Posted on Categories:Quantum mechanics, 物理代写, 量子力学

# 物理代写|量子力学代考Quantum mechanics代考|PHY401 Time Dependence of Expectation Values

avatest.org™量子力学Quantum mechanics代写，免费提交作业要求， 满意后付款，成绩80\%以下全额退款，安全省心无顾虑。专业硕 博写手团队，所有订单可靠准时，保证 100% 原创。avatest.org™， 最高质量的量子力学Quantum mechanics作业代写，服务覆盖北美、欧洲、澳洲等 国家。 在代写价格方面，考虑到同学们的经济条件，在保障代写质量的前提下，我们为客户提供最合理的价格。 由于统计Statistics作业种类很多，同时其中的大部分作业在字数上都没有具体要求，因此量子力学Quantum mechanics作业代写的价格不固定。通常在经济学专家查看完作业要求之后会给出报价。作业难度和截止日期对价格也有很大的影响。

my-assignmentexpert™ 为您的留学生涯保驾护航 在网课代写方面已经树立了自己的口碑, 保证靠谱, 高质且原创的网课代写服务。我们的专家在量子力学Quantum mechanics代写方面经验极为丰富，各种量子力学Quantum mechanics相关的作业也就用不着 说。

avatest™

## 物理代写|量子力学代考QUANTUM MECHANICS代考|Time Dependence of Expectation Values

It is instructive to study how the expectation value of an observable changes as a function of time. Suppose that at $t=0$ the initial state is one of the eigenstates of anservable $A$ that commutes with $H$, as in (2.40). We now look at the expectation value of some other observable $B$, which need not commute with $A$ nor with $H$. Because at a later time we have
$$\left|a^{\prime}, t_{0}=0 ; t\right\rangle=\mathscr{U}(t, 0)\left|a^{\prime}\right\rangle$$
for the state ket, $\langle B\rangle$ is given by
\begin{aligned} \langle B\rangle &=\left(\left\langle a^{\prime}\right| \mathscr{U}^{\dagger}(t, 0)\right) \cdot B \cdot\left(\mathscr{U}(t, 0)\left|a^{\prime}\right\rangle\right) \ &=\left\langle a^{\prime}\left|\exp \left(\frac{i E_{a^{\prime}} t}{\hbar}\right) B \exp \left(\frac{-i E_{a^{\prime}} t}{\hbar}\right)\right| a^{\prime}\right\rangle \ &=\left\langle a^{\prime}|B| a^{\prime}\right\rangle \end{aligned}
which is independent of $t$. So the expectation value of an observable taken with respect to an energy eigenstate does not change with time. For this reason an energy eigenstate is often referred to as a stationary state.

The situation is more interesting when the expectation value is taken with respect to a superposition of energy eigenstates, or a nonstationary state. Suppose that initially we have
$$\left|\alpha, t_{0}=0\right\rangle=\sum_{a^{\prime}} c_{\alpha^{\prime}}\left|a^{\prime}\right\rangle .$$
We easily compute the expectation value of $B$ to be
\begin{aligned} \langle B\rangle &=\left[\sum_{a^{\prime}} c_{\alpha^{\prime}}^{}\left\langle a^{\prime}\right| \exp \left(\frac{i E_{a^{\prime}} t}{\hbar}\right)\right] \cdot B \cdot\left[\sum_{a^{\prime \prime}} c_{d^{\prime \prime}} \exp \left(\frac{-i E_{a^{\prime \prime}} t}{\hbar}\right)\left|a^{\prime \prime}\right\rangle\right] \ &=\sum_{d^{\prime}} \sum_{d^{\prime \prime}} c_{d^{\prime}}^{} c_{d^{\prime \prime}}\left\langle a^{\prime}|B| a^{\prime \prime}\right\rangle \exp \left[\frac{-i\left(E_{d^{\prime \prime}}-E_{d^{\prime}}\right) t}{\hbar}\right] \end{aligned}
So this time the expectation value consists of oscillating terms whose angular frequencies are determined by N. Bohr’s frequency condition
$$\omega_{d^{\prime \prime} \alpha^{\prime}}=\frac{\left(E_{a^{\prime \prime}}-E_{\alpha^{\prime}}\right)}{\hbar} .$$

## 物理代写|量子力学代考QUANTUM MECHANICS代考|Spin Precession

It is appropriate to treat an example here. We consider an extremely simple system which, however, illustrates the basic formalism we have developed.

We start with a Hamiltonian of a spin $\frac{1}{2}$ system with magnetic moment $e \hbar / 2 m_{e} c$ subjected to an external magnetic field $\mathbf{B}$ :
$$H=-\left(\frac{e}{m_{e} c}\right) \mathbf{S} \cdot \mathbf{B}$$
( $e<0$ for the electron). Furthermore, we take $\mathbf{B}$ to be a static, uniform magnetic field in the $z$-direction. We can then write $H$ as
$$H=-\left(\frac{e B}{m_{e} c}\right) S_{z} .$$
Because $S_{z}$ and $H$ differ just by a multiplicative constant, they obviously commute. The $S_{z}$ eigenstates are also energy eigenstates, and the corresponding energy eigenvalues are
$$E_{\pm}=\mp \frac{e \hbar B}{2 m_{e} c}, \text { for } S_{z} \pm .$$
It is convenient to define $\omega$ in such a way that the difference in the two energy eigenvalues is $\hbar \omega$ :
$$\omega \equiv \frac{|e| B}{m_{e} c}$$

## 物理代写|量子力学代考QUANTUM MECHANICS代考|Time Dependence of Expectation Values

$$\left|a^{\prime}, t_{0}=0 ; t\right\rangle=\mathscr{U}(t, 0)\left|a^{\prime}\right\rangle$$

$$\langle B\rangle=\left(\left\langle a^{\prime}\right| \mathscr{U}^{\dagger}(t, 0)\right) \cdot B \cdot\left(\mathscr{U}(t, 0)\left|a^{\prime}\right\rangle\right) \quad=\left\langle a^{\prime}\left|\exp \left(\frac{i E_{a^{\prime}} t}{\hbar}\right) B \exp \left(\frac{-i E_{a^{\prime}} t}{\hbar}\right)\right| a^{\prime}\right\rangle=\left\langle a^{\prime}|B| a^{\prime}\right\rangle$$

$$\left|\alpha, t_{0}=0\right\rangle=\sum_{a^{\prime}} c_{\alpha^{\prime}}\left|a^{\prime}\right\rangle$$

$$\langle B\rangle=\left[\sum_{a^{\prime}} c_{\alpha^{\prime}}\left\langle a^{\prime}\right| \exp \left(\frac{i E_{a^{\prime}} t}{\hbar}\right)\right] \cdot B \cdot\left[\sum_{a^{\prime \prime}} c_{d^{\prime \prime}} \exp \left(\frac{-i E_{a^{\prime \prime}} t}{\hbar}\right)\left|a^{\prime \prime}\right\rangle\right] \quad=\sum_{d^{\prime}} \sum_{d^{\prime \prime}} c_{d^{\prime}} c_{d^{\prime \prime}}\left\langle a^{\prime}|B| a^{\prime \prime}\right\rangle \exp \left[\frac{-i\left(E_{d^{\prime \prime}}-E_{d^{\prime}}\right) t}{\hbar}\right]$$

$$\omega_{d^{\prime \prime} \alpha^{\prime}}=\frac{\left(E_{a^{\prime \prime}}-E_{\alpha^{\prime}}\right)}{\hbar}$$

## 物理代写|量子力学代考QUANTUM MECHANICS代考|Spin Precession

$$H=-\left(\frac{e}{m_{e} c}\right) \mathbf{S} \cdot \mathbf{B}$$
$(e<0$ 为电子) 。此外，我们取 $\mathbf{B}$ 是一个静态的、均匀的磁场 $z$-方向。然后我们可以写 $H$ 作为
$$H=-\left(\frac{e B}{m_{e} c}\right) S_{z} .$$

$$E_{\pm}=\mp \frac{e \hbar B}{2 m_{e} c}, \text { for } S_{z} \pm$$

$$\omega \equiv \frac{|e| B}{m_{e} c}$$

## 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。