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• Foundations of Data Science 数据科学基础

## 物理代写|固体物理代写Solid-state physics代考|The ground-state

Let us consider a metal specimen at zero temperature. Since its bulk properties do not depend on the shape, for mathematical convenience we will consider a cubic sample with side $L$ and faces normal to the $x, y$, and $z$ Cartesian axes. The singleparticle wavefunction for any (free and independent) electron of the conduction gas is obtained by solving the Schrödinger equation $$-\frac{\hbar^2}{2 m_e} \nabla^2 \psi(\mathbf{r})=E \psi(\mathbf{r}),$$
where $E$ is the electron energy. By imposing the Born-von Karman condition stated in equation (1.4), we easily get the normalised wavefunction
$$\psi_{\mathbf{k}}(\mathbf{r})=\frac{1}{L^{3 / 2}} \exp (i \mathbf{k} \cdot \mathbf{r})=\frac{1}{\sqrt{V}} \exp (i \mathbf{k} \cdot \mathbf{r}),$$
where $V=L^3$ is the system volume and the electron wavevector $\mathbf{k}$ has the following Cartesian components ${ }^{11}$
$$k_x=\frac{2 \pi}{L} \xi_x \quad k_x=\frac{2 \pi}{L} \xi_y \quad k_x=\frac{2 \pi}{L} \xi_z,$$
with $\xi_x, \xi_y, \xi_z=0, \pm 1, \pm 2, \pm 3, \ldots$ We stress that (i) the free electron wavefunction given in equation (7.23) has been labelled by $\mathbf{k}$ which plays the role of a quantum number for the crystalline states ${ }^{12}$, (ii) the wavefunction given in equation (7.23) does obey the Bloch theorem discussed in section 6.3: in this specific case we simply have $u_{\mathbf{k}}(\mathbf{r})=1$. The electron energy is
$$E=\frac{\hbar^2 k^2}{2 m_e}=\frac{\hbar^2}{2 m_e}\left(k_x^2+k_y^2+k_z^2\right),$$
a result which makes quite evident the function of quantum numbers associated with $k_x, k_y$, and $k_z$. By using the quantum mechanical operator $\hat{\mathbf{p}}=-i \hbar \nabla$, we easily obtain the electron momentum
$$\mathbf{p}=\hbar \mathbf{k},$$
and the corresponding electron velocity $\mathbf{v}=\hbar \mathbf{k} / m_e$.

## 物理代写|固体物理代写Solid-state physics代考|Finite temperature properties

Let us now consider a metal in equilibrium at temperature $T>0 \mathrm{~K}$. In this case the eDOS is written as
\begin{aligned} G(E, T) &=G(E) n_{\mathrm{FD}}(E, T) \ &=\frac{V}{2 \pi^2 \hbar^3}\left(2 m_{\mathrm{e}}\right)^{3 / 2} \frac{1}{1+\exp \left[\left(E-\mu_{\mathrm{c}}\right)\right] / k_{\mathrm{B}} T} E^{\mathrm{1} / 2}, \end{aligned}
where we have combined the expression given in equation (7.28), which is a mere counting of states, with the finite-temperature probability $n_{\mathrm{FD}}(E, T)$ that the quantum level $E$ is occupied, a correction entering our theory through equation (6.7). The $G(E, T)$ function is plotted in figure $7.5$ (thick blue line), together with its zero-temperature counterpart (thin black line). We remark that in plotting this figure we have neglected the temperature dependence of the chemical potential and, accordingly, we have set $\mu_{\mathrm{c}}=E_{\mathrm{F}}$ at any $T \geqslant 0$. We will very soon critically re-address this assumption, proving that it is valid to a very good extent.

The number $N$ of electrons is obviously unaffected by temperature and we can therefore cast the normalisation condition (previously expressed as in equation (7.29)) in a new form $$N=\int_0^{+\infty} G(E, T) d E=\int_0^{+\infty} G(E) n_{\mathrm{FD}}(E, T) d E$$
which allows us to interpret the shaded area of figure $7.5$ as the conserved number of electrons. Since this notion is valid for any selected range of energy, we can develop a new interesting concept.

At first we remark that the area under the $G(E, T)$ function corresponding to the energy interval $E_{\mathrm{F}} \leqslant E \leqslant+\infty$ represents the number of those few electrons that, upon increasing temperature from zero to $T$, have been promoted to energies above the Fermi energy. Obviously, this promotion has emptied an equal number of states just below $E_{\mathrm{F}}$ : this number corresponds to the area in between the function $G(E, T)$ and its zero-temperature counterpart and calculated for $0 \leqslant E \leqslant E_{\mathrm{F}}$. This phenomenon is usually referred to as thermal excitation of electrons and it is summarised by stating that at any finite temperature a number of electrons of the conduction gas are promoted from quantum states just below the Fermi energy to states just above it. The thermal excitation phenomenon mostly involves electrons with energy in the interval $\sim k_{\mathrm{B}} T$ centred at $E_{\mathrm{F}}$.

# 固体物理代写

## 物理代写|固体物理代写Solid-state physics代考|基态

$$\psi_{\mathbf{k}}(\mathbf{r})=\frac{1}{L^{3 / 2}} \exp (i \mathbf{k} \cdot \mathbf{r})=\frac{1}{\sqrt{V}} \exp (i \mathbf{k} \cdot \mathbf{r}),$$

$$k_x=\frac{2 \pi}{L} \xi_x \quad k_x=\frac{2 \pi}{L} \xi_y \quad k_x=\frac{2 \pi}{L} \xi_z,$$
with $\xi_x, \xi_y, \xi_z=0, \pm 1, \pm 2, \pm 3, \ldots$我们强调(i)式(7.23)中给出的自由电子波函数被$\mathbf{k}$标记，它扮演晶体态的量子数的角色${ }^{12}$，(ii)式(7.23)中给出的波函数符合第6.3节中讨论的布洛赫定理:在这种具体情况下，我们只需得到$u_{\mathbf{k}}(\mathbf{r})=1$。电子能量
$$E=\frac{\hbar^2 k^2}{2 m_e}=\frac{\hbar^2}{2 m_e}\left(k_x^2+k_y^2+k_z^2\right),$$
，这个结果很明显地说明了与$k_x, k_y$和$k_z$相关的量子数的函数。利用量子力学算符$\hat{\mathbf{p}}=-i \hbar \nabla$，我们很容易得到电子动量
$$\mathbf{p}=\hbar \mathbf{k},$$

## 物理代写|固体物理代写固相物理代考|有限温度特性

. . >物理代写|

\begin{aligned} G(E, T) &=G(E) n_{\mathrm{FD}}(E, T) \ &=\frac{V}{2 \pi^2 \hbar^3}\left(2 m_{\mathrm{e}}\right)^{3 / 2} \frac{1}{1+\exp \left[\left(E-\mu_{\mathrm{c}}\right)\right] / k_{\mathrm{B}} T} E^{\mathrm{1} / 2}, \end{aligned}
，其中我们将公式(7.28)中给出的表达式(仅仅是状态计数)与量子能级$E$被占据的有限温度概率$n_{\mathrm{FD}}(E, T)$结合起来，通过公式(6.7)进入我们的理论修正。$G(E, T)$函数被绘制在图$7.5$(粗蓝线)中，以及它的零温度对应函数(细黑线)。我们注意到，在绘制这个图时，我们忽略了化学势的温度依赖性，因此，我们将$\mu_{\mathrm{c}}=E_{\mathrm{F}}$设为任何$T \geqslant 0$。我们很快就会批判性地重新处理这个假设，证明它在很大程度上是有效的

，这允许我们将图$7.5$的阴影区域解释为电子的保守数量。由于这个概念对任何选定的能量范围都是有效的，我们可以发展出一个新的有趣的概念

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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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