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## 物理代写|量子力学代写Quantum mechanics代考|Quantum Statistics and Hidden Variables

In quantum mechanics it is meaningful to talk about the value of a certain observable, if the system has been prepared in an eigenstate of that observable with some eigenvalue (which is done by measuring that observable and keeping only those systems with a particular outcome). Then subsequent measurements of that observable will return the given eigenvalue with $100 \%$ probability, as explained in Sec. 2.8. More generally, we can prepare a system in a simultaneous eigenstate of commuting observables, and we can talk about the values of those observables. If the observables in question are not constant in time (if they do not commute with the Hamiltonian), then the system will not remain in the given eigenstate, and in order to obtain the $100 \%$ probability quoted it will be necessary to make the subsequent measurements immediately after the preparation. In particular, the Hamiltonian for an isolated system commutes with itself, so if a system is measured to have a certain energy, then it is meaningful afterwards (assuming the system remains isolated) to talk about its energy.

There is, however, no role played in the orthodox interpretation of quantum mechanics for the simultaneous values of noncommuting observables. These are in principle not measurable. We are of course tempted by the analogy with classical mechanics to think in such terms, because in classical mechanics such simultaneous values of noncommuting observables are meaningful. But to do so means that we are using concepts for understanding physical reality that have no physical consequences. One is reminded of Newton’s ideas of absolute space and time, which likewise had no physical consequences, and which were eliminated from physics with the advent of relativity theory. The idea of basing quantum mechanics on strictly measurable quantities seems to be due to Heisenberg, who was apparently influenced by Einstein’s similar reasoning in his development of relativity theory.

## 物理代写|量子力学代写Quantum mechanics代考|The Properties of the Density Operator

We return now to the density operator and describe its characteristic properties, of which there are three. First, $\rho$ is Hermitian, as follows immediately from the definitions (15) and (16). Second, $\rho$ is nonnegative definite (see Eq. (1.65)), as follows by computing the expectation value of $\rho$ with respect to an arbitrary ket $|\phi\rangle$,
$$\langle\phi|\rho| \phi\rangle=\sum_i f_i\left|\left\langle\psi_i \mid \phi\right\rangle\right|^2 \geq 0,$$
where for simplicity we work with the discrete case. The third characteristic property of a density operator is that it has unit trace,
$$\operatorname{tr} \rho=1 .$$
This property is equivalent to the normalization condition on the probabilities, Eq. (11) or (13), as one can easily show. Conversely, as we shall show below, every nonnegative definite, Hermitian operator with unit trace can be interpreted as a density operator, that is, there exist kets and corresponding statistical weights such that the operator can be written in the form (15) or (16).

## 物理代写|量子力学代写Quantum mechanics代考|The Properties of the Density Operator

$$\langle\phi|\rho| \phi\rangle=\sum_i f_i\left|\left\langle\psi_i \mid \phi\right\rangle\right|^2 \geq 0,$$

$$\operatorname{tr} \rho=1 .$$

## 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。

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## 物理代写|量子力学代写Quantum mechanics代考|Young diagrams

A Young diagram consists of an array of boxes (or some other symbol) arranged in one or more left-justified rows, with each row being at least as long as the row beneath. The correspondence between a diagram and a multiplet label is: The top row juts out $\alpha$ boxes to the right past the end of the second row, the second row juts out $\beta$ boxes to the right past the end of the third row, etc. A diagram in $\mathrm{SU}(n)$ has at most $n$ rows. There can be any number of “completed” columns of $n$ boxes buttressing the left of a diagram; these don’t affect the label. Thus in $\mathrm{SU}(3)$ the diagrams
represent the multiplets $(1,0),(0,1),(0,0),(1,1)$, and $(3,0)$. In any $\mathrm{SU}(n)$, the quark multiplet is represented by a single box, the antiquark multiplet by a column of $(n-1)$ boxes, and a singlet by a completed column of $n$ boxes.

## 物理代写|量子力学代写Quantum mechanics代考|Coupling multiplets together

The following recipe tells how to find the multiplets that occur in coupling two multiplets together. To couple together more than two multiplets, first couple two, then couple a third with each of the multiplets obtained from the first two, etc.

First a definition: A sequence of the letters $a, b, c, \ldots$ is admissible if at any point in the sequence at least as many $a$ ‘s have occurred as $b$ ‘s, at least as many $b$ ‘s have occurred as $c$ ‘s, etc. Thus $a b c d$ and $a a b c b$ are admissible sequences and $a b b$ and $a c b$ are not. Now the recipe:
(a) Draw the Young diagrams for the two multiplets, but in one of the diagrams replace the boxes in the first row with a’s, the boxes in the second row with b’s, etc. Thus, to couple two SU(3) octets (such as the $\pi$-meson octet and the baryon octet), we start with $\square$ and a a $b$ . The unlettered diagram forms the upper left-hand corner of all the enlarged diagrams constructed below.
(b) Add the a’s from the lettered diagram to the right-hand ends of the rows of the unlettered diagram to form all possible legitimate Young diagrams that have no more than one $a$ per column. In general, there will be several distinct diagrams, and all the $a$ ‘s appear in each diagram. At this stage, for the coupling of the two SU(3) octets, we have:
(c) Use the b’s to further enlarge the diagrams already obtained, subject to the same rules. Then throw away any diagram in which the full sequence of letters formed by reading right to left in the first row, then the second row, etc., is not admissible.
(d) Proceed as in (c) with the $c$ ‘s (if any), etc.
The final result of the coupling of the two $\mathrm{SU}(3)$ octets is:
Here only the diagrams with admissible sequences of $a$ ‘s and $b$ ‘s and with fewer than four rows (since $n=3$ ) have been kept. In terms of multiplet labels, the above may be written
$$(1,1) \otimes(1,1)=(2,2) \oplus(3,0) \oplus(0,3) \oplus(1,1) \oplus(1,1) \oplus(0,0) .$$

## 物理代写|量子力学代写|量子力学代考|将多子耦合在一起

(a) 画出这两个多子的杨氏图，但在其中一个图中用a替换第一行的方框，用b替换第二行的方框，等等。因此，为了耦合两个SU(3)八重体（如$pi$介子八重体和重子八重体），我们从$/square$和a a $b$开始。这个无字图构成了下面构建的所有放大图的左上角。
(b) 将有字图中的a添加到无字图各行的右端，形成所有可能的合法杨氏图，每列不超过一个$a$。一般来说，会有几个不同的图，所有的$a$都出现在每个图中。在这个阶段，对于两个sU(3)八边形的耦合，我们有。
(c) 使用b来进一步扩大已经得到的图，并遵守同样的规则。然后扔掉任何图，在其中通过从右到左读第一行，然后读第二行等形成的完整字母序列是不被允许的。
(d) 按(c)的方法处理$c$’s（如果有的话），等等。

$$(1,1) \otimes(1,1)=(2,2) \oplus(3,0) \oplus(0,3) \oplus(1,1) \oplus(0,0) 。$$

## 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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## 物理代写|量子力学代写Quantum mechanics代考|Symmetry in classical physics

Observers A and B use their own coordinate systems to keep track of the particles. For the particle labelled by the index $i$ let us define A’s coordinates by $\mathbf{r}i$ and B’s coordinates by $\mathbf{r}_i^{\prime}$. These are related to each other by coordinate transformations that involve several parameters. For example in the case of translations $\mathbf{r}_i^{\prime}=\mathbf{r}_i+\mathbf{a}_i$, where $\mathbf{a}_i$ are the parameters. It is useful to consider nearby observers which are related to each other by infinitesimal coordinate transformations. If the infinitesimal parameters for N symmetries are $\epsilon_a, a=1,2, \cdots, N$, we may expand the relation between $\mathbf{r}_i$ and $\mathbf{r}_i^{\prime}$ to first order in the $\epsilon_a$ ‘s, and write $$r_i^{\prime I}=r_i^I+\delta\epsilon r_i^I, \quad \delta_\epsilon r_i^I=\sum_a \epsilon_a f_i^{I a}(\mathbf{r}, \dot{\mathbf{r}})$$
where $I=1,2,3$ denotes the vector index.

If the two observers $\mathrm{A}$ and $\mathrm{B}$ see identical physical phenomena and measure the same results, it must be that the equations that they use in terms of $\mathbf{r}_i$ and $\mathbf{r}_i^{\prime}$ respectively have the same form. If one takes A’s equations and substitutes $\mathbf{r}_i^{\prime}$ instead of $\mathbf{r}_i$ the resulting equations are B’s equations. Only if there is a symmetry, B’s equations, rewritten in terms of $\mathbf{r}_i$, will yield A’s equations in identical form, not otherwise.

Instead of discussing the symmetries of the equations of motion, it is more efficient to consider the action from which they are derived by a variational principle. The action $S$ is constructed from a Lagrangian in the form $S\left(\mathbf{r}_i\right)=$ $\int_1^2 d t L\left(\mathbf{r}_i(t), \dot{\mathbf{r}}_i(t)\right)$. The Euler equations are then
$$\frac{\partial}{\partial t} \frac{\partial L}{\partial \dot{\mathbf{r}}_i(t)}-\frac{\partial L}{\partial \mathbf{r}_i(t)}=0$$
There will be a symmetry provided, under the substitution $\mathbf{r}_i \rightarrow \mathbf{r}_i^{\prime}$, the form of the action remains invariant up to a “constant”
$$S(\mathbf{r}) \rightarrow S\left(\mathbf{r}^{\prime}\right)=S(\mathbf{r})+\operatorname{constant}(1,2)$$

## 物理代写|量子力学代写Quantum mechanics代考|Symmetry and classical conservation laws

The above examples illustrate some simple physical systems with symmetries. Now consider a general Lagrangian describing an arbitrary system of particles located at $\mathbf{r}_i(t)$ at time $t$. Suppose the Lagrangian has a symmetry under the infinitesimal transformation of (8.1) with some specific functions $f_i^{I a}\left(\mathbf{r}_j, \dot{\mathbf{r}}_j\right)$. According to Noether’s theorem, that we will prove below, corresponding to every symmetry parameter $\epsilon_a$ there exists a conserved quantity $Q_a\left(\mathbf{r}_i, \dot{\mathbf{r}}_i\right)$ that is time independent. That is, even though the location and velocities of the particles may be changing with time, the conserved quantities $Q_a$, which are constructed from them, remain unchanged, i.e. $d Q_a / d t=0$. The conservation of energy, momentum and angular momentum are some examples of consequences of symmetry. There are many more interesting cases in specific physical systems.
To construct the explicit form of $Q_a\left(\mathbf{r}_i, \dot{\mathbf{r}}_i\right)$ and prove Noether’s theorem, first note that the symmetry of the action (8.3) is satisfied most generally provided the Lagrangian behaves as follows
$$L\left(\mathbf{r}_i^{\prime}(t), \dot{\mathbf{r}}_i^{\prime}(t)\right)=\frac{\partial t^{\prime}}{\partial t} L\left(\mathbf{r}_i\left(t^{\prime}\right), \dot{\mathbf{r}}_i\left(t^{\prime}\right)\right)+\frac{\partial}{\partial t} \alpha\left(\mathbf{r}_i(t), \dot{\mathbf{r}}_i(t)\right) .$$
Here $t^{\prime}(t, \epsilon)$ is a change of variables that generally may depend on the parameters of the symmetry transformation, and $\frac{\partial t^{\prime}}{\partial t}$ is the Jacobian for the change of variables. $\alpha$ is some function of the dynamical variables and the parameters, which vanishes as $\epsilon_a \rightarrow 0$. The function $\alpha$ is zero in most cases, but not for every case, as will be seen in examples below. Also, in most cases $t^{\prime}(t, \epsilon)=t$, otherwise the infinitesimal expansion gives $t^{\prime}=t+\sum_a \epsilon_a \gamma^a\left(\mathbf{r}_i(t), \dot{\mathbf{r}}_i(t)\right)$ with some functions $\gamma^a$. When equation (8.10) is integrated, the left side yields $\int_1^2 L\left(\mathbf{r}_i^{\prime}(t), \dot{\mathbf{r}}_i^{\prime}(t)\right)=S\left(\mathbf{r}_i^{\prime}\right)$, and the right side gives
\begin{aligned} & \int_1^2 d t \frac{\partial t^{\prime}}{\partial t} L\left(\mathbf{r}_i\left(t^{\prime}\right), \dot{\mathbf{r}}_i\left(t^{\prime}\right)\right)+\int_1^2 d t \frac{\partial}{\partial t} \Lambda\left(\mathbf{r}_i(t), \dot{\mathbf{r}}_i(t)\right) \ & =\int d t^{\prime} L\left(\mathbf{r}_i\left(t^{\prime}\right), \dot{\mathbf{r}}_i\left(t^{\prime}\right)\right)+\Lambda(1)-\Lambda(2) \ & =S\left(\mathbf{r}_i\right)+\text { constant }(1,2) \end{aligned}
Thus, the condition of symmetry (8.3) is equivalent to (8.10).

## 物理代写|量子力学代写Quantum mechanics代考|Symmetry in classical physics

$$\frac{\partial}{\partial t} \frac{\partial L}{\partial \dot{\mathbf{r}}_i(t)}-\frac{\partial L}{\partial \mathbf{r}_i(t)}=0$$

$$S(\mathbf{r}) \rightarrow S\left(\mathbf{r}^{\prime}\right)=S(\mathbf{r})+\text { constant }(1,2)$$

## 物理代写|量子力学代写Quantum mechanics代考|Symmetry and classical conservation laws

$$L\left(\mathbf{r}_i^{\prime}(t), \dot{\mathbf{r}}_i^{\prime}(t)\right)=\frac{\partial t^{\prime}}{\partial t} L\left(\mathbf{r}_i\left(t^{\prime}\right), \dot{\mathbf{r}}_i\left(t^{\prime}\right)\right)+\frac{\partial}{\partial t} \alpha\left(\mathbf{r}_i(t), \dot{\mathbf{r}}_i(t)\right) .$$

$$\int_1^2 d t \frac{\partial t^{\prime}}{\partial t} L\left(\mathbf{r}_i\left(t^{\prime}\right), \dot{\mathbf{r}}_i\left(t^{\prime}\right)\right)+\int_1^2 d t \frac{\partial}{\partial t} \Lambda\left(\mathbf{r}_i(t), \dot{\mathbf{r}}_i(t)\right) \quad=\int d t^{\prime} L\left(\mathbf{r}_i\left(t^{\prime}\right), \dot{\mathbf{r}}_i\left(t^{\prime}\right)\right)+\Lambda(1)-\Lambda(2)=S\left(\mathbf{r}_i\right)+\operatorname{constant}(1,2)$$

## 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。

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

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## 物理代写|量子力学代写Quantum mechanics代考|Quadratic interactions for N particles

Consider the problem of $\mathrm{N}$ particles moving in $d$-dimensions and interacting through ideal springs as in Fig.(5.1).

The potential energy stored in the springs is proportional to the square of the distance between the particles at its two end points. For example for three springs coupled to three particles in all possible ways the potential energy of the system is
$$V=\frac{k_{12}}{2}\left(\vec{x}1-\vec{x}_2\right)^2+\frac{k{23}}{2}\left(\vec{x}2-\vec{x}_3\right)^2+\frac{k{31}}{2}\left(\vec{x}3-\vec{x}_1\right)^2$$ where $k{i j}$ are the spring constants, and the kinetic energy is
$$K=\frac{\vec{p}1^2}{2 m_1}+\frac{\vec{p}_2^2}{2 m_2}+\frac{\vec{p}_3^2}{2 m_3} .$$ More generally the springs may not be isotropic and may pull differently in various directions. To cover all possibilities we will consider a Hamiltonian of the form $$\left.H=\frac{1}{2} \sum{i, j=1}^N\left(K_{i j} p_i p_j+V_{i j} x_i x_j+W_{i j}^T x_i p_j+W_{i j} p_i x_j\right)\right)+\sum_{i=1}^N\left(\alpha_i p_i+\beta_i x_i\right)$$
where the indices $i, j$ run over the particle types and the various directions, and we will assume a real general matrix $W$, arbitrary symmetric matrices $K, V$, and coefficients $\alpha, \beta$ which may be considered column or row matrices (vectors). The mathematics of this system could model a variety of other physical situations besides the coupled spring problem which we used to motivate this Hamiltonian.. This general problem has an exact solution in both classical and quantum mechanics.

## 物理代写|量子力学代写Quantum mechanics代考|An infinite number of particles as a string

Consider a system of coupled particles and springs as in the previous section, but with only nearest neighbor interactions with $N$ springs whose strengths are the same, and $N+1$ particles whose masses are the same. The Lagrangian in $d$-dimensions written in vector notation is
$$L\left(\mathbf{x}i, \dot{\mathbf{x}}_i\right)=\frac{1}{2} m \sum{i=0}^N\left(\dot{\mathbf{x}}i \cdot \dot{\mathbf{x}}_i\right)-\frac{k}{2} \sum{i=1}^N\left(\mathbf{x}i-\mathbf{x}{i-1}\right)^2 .$$
The index $i=0, \cdots, N$ refers to the $i-$ th particle. We have argued above that one can always solve a problem like this. We will see, in fact, that the solution of such a system for $N \rightarrow \infty$ will describe the motion of a string moving in $d$ dimensions. Let us visualize the system in $d=3$. We have an array of particles whose motion is described by the solution of the coupled equations for $\mathbf{x}_i(t)$. Suppose that the $N$ particles are initially arranged as in Fig.(5.2) at $t=t_0$

As $t$ increases, the configuration of such an array of particles changes. Taking pictures at $t=t_1, t=t_2, t=t_3$ we can trace the trajectories as in Fig.(5.3).

## 物理代写|量子力学代写Quantum mechanics代考|Energy-Parity representation

$$S_P \hat{x} S_P^{-1}=-\hat{x}, \quad S_P \hat{p} S_P^{-1}=-\hat{p} .$$

$$S_P \hat{H} S_P^{-1}=+\hat{H} .$$

$S P|x>=|-x>$ ，根据其对位置云算符的操作要求。因此，它的平方充当身份运算符 $S_P^2|x>=| x>$. 由于位置空间是完备 的， $S_P^2$ 也是任何状态的身份 $S_P^2|\psi\rangle=|\psi\rangle$. 因此，逆 $S_P$ 本身就是 $S_P^{-1}=S_P$ ，这要求其特征值满足 $\lambda_P^2=1$. 唯一的可能是 $\lambda_P=\pm 1$. 因此，在目前的问题中，能量和奇偶校验特征值的结合提供了一套完整的标签 $\mid E, \pm>$ 为一个完整的脪尔伯特空间。

## 物理代写|量子力学代写Quantum mechanics代考|Finite square well

$$V(x)=-V_0 \theta(a-|x|)$$

$$\psi_E(x)=\psi_E^L(x) \theta(-x-a)+\psi_E^0(x) \theta(a-|x|)+\psi_E^R(x) \theta(x-a)$$

$$\left(-\frac{\hbar^2}{2 m} \partial_x^2\right) \psi_E^{L, R}=E \psi_E^{L, R} \quad|x|>a\left(-\frac{\hbar^2}{2 m} \partial_x^2-V_0\right) \psi_E^0=E \psi_E^0 \quad|x|<a$$

## 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。

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

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## 物理代写|量子力学代写Quantum mechanics代考|Energy-Parity representation

The solutions are naturally classified as even and odd under the parity transformation $x \rightarrow-x$. This is to be expected for the following reasons. In Quantum Mechanics we define the unitary parity operator $S_P$ and its inverse $S_P^{-1}$ such that
$$S_P \hat{x} S_P^{-1}=-\hat{x}, \quad S_P \hat{p} S_P^{-1}=-\hat{p} .$$
Then, any Hamiltonian with a potential energy that is an even function , $V(-x)=V(x)$, is automatically invariant under parity transformations,
$$S_P \hat{H} S_P^{-1}=+\hat{H} .$$
This means that the Hamiltonian commutes with the parity operator, $\left[\hat{H}, S_P\right]=$ 0 , and therefore they are simultaneous observables. The eigenvalues of the parity operator must label the complete set of states along with the eigenvalues of the Hamiltonian. What are the eigenvalues of the parity operator $S_P \mid \psi>=$ $\lambda_P \mid \psi>$ ? First notice that its action on position space is $S_P|x>=|-x>$, as required by its action on the position operator. So, its square acts as the identity operator $S_P^2|x>=| x>$. Since position space is complete, $S_P^2$ is also identity on any state $S_P^2|\psi\rangle=|\psi\rangle$. Therefore the inverse of $S_P$ is itself $S_P^{-1}=S_P$, and this requires that its eigenvalues satisfy $\lambda_P^2=1$. The only possibility is $\lambda_P=\pm 1$. Therefore in the present problem energy and parity eigenvalues combined provide a complete set of labels $\mid E, \pm>$ for a complete Hilbert space.

## 物理代写|量子力学代写Quantum mechanics代考|Finite square well

For the finite square well we will choose the origin of the energy axis such that the potential energy is either zero or negative. Then we have the potential energy
$$V(x)=-V_0 \theta(a-|x|)$$
corresponding to Fig. (4.5).

The Schrödinger wavefunction may be written in the form
$$\psi_E(x)=\psi_E^L(x) \theta(-x-a)+\psi_E^0(x) \theta(a-|x|)+\psi_E^R(x) \theta(x-a)$$
where the various functions satisfy the Schrödinger equation in the left (L) right (R) and middle (o) regions
$$\begin{array}{ll} \left(-\frac{\hbar^2}{2 m} \partial_x^2\right) \psi_E^{L, R}=E \psi_E^{L, R} & |x|>a \ \left(-\frac{\hbar^2}{2 m} \partial_x^2-V_0\right) \psi_E^0=E \psi_E^0 & |x|<a \end{array}$$

.

## 物理代写|量子力学代写Quantum mechanics代考|Energy-Parity representation

$$S_P \hat{x} S_P^{-1}=-\hat{x}, \quad S_P \hat{p} S_P^{-1}=-\hat{p} .$$

$$S_P \hat{H} S_P^{-1}=+\hat{H} .$$

$S P|x>=|-x>$ ，根据其对位置云算符的操作要求。因此，它的平方充当身份运算符 $S_P^2|x>=| x>$. 由于位置空间是完备 的， $S_P^2$ 也是任何状态的身份 $S_P^2|\psi\rangle=|\psi\rangle$. 因此，逆 $S_P$ 本身就是 $S_P^{-1}=S_P$ ，这要求其特征值满足 $\lambda_P^2=1$. 唯一的可能是 $\lambda_P=\pm 1$. 因此，在目前的问题中，能量和奇偶校验特征值的结合提供了一套完整的标签 $\mid E, \pm>$ 为一个完整的脪尔伯特空间。

## 物理代写|量子力学代写Quantum mechanics代考|Finite square well

$$V(x)=-V_0 \theta(a-|x|)$$

$$\psi_E(x)=\psi_E^L(x) \theta(-x-a)+\psi_E^0(x) \theta(a-|x|)+\psi_E^R(x) \theta(x-a)$$

$$\left(-\frac{\hbar^2}{2 m} \partial_x^2\right) \psi_E^{L, R}=E \psi_E^{L, R} \quad|x|>a\left(-\frac{\hbar^2}{2 m} \partial_x^2-V_0\right) \psi_E^0=E \psi_E^0 \quad|x|<a$$

## 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。

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

## avatest™帮您通过考试

avatest™的各个学科专家已帮了学生顺利通过达上千场考试。我们保证您快速准时完成各时长和类型的考试，包括in class、take home、online、proctor。写手整理各样的资源来或按照您学校的资料教您，创造模拟试题，提供所有的问题例子，以保证您在真实考试中取得的通过率是85%以上。如果您有即将到来的每周、季考、期中或期末考试，我们都能帮助您！

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## 物理代写|量子力学代写Quantum mechanics代考|Compatible observables and complete vector space

Given a physical system, there is a maximal number of compatible observables. These form a set of operators $\left{A_1, A_2, \cdots A_N\right}$ that commute with each other
$$\left[A_i, A_j\right]=0 .$$
In principle the number $N$ corresponds to the number of degrees of freedom of a system in its formulation in Classical Mechanics. This corresponds to the number of canonical position-momentum pairs, plus spin, charge, color, flavor, etc. degrees of freedom, if any. Thus, for a system of $k$ spinless neutral particles in $d$ dimensions there are $N=k \times d$ degrees of freedom that correspond to compatible positions or momenta needed to describe the system. The operators $\left{A_1, A_2, \cdots A_N\right}$ could be chosen as the $N$ positions or the $N$ momenta or some other $N$ compatible operators constructed from them, such as energy, angular momentum, etc.. As explained below, for each choice there is a corresponding complete set of eigenstates that define a basis for the same physical system. A physical state may be written as a linear combination of basis vectors for any one choice of basis corresponding to the eigenstates of compatible observables. Furthermore any basis vector may be expanded in terms of the vectors of some other basis since they all describe the same Hilbert space.

Let us consider the matrix elements of two compatible observables $A_1$ and $A_2$ in the basis that diagonalizes $\left(A_1\right){i j}=\alpha{1 i} \delta_{i j}$. The matrix elements of the zero commutator give
$$0==\left(\alpha_{1 i}-\alpha_{1 j}\right)\left(A_2\right)_{i j}$$

## 物理代写|量子力学代写Quantum mechanics代考|Incompatible observables

Let us now consider an observable $B$ that is not compatible with the set $\left{A_1, A_2, \cdots A_N\right}$, that is $\left[B, A_i\right] \neq 0$. Then we may derive
$$\left(\alpha_{k i_k}-\alpha_{k j_k}\right)(B)_{i j}==0 \text { (?) }$$
The right hand side may or may not be zero; this depends on the states $i, j$ and operators $A, B$. Consider the subset of states for which the result is non-zero. In general when $i \neq j$ the eigenvalues of the $A_k$ ‘s are different, and the nonzero result on the right implies that $B$ cannot be diagonal. Therefore, there cannot exist a basis in which incompatible observables would be simultaneously diagonal. If there were such a basis, they would commute by virtue of being diagonal matrices, and this contradicts the assumption.

Let us consider two incompatible observables and their respective sets of eigenstates
$$A\left|\alpha_i\right\rangle=\alpha_i\left|\alpha_i\right\rangle, \quad B\left|\beta_i>=\beta_i\right| \beta_i>.$$
Since each set of eigenstates is complete and orthonormal, and they span the same vector space, they must satisfy
$$\begin{array}{ccc} \sum_k\left|\alpha_k><\alpha_k\right| & =\mathbf{1}= & \sum_k\left|\beta_k><\beta_k\right| \ <\alpha_i \mid \alpha_j> & =\delta_{i j}= & <\beta_i \mid \beta_j> \end{array} .$$

.

## 物理代写|量子力学代写Quantum mechanics代考|Compatible observables and complete vector space

$$\left[A_i, A_j\right]=0 .$$

$$0==\left(\alpha_{1 i}-\alpha_{1 j}\right)\left(A_2\right){i j}$$

## 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。