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# 物理代写|多体物理作业代写many-body physics代考|Conventions, Notation, Reminders

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## 物理代写|多体物理作业代写many-body physics代考|Units, Physical Constants

We will use a system of units in which
$$\hbar=k_{B}=e=1$$
In such a system of units, we measure energies, temperatures, and frequencies in electron volts. The basic rule of thumb is that $1 \mathrm{eV} \sim 10,000 \mathrm{~K}$ or $1 \mathrm{meV} \sim 10 \mathrm{~K}$, while a frequency of $1 \mathrm{~Hz}$ corresponds to $\sim 6 \times 10^{-16} \mathrm{eV}$. The Fermi energy in a typical metal is $\sim 1 e V$. In a conventional, ‘low-temperature’ superconductor, $T_{c} \sim 0.1-1 m e V$. This corresponds to a frequency of $10^{11}-10^{12} \mathrm{~Hz}$ or a wavelength of light of $\sim 1 \mathrm{~cm}$. We could set the speed of light to 1 and measure distances in $(e v)^{-1}$, but most of the velocities which we will be dealing with are much smaller than the speed of light, so this is not very useful. The basic unit of length is the angstrom, $1 \AA\left(10^{-10} \mathrm{~m}\right.$. The lattice spacing in a typical crystal is $\sim 1-10 \AA$.

## 物理代写|多体物理作业代写many-body physics代考|Mathematical Conventions

Vectors will be denoted in boldface, $\mathbf{x}, \mathbf{E}$, or with a Latin subscript $x_{i}, E_{i}, i=$ $1,2, \ldots, d$. Unless otherwise specified, we will work in $d=3$ dimensions. Occasionally,we will use Greek subscripts, e.g. $j_{\mu}, \mu=0,1, \ldots, d$ where the 0 -component is the time-component as in $x_{\mu}=(t, x, y, z)$. Unless otherwise noted, repeated indices are summed over, e.g. $a_{i} b_{i}=a_{1} b_{1}+a_{2} b_{2}+a_{3} b_{3}=\mathbf{a} \cdot \mathbf{b}$
We will use the following Fourier transform convention:
\begin{aligned} f(t) &=\int_{-\infty}^{\infty} \frac{d \omega}{(2 \pi)^{1 / 2}} \tilde{f}(\omega) e^{-i \omega t} \ \tilde{f}(\omega) &=\int_{-\infty}^{\infty} \frac{d t}{(2 \pi)^{1 / 2}} f(t) e^{i \omega t} \end{aligned}

## 物理代写|多体物理作业代写MANY-BODY PHYSICS代考|Quantum Mechanics

A quantum mechanical system is defined by a Hilbert space, $\mathcal{H}$, whose vectors are states, $|\psi\rangle$. There are linear operators, $\mathcal{O}{i}$ which act on this Hilbert space. These operators correspond to physical observables. Finally, there is an inner product, which assigns a complex number, $\langle\chi \mid \psi\rangle$, to any pair of states, $|\psi\rangle,|\chi\rangle$. A state vector, $|\psi\rangle$ gives a complete description of a system through the expectation values, $\left\langle\psi\left|\mathcal{O}{i}\right| \psi\right\rangle$ (assuming that $|\psi\rangle$ is normalized so that $\langle\psi \mid \psi\rangle=1$ ), which would be the average values of the corresponding physical observables if we could measure them on an infinite collection of identical systems each in the state $|\psi\rangle$.
The adjoint, $\mathcal{O}^{\dagger}$, of an operator is defined according to
$$\langle\chi|(\mathcal{O}|\psi\rangle)=\left(\langle\chi| \mathcal{O}^{\dagger}\right)|\psi\rangle$$
In other words, the inner product between $|\chi\rangle$ and $\mathcal{O}|\psi\rangle$ is the same as that between $\mathcal{O}^{\dagger}|\chi\rangle$ and $|\psi\rangle$. An Hermitian operator satisfies
$$\mathcal{O}=\mathcal{O}^{\dagger}$$
while a unitary operator satisfies
$$\mathcal{O} \mathcal{O}^{\dagger}=\mathcal{O}^{\dagger} \mathcal{O}=1$$

⁇=ķ乙=和=1

## 物理代写|多体物理作业代写MANY-BODY PHYSICS代考|MATHEMATICAL CONVENTIONS

F(吨)=∫−∞∞dω(2圆周率)1/2F~(ω)和−一世ω吨 F~(ω)=∫−∞∞d吨(2圆周率)1/2F(吨)和一世ω吨

## 物理代写|多体物理作业代写MANY-BODY PHYSICS代考|QUANTUM MECHANICS

The adjoint, $\mathcal{O}^{\dagger}$, of an operator is defined according to
$$\langle\chi|(\mathcal{O}|\psi\rangle)=\left(\langle\chi| \mathcal{O}^{\dagger}\right)|\psi\rangle$$
In other words, the inner product between $|\chi\rangle$ and $\mathcal{O}|\psi\rangle$ is the same as that between $\mathcal{O}^{\dagger}|\chi\rangle$ and $|\psi\rangle$. An Hermitian operator satisfies
$$\mathcal{O}=\mathcal{O}^{\dagger}$$
while a unitary operator satisfies
$$\mathcal{O} \mathcal{O}^{\dagger}=\mathcal{O}^{\dagger} \mathcal{O}=1$$

## 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。