assumed scale fineness to the results of our measurements.

## 物理代写|理论力学代写theoretical mechanics代考|Mean Value and Average Error

After $N$ measurements of $x$, we have a sequence of measured values $\left{x_{1}, \ldots, x_{N}\right}$. These values are generally not all equal, but we want to assume that their fluctuations are purely random, and we shall only deal with such errors in the following.

Since none of the measurement readings should be preferred, the true value $x_{0}$ is assumed to be near the mean value
$$\bar{x} \equiv \frac{1}{N} \sum_{n=1}^{N} x_{n},$$
because deviations may occur equally often to higher or lower values: $x_{0} \approx \bar{x}$. Our best estimate for the true value $x_{0}$ is the mean value $\bar{x}$.

Here, the less the values $x_{n}$ deviate from $\bar{x}$, the more we trust the approximation $x_{0} \approx \bar{x}$. From the fluctuations, we deduce a measure $\Delta x$ for the uncertainty in our estimate. To do this, we take the squares $\left(x_{n}-\bar{x}\right)^{2}$ of the deviations rather than their absolute values $\left|x_{n}-\bar{x}\right|$, because the squares are differentiable, while the absolute values are not, something we shall exploit in Sect. 1.3.7. However, we may take their mean value
$$\overline{(x-\bar{x})^{2}}=\overline{x^{2}-2 \bar{x} x+\bar{x}^{2}}=\overline{x^{2}}-2 \bar{x} \bar{x}+\bar{x}^{2}=\overline{x^{2}}-\bar{x}^{2}$$
as a measure for the uncertainty only in the limit of many measurements, not just a small number of measurements. So, for a single measurement nothing whatsoever can be said about the fluctuations. For a second measurement, we would have only a first clue about the fluctuations. In fact, we shall set
$$(\Delta x)^{2}=\frac{1}{N-1} \sum_{n=1}^{N}\left(x_{n}-\bar{x}\right)^{2}=\frac{N}{N-1} \overline{(x-\bar{x})^{2}},$$
as will be justified in the following sections. Here we shall rely on a simple special case of the law of error propagation. But this law can also be proven rather easily in its general form and will be needed for other purposes. Therefore, we prove it generally now, whereupon thé last equation can be derivèd easily. To this end, however, we have to consider general properties of error distributions.

## 物理代写|理论力学代写theoretical mechanics代考|Error Distribution

We presume that the errors are distributed in a purely random manner. Then the error probability can be derived from sufficiently many readings of the measurement $(N \gg 1$ ). From the relative occurrences of the single values, we can determine the probability $\rho(\varepsilon) \mathrm{d} \varepsilon$ that the error lies between $\varepsilon$ and $\varepsilon+\mathrm{d} \varepsilon$. The probability density $\rho(\varepsilon)$ is characterized essentially by the average error $\sigma$, as the following considerations show.

Each probability distribution $\rho$ has to be normalized to unity and may not take negative values: $\int \rho(\varepsilon) \mathrm{d} \varepsilon=1$ and $\rho(\varepsilon) \geq 0$ for all $\varepsilon(\in \mathrm{R})$. In addition, we expect $\rho(\varepsilon)$ to be essentially different from zero only for $\varepsilon \approx 0$ and to tend to zero monotonically with increasing $|\varepsilon|$. The distribution is also assumed to be an even function, at least in the important region around the zero point: $\rho(\varepsilon)=\rho(-\varepsilon)$. Hence, $\int \varepsilon \rho(\varepsilon) \mathrm{d} \varepsilon=0$. The next important feature is the width of the distribution. It can be measured with the second moment, the average of the squared errors $\sigma(\geq 0)$, also called the mean square fluctuation or variance,
$$\sigma^{2} \equiv \int \varepsilon^{2} \rho(\varepsilon) \mathrm{d} \varepsilon$$
Note, however, that the mean square error is not finite for all allowable error distributions, e.g., for the Lorentz distribution $\rho(\varepsilon)=\gamma /\left{\pi\left(\varepsilon^{2}+\gamma^{2}\right)\right}$, which is instead characterized by half the Lorentz half-width $\gamma$-more on that in the discussion around Fig. 5.6.

## 物理代写|理论力学代写theoretical mechanics代考|Mean Value and Average Error

X¯≡1ñ∑n=1ñXn,

(X−X¯)2―=X2−2X¯X+X¯2―=X2―−2X¯X¯+X¯2=X2―−X¯2

(ΔX)2=1ñ−1∑n=1ñ(Xn−X¯)2=ññ−1(X−X¯)2―,

σ2≡∫e2ρ(e)de

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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。