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# 物理代写|量子力学代写Quantum mechanics代考|PHYS2041 Quadratic interactions for N particles

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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

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

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

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

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