# 第13章 氢原子

## 13.1 本征值问题

**习题 13.1.1** 从方程 (13.1.8)–(13.1.10) 出发，推导方程 (13.1.11) 和 (13.1.14)。



**习题 13.1.2** 推导简并度公式，即方程 (13.1.18)。

**习题 13.1.3** 从递推关系出发，求出归一化的 $\psi_{210}$。

**习题 13.1.4** 回忆上一章中的结果 [方程 (12.6.19)]：在 Coulomb 势
$V=-\mathrm{e}^2/r$
中，当 $r\rightarrow\infty$ 时，
$U_E\sim(r)^{m e^2/\kappa\hbar^2}\mathrm{e}^{-\kappa r}$，
其中
$\kappa=\left(2mW/\hbar^2\right)^{1/2}$。
证明这一结果与方程 (13.1.26) 一致。

**习题 13.1.5** （Virial 定理）由于 $|n,l,m\rangle$ 是定态，对于任意 $\mathbf{\Omega}$，都有
$\langle\dot{\mathbf{\Omega}}\rangle=0$。\
取
$\mathbf{\Omega}=\mathbf{R}\cdot\mathbf{P}$，
并利用 Ehrenfest 定理，证明在状态 $|n,l,m\rangle$ 中
$\langle T\rangle=(-1/2)\langle V\rangle$。

## 13.2 氢原子能谱的简并

**习题 13.2.1** 让我们看看为什么 Runge-Lenz 矢量 $\mathbf{n}$ 的守恒意味着轨道是闭合的。

(1) 仅用 $\mathbf{r}$ 和 $\mathbf{p}$ 表示 $\mathbf{n}$（消去 $\mathbf{1}$）。

(2) 由于粒子处于束缚态，它不可能逃到无穷远。因此，从某个任意时刻开始跟踪粒子的运动，它必然会到达某一点 $\mathbf{r}_{\max}$，此时它到原点的距离不再增大。证明

$$
\mathbf{n}=\mathbf{r}_{\max}\left(2 E+\frac{\mathrm{e}^2}{r_{\max}}\right)
$$

## 13.3 数值估算及与实验的比较

**习题 13.3.1** π 介子作为核力的媒介，其作用范围为
$1~\mathrm{Fermi}=10^{-5}\text{\AA}$。
估算它的静止能量。

**习题 13.3.2** 估算动能为
$200~\mathrm{eV}$
的电子的 de Broglie 波长。（回忆
$\lambda=2\pi\hbar/p$。）

## 13.4 多电子原子与元素周期表

**习题 13.4.1** 证明：如果忽略电子之间的相互作用，多电子原子的能级按 $Z^2$ 标度。既然 Coulomb 势为 $Ze/r$，为什么能量却正比于 $Z^2$？

**习题 13.4.2** 粗略比较铀原子和氢原子的大小。假设能级按照 $n$ 递增的顺序依次填充，并且非相对论描述仍然适用。忽略电子之间的相互作用。

**习题 13.4.3** 可见光的波长约为
$5000\text{\AA}$。
你认为 Lyman、Balmer、Paschen 这几个谱系中，哪一个最先被发现？

# Chapter 13 The Hydrogen Atom
## 13.1 The Eigenvalue Problem
**Exercise 13.1.1** Derive Eqs. (13.1.11) and (13.1.14) starting from Eqs. (13.1.8)-(13.1.10).



**Exercise 13.1.2** Derive the degeneracy formula, Eq. (13.1.18).

**Exercise 13.1.3** Starting from the recursion relation, obtain $\psi_{210}$ (normalized).

**Exercise 13.1.4** Recall from the last chapter [Eq. (12.6.19)] that as $r \rightarrow \infty$, $U_E \sim(r)^{m e^2 / \kappa \hbar^2} \mathrm{e}^{-\kappa r}$ in a Coulomb potential $V=-\mathrm{e}^2 / r$ $\left[\kappa=\left(2 m W / \hbar^2\right)^{1 / 2}\right]$. Show that this agrees with Eq. (13.1.26).

**Exercise 13.1.5** (Virial Theorem) Since $|n, l, m\rangle$ is a stationary state, $\langle\dot{\mathbf{\Omega}}\rangle=0$ for any $\mathbf{\Omega}$.\\ Consider $\mathbf{\Omega}=\mathbf{R} \cdot \mathbf{P}$ and use Ehrenfest's theorem to show that $\langle T\rangle=(-1 / 2)\langle V\rangle$ in the state $|n, l, m\rangle$.

## 13.2 The Degeneracy of the Hydrogen Spectrum
**Exercise 13.2.1** Let us see why the conservation of the Runge-Lenz vector $\mathbf{n}$ implies closed orbits.

(1) Express $\mathbf{n}$ in terms of $\mathbf{r}$ and $\mathbf{p}$ alone (get rid of $\mathbf{1}$).

(2) Since the particle is bound, it cannot escape to infinity. So, as we follow it from some arbitrary time onward, it must reach a point $\mathbf{r}_{\max}$ where its distance from the origin stops growing. Show that
$$
\mathbf{n}=\mathbf{r}_{\max}\left(2 E+\frac{\mathrm{e}^2}{r_{\max}}\right)
$$
## 13.3 Numerical Estimates and Comparison with Experiment
**Exercise 13.3.1** The pion has a range of $1~\mathrm{Fermi}=10^{-5} \text{\AA}$ as a mediator of nuclear force. Estimate its rest energy.

**Exercise 13.3.2** Estimate the de Broglie wavelength of an electron of kinetic energy $200~\mathrm{eV}$. (Recall $\lambda=2 \pi \hbar / p$.)
## 13.4 Multielectron Atoms and the Periodic Table
**Exercise 13.4.1** Show that if we ignore interelectron interactions, the energy levels of a multielectron atom go as $Z^2$. Since the Coulomb potential is $Z e / r$, why is the energy $\propto Z^2$?

**Exercise 13.4.2** Compare (roughly) the sizes of the uranium atom and the hydrogen atom. Assume levels fill in the order of increasing $n$, and that the nonrelativistic description holds. Ignore interelectron effects.

**Exercise 13.4.3** Visible light has a wavelength of approximately $5000 \text{\AA}$. Which of the series-Lyman, Balmer, Paschen-do you think was discovered first?

§ QM-13 氢原子

§ QM-13 The Hydrogen Atom

§ QM-12 旋转不变性和角动量

§ QM-12 Rotational Invariance and Angular Momentum

# 第11章 对称性及其推论

## 11.1 概述

## 11.2 量子理论中的平移不变性

**习题 11.2.1** 验证方程 (11.2.11b)。



**习题 11.2.2** 利用
$T^{\dagger}(\varepsilon)T(\varepsilon)=I$
在 $\varepsilon$ 一阶近似下成立这一事实，推出
$G^{\dagger}=G$。

**习题 11.2.3** 回忆我们曾通过求解微分方程（第 2.8 节），从无穷小旋转变换得到有限旋转变换。验证：如果改为利用无穷串 Poisson 括号把变换后的坐标 $\overline{x}$ 和 $\overline{y}$ 与 $x$、$y$ 联系起来，也会得到相同的结果，
$\overline{x}=x \cos \theta-y \sin \theta$
等。（回忆 $\sin\theta$ 等函数的级数展开。）

## 11.3 时间平移不变性

## 11.4 宇称不变性

**习题 11.4.1** 证明：如果
$[\Pi,H]=0$，
那么一个初始处于偶/奇宇称状态的系统将保持其宇称不变。（注意，由于宇称是一个离散操作，因此在经典力学中没有与之对应的守恒定律。）

**习题 11.4.2** 一个粒子处于势

$$
V(x)=V_0 \sin (2 \pi x / a)
$$

中，该势在平移
$x \rightarrow x+m a$
下保持不变，其中 $m$ 为整数。动量守恒吗？为什么不守恒？

**习题 11.4.3** 已知在某个反应中，电子出来时其自旋总是与动量平行。说明宇称被破坏。

**习题 11.4.4** 我们一直把宇称看作镜面反射。在一维中这当然是正确的，因为
$x\rightarrow -x$
可以看成关于原点处一个（点）镜面的反射结果。但在更高维中，如果使用一个平面镜（例如位于 $x-y$ 平面内），只有一个坐标（$z$）会反号，而宇称变换则会使三个坐标全部反号。

验证：关于 $x-y$ 平面的镜面反射，等价于先做宇称变换，再绕 $z$ 轴旋转 $180^{\circ}$。由于弱相互作用具有旋转不变性，因此镜面反射不变性的破坏意味着宇称不变性的破坏。

## 11.5 时间反演对称性

# Chapter 11 Symmetries and Their Consequences
## 11.1 Overview
## 11.2 Translational Invariance in Quantum Theory
**Exercise 11.2.1** Verify Eq. (11.2.11b).



**Exercise 11.2.2** Using $T^{\dagger}(\varepsilon)T(\varepsilon)=I$ to order $\varepsilon$, deduce that $G^{\dagger}=G$.

**Exercise 11.2.3** Recall that we found the finite rotation transformation from the infinitesimal one, by solving differential equations (Section 2.8). Verify that if, instead, you relate the transformed coordinates $\overline{x}$ and $\overline{y}$ to $x$ and $y$ by the infinite string of Poisson brackets, you get the same result, $\overline{x}=x \cos \theta-y \sin \theta$, etc. (Recall the series for $\sin \theta$, etc.)
## 11.3 Time Translational Invariance
## 11.4 Parity Invariance
**Exercise 11.4.1** Prove that if $[\Pi, H]=0$, a system that starts out in a state of even/odd parity maintains its parity. (Note that since parity is a discrete operation, it has no associated conservation law in classical mechanics.)

**Exercise 11.4.2** A particle is in a potential
$$
V(x)=V_0 \sin (2 \pi x / a)
$$
which is invariant under the translations $x \rightarrow x+m a$, where $m$ is an integer. Is momentum conserved? Why not?

**Exercise 11.4.3** You are told that in a certain reaction, the electron comes out with its spin always parallel to its momentum. Argue that parity is violated.

**Exercise 11.4.4** We have treated parity as a mirror reflection. This is certainly true in one dimension, where $x \rightarrow-x$ may be viewed as the effect of reflecting through a (point) mirror at the origin. In higher dimensions when we use a plane mirror (say lying on the $x-y$ plane), only one ($z$) coordinate gets reversed, whereas the parity transformation reverses all three coordinates.

Verify that reflection on a mirror in the $x-y$ plane is the same as parity followed by $180^{\circ}$ rotation about the $z$ axis. Since rotational invariance holds for weak interactions, noninvariance under mirror reflection implies noninvariance under parity.
## 11.5 Time-Reversal Symmetry

§ QM-11 对称性与其结论

§ QM-11 Symmetries and Their Consequences

# 第10章 具有 $N$ 个自由度的系统
## 10.1 一维中的 $N$ 个粒子
**习题 10.1.1** 证明下列关系：

(1)
$$
[\Omega_{1}^{(1)}\otimes I^{(2)}, I^{(1)}\otimes \Lambda_{2}^{(2)}]=0~\text{for any}~\Omega_{1}^{(1)}~\text{and}~\Lambda_{2}^{(2)}\tag{10.1.17a}
$$

(2)
$$
(\Omega_{1}^{(1)}\otimes\Gamma_{2}^{(2)})(\theta_{1}^{(1)}\otimes\Lambda_{2}^{(2)})=(\Omega\theta)_{1}^{(1)}\otimes(\Gamma\Lambda)_{2}^{(2)}\tag{10.1.17b}
$$

(3) 如果
$$
[\Omega_{1}^{(1)},\Lambda_{1}^{(1)}]=\Gamma_{1}^{(1)}
$$
那么
$$
[\Omega_{1}^{(1)\otimes(2)},\Lambda_{1}^{(1)\otimes(2)}]=\Gamma_{1}^{(1)}\otimes I^{(2)}\tag{10.1.17c}
$$
并且将 $1\to 2$ 时也有类似关系。

(4)
$$
(\Omega_{1}^{(1)\otimes (2)}+\Omega_{2}^{(1)\otimes(2)})^{2}=(\Omega_{1}^{2})^{(1)}\otimes I^{(2)}+I^{(1)}\otimes (\Omega_{2}^{2})^{(2)}+2\Omega_{1}^{(1)}\otimes\Omega_{2}^{(2)}\tag{10.1.17d}
$$



**习题 10.1.2** 设想一个虚构的世界，其中单粒子 Hilbert 空间是二维的。我们用 $|+\rangle$ 和 $|-\rangle$ 表示基矢。令
![image](https://i.upmath.me/svgb/U1HhiinOTM9NjDeMq9Yw1Ky15YpJyi9KSS3KTSwpyqyo5lJT0FZQU9BViEku4gKxEoE4CczTBbKSgTiFqzamsDQxJaYktaKkOjEvBcqFmGsENNcIaC4BY1OBOA1ubDoQZ3DVcqmoAAA)

分别是在 $\mathbb{V}_1$ 和 $\mathbb{V}_2$ 上的算符（其中 $\pm$ 号标记基矢。例如 $b=\langle+| \sigma_1^{(1)}|-\rangle$ 等）。空间 $\mathbb{V}_1 \otimes \mathbb{V}_2$ 由四个矢量
$|+\rangle \otimes|+\rangle$、$|+\rangle \otimes|-\rangle$、$|-\rangle\otimes|+\rangle$、$|-\rangle\otimes|-\rangle$
张成。证明（可以使用图像法或其他方法）：

(1) ![image](https://i.upmath.me/svgb/U4kpzkzPTYyvNqyNq9Yw1IzJL8nMTS3WMNKstUWTqoXKKXgCuWD5pPyilNSi3MSSosyKai41BW1tBSChCyR0QSxd3ZjkIi6wYCIQGwBxEpgGC-tChRBSIGGwxmSoUApctS5MNVyKq1YFAA)

（回忆 $\langle\alpha|\otimes\langle\beta|$ 是与 $|\alpha\rangle\otimes|\beta\rangle$ 对应的 bra。）

(2)
$$
\sigma_{2}^{(1)\otimes(2)}=\begin{pmatrix}
e & f & 0 & 0\\
g & h & 0 & 0\\
0 & 0 & e & f\\
0 & 0 & g & h
\end{pmatrix}
$$

(3)
$$
(\sigma_{1}\sigma_{2})^{(1)\otimes(2)}=\sigma_{1}^{(1)}\otimes\sigma_{2}^{(2)}=\begin{pmatrix}
ae & af & be & bf\\
ag & ah & bg & bh\\
ce & cf & de & df\\
cg & ch & dg & dh
\end{pmatrix}
$$

用两种方法完成第 (3) 问：一种是直接计算
$\sigma_{1}^{(1)\otimes (2)}$
与
$\sigma_{2}^{(1)\otimes (2)}$
的矩阵乘积；另一种是直接计算
$\sigma_{1}^{(1)}\otimes\sigma_{2}^{(2)}$
的矩阵元。

**习题 10.1.3** 考虑耦合质量块系统的 Hamilton量：
$$
\mathscr{H}=\dfrac{p_{1}^{2}}{2m}+\dfrac{p_{2}^{2}}{2m}+\dfrac{1}{2}m\omega^{2}[x_{1}^{2}+x_{2}^{2}+(x_{1}-x_{2})^{2}]
$$
我们从例 1.8.6 已知，如果采用简正坐标
$$
x_{\text{I},\text{II}}=\dfrac{x_{1}\pm x_{2}}{2^{1/2}}
$$
以及相应的动量
$$
p_{\text{I},\text{II}}=\dfrac{p_{1}\pm p_{2}}{2^{1/2}}
$$
则可以将 $\mathscr{H}$ 解耦。

(1) 用简正坐标重写 $\mathscr{H}$。验证这些简正坐标也是正则坐标，即
$$
\{x_{i},p_{j}\}=\delta_{ij}~\text{etc.;}\qquad i,j=\text{I},\text{II}
$$
现在将该系统量子化，把这些变量提升为满足
$$
[X_{i},P_{j}]=\mathrm{i}\hbar\delta_{ij}~\text{etc.;}\qquad i,j=\text{I},\text{II}
$$
的算符。在 $X_{\text{I}}$ 与 $X_{\text{II}}$ 的共同本征基底中写出 $H$ 的本征值方程。

(2) 直接量子化该系统，即把 $x_1$、$x_2$、$p_1$ 和 $p_2$ 提升为量子算符。在 $X_1$ 和 $X_2$ 的共同本征基底中写出 $H$ 的本征值方程。接着在该微分方程中，从 $x_1$、$x_2$（以及当然还有 $\partial/\partial x_1$、$\partial/\partial x_2$）变换到 $x_{\text{I}}$、$x_{\text{II}}$（以及 $\partial/\partial x_{\text{I}}$、$\partial/\partial x_{\text{II}}$）。最终应得到与第 (1) 问相同的结果。

## 10.2 更多维数中的更多粒子
**习题 10.2.1** （三维势箱中的粒子）
回忆一个一维势箱中的粒子，当势箱从 $x=0$ 延伸到 $L$ 时，粒子被限制在区域 $0 \leqslant x \leqslant L$ 内；其波函数在边界 $x=0$、$L$ 处以及势箱外都为零（见习题 5.2.5）。现在考虑一个被限制在体积为 $L^3$ 的三维立方势箱中的粒子。取立方体的一个顶点为原点，并令 $x$、$y$、$z$ 轴沿该顶点相交的三条棱方向，证明归一化能量本征函数为
$$
\psi_E(x, y, z)=\left(\frac{2}{L}\right)^{1 / 2} \sin \left(\frac{n_x \pi x}{L}\right)\left(\frac{2}{L}\right)^{1 / 2} \sin \left(\frac{n_y \pi y}{L}\right)\left(\frac{2}{L}\right)^{1 / 2} \sin \left(\frac{n_z \pi z}{L}\right)
$$
其中
$$
E=\frac{\hbar^2 \pi^2}{2 M L^2}\left(n_x^2+n_y^2+n_z^2\right)
$$
并且 $n_i$ 为正整数。

**习题 10.2.2** 量子化二维谐振子，其
$$
\mathscr{H}=\frac{p_x^2+p_y^2}{2 m}+\frac{1}{2} m \omega_x^2 x^2+\frac{1}{2} m \omega_y^2 y^2
$$

(1) 证明允许的能量为
$$
E=\left(n_x+1 / 2\right) \hbar \omega_x+\left(n_y+1 / 2\right) \hbar \omega_y, \quad n_x, n_y=0,1,2, \ldots
$$

(2) 用单个谐振子的波函数写出相应的波函数。验证它们在
$x\rightarrow -x$、$y\rightarrow -y$
下具有确定的宇称（偶/奇），并且宇称只依赖于
$n=n_x+n_y$。

(3) 接着考虑各向同性谐振子
$\left(\omega_x=\omega_y\right)$。
写出前三个状态（即 $n=0$ 和 $1$ 的情形）的显式归一化本征函数。为了后续使用，将结果改写成极坐标 $\rho$ 和 $\phi$ 的形式。证明能量为
$E=(n+1)\hbar\omega$
的能级简并度为 $(n+1)$。

**习题 10.2.3** 量子化三维各向同性谐振子，其
$$
\mathscr{H}=\frac{p_x^2+p_y^2+p_z^2}{2 m}+\frac{1}{2} m \omega^2\left(x^2+y^2+z^2\right)
$$

(1) 证明
$E=(n+3 / 2) \hbar \omega$；
$n=n_x+n_y+n_z$；
$n_x,n_y,n_z=0,1,2,\ldots$。

(2) 用单个谐振子的波函数写出相应的本征函数，并验证给定 $n$ 的能级宇称为 $(-1)^n$。将前四个状态改写成球坐标形式。证明能量为
$E=(n+3 / 2)\hbar\omega$
的能级简并度为
$(n+1)(n+2)/2$。

## 10.3 全同粒子
**习题 10.3.1** 两个全同玻色子分别处于状态 $|\phi\rangle$ 和 $|\psi\rangle$。当
$\langle\phi \mid \psi\rangle \neq 0$
时，写出描述该系统的归一化状态矢量。

**习题 10.3.2** 对一个势箱中的三个玻色子系统进行能量测量，得到的 $n$ 值分别为 $3$、$3$ 和 $4$。写出一个对称化且归一化的状态矢量。

**习题 10.3.3** 设想这样一种情形：有三个粒子，并且只有三个状态 $a$、$b$、$c$ 可供它们占据。证明该系统允许的不同组态总数为：

(1) 如果粒子可区分，则为 27；

(2) 如果粒子是玻色子，则为 10；

(3) 如果粒子是费米子，则为 1。

**习题 10.3.4** 两个质量均为 $m$ 的全同粒子处于长度为 $L$ 的一维势箱中。对系统进行能量测量，得到
$E_{\text{sys}}=\hbar^2 \pi^2/mL^2$。
写出系统的状态矢量。

再对
$E_{\text{sys}}=5 \hbar^2 \pi^2 / 2 m L^2$
的情形重复上述过程。（此时有两个可能的状态矢量。）题目没有告诉你这些粒子是玻色子还是费米子。可以假设系统唯一的自由度是轨道自由度。

**习题 10.3.5** 考虑交换算符 $P_{12}$，它在 $X$ 基底上的作用为
$$
P_{12}\left|x_1, x_2\right\rangle=\left|x_2, x_1\right\rangle
$$

(1) 证明 $P_{12}$ 的本征值为 $\pm 1$。（它既是 Hermitian 的，也是酉的。）

(2) 证明它作用在基矢
$\left|\omega_1,\omega_2\right\rangle$
上时，同样只是交换标号 $1$ 和 $2$，从而证明
$\mathbb{V}_{S/A}$
分别是其本征值为 $\pm 1$ 的本征子空间。

(3) 证明
$P_{12}X_1P_{12}=X_2$、
$P_{12}X_2P_{12}=X_1$，
并且对 $P_1$ 和 $P_2$ 也有类似关系。接着证明
$$
P_{12}\Omega\left(X_1,P_1;X_2,P_2\right)P_{12}
=
\Omega\left(X_2,P_2;X_1,P_1\right)
$$
[考虑其在
$\left|x_1,x_2\right\rangle$
或
$\left|p_1,p_2\right\rangle$
上的作用。对于 $X$ 和 $P$ 的函数，假设它们可以展开为幂级数，并考察级数中的任意一项。如果需要帮助，可以参考通向方程 (11.2.22) 的讨论。]

(4) 证明对于两个全同粒子，Hamilton量和传播子在变换
$H\rightarrow P_{12}HP_{12}$
以及
$U\rightarrow P_{12}UP_{12}$
下保持不变。由此证明，$P_{12}$ 的任一本征态在随时间演化时，仍然保持为具有相同本征值的本征态；也就是说，
$\mathbb{V}_{S/A}$
中的状态永远不会离开它们最初所在的对称或反对称子空间。

**习题 10.3.6** 考虑像氢原子这样的复合对象。它会表现为玻色子还是费米子？一般性地说明：由偶数/奇数个费米子组成的对象将分别表现为玻色子/费米子。

# Chapter 10 Systems with $N$ Degrees of Freedom
## 10.1 $N$ Particles in One Dimension
**Exercise 10.1.1** Show the following:

(1)
$$
[\Omega_{1}^{(1)}\otimes I^{(2)}, I^{(1)}\otimes \Lambda_{2}^{(2)}]=0~\text{for any}~\Omega_{1}^{(1)}~\text{and}~\Lambda_{2}^{(2)}\tag{10.1.17a}
$$

(2)
$$
(\Omega_{1}^{(1)}\otimes\Gamma_{2}^{(2)})(\theta_{1}^{(1)}\otimes\Lambda_{2}^{(2)})=(\Omega\theta)_{1}^{(1)}\otimes(\Gamma\Lambda)_{2}^{(2)}\tag{10.1.17b}
$$

(3)If
$$
[\Omega_{1}^{(1)},\Lambda_{1}^{(1)}]=\Gamma_{1}^{(1)}
$$
then
$$
[\Omega_{1}^{(1)\otimes(2)},\Lambda_{1}^{(1)\otimes(2)}]=\Gamma_{1}^{(1)}\otimes I^{(2)}\tag{10.1.17c}
$$
and similarly with $1\to 2$.

(4)
$$
(\Omega_{1}^{(1)\otimes (2)}+\Omega_{2}^{(1)\otimes(2)})^{2}=(\Omega_{1}^{2})^{(1)}\otimes I^{(2)}+I^{(1)}\otimes (\Omega_{2}^{2})^{(2)}+2\Omega_{1}^{(1)}\otimes\Omega_{2}^{(2)}\tag{10.1.17d}
$$



**Exercise 10.1.2** Imagine a fictitious world in which the single-particle Hilbert space is two-dimensional. Let us denote the basis vectors by $|+\rangle$ and $|-\rangle$. Let
![image](https://i.upmath.me/svgb/U1HhiinOTM9NjDeMq9Yw1Ky15YpJyi9KSS3KTSwpyqyo5lJT0FZQU9BViEku4gKxEoE4CczTBbKSgTiFqzamsDQxJaYktaKkOjEvBcqFmGsENNcIaC4BY1OBOA1ubDoQZ3DVcqmoAAA)

be operators in $\mathbb{V}_1$ and $\mathbb{V}_2$, respectively (the $\pm$ signs label the basis vectors. Thus $b=\langle+| \sigma_1^{(1)}|-\rangle$ etc.) The space $\mathbb{V}_1 \otimes \mathbb{V}_2$ is spanned by four vectors $|+\rangle \otimes|+\rangle$, $|+\rangle \otimes|-\rangle$, $|-\rangle\otimes|+\rangle$, $|-\rangle\otimes|-\rangle$. Show (using the method of images or otherwise) that

(1) ![image](https://i.upmath.me/svgb/U4kpzkzPTYyvNqyNq9Yw1IzJL8nMTS3WMNKstUWTqoXKKXgCuWD5pPyilNSi3MSSosyKai41BW1tBSChCyR0QSxd3ZjkIi6wYCIQGwBxEpgGC-tChRBSIGGwxmSoUApctS5MNVyKq1YFAA)

(Recall that $\langle\alpha|\otimes\langle\beta|$ is the bra corresponding to $|\alpha\rangle\otimes|\beta\rangle$.)

(2) $\sigma_{2}^{(1)\otimes(2)}=\begin{pmatrix}
e & f & 0 & 0\\
g & h & 0 & 0\\
0 & 0 & e & f\\
0 & 0 & g & h
\end{pmatrix}$

(3) $(\sigma_{1}\sigma_{2})^{(1)\otimes(2)}=\sigma_{1}^{(1)}\otimes\sigma_{2}^{(2)}=\begin{pmatrix}
ae & af & be & bf\\
ag & ah & bg & bh\\
ce & cf & de & df\\
cg & ch & dg & dh
\end{pmatrix}$

Do part (3) in two ways, by taking the matrix product of $\sigma_{1}^{(1)\otimes (2)}$ and $\sigma_{2}^{(1)\otimes (2)}$ and by directly computing the matrix elements of $\sigma_{1}^{(1)}\otimes\sigma_{2}^{(2)}$.

**Exercise 10.1.3** Consider the Hamiltonian of the coupled mass system:
$$
\mathscr{H}=\dfrac{p_{1}^{2}}{2m}+\dfrac{p_{2}^{2}}{2m}+\dfrac{1}{2}m\omega^{2}[x_{1}^{2}+x_{2}^{2}+(x_{1}-x_{2})^{2}]
$$
We know from Example 1.8.6 that $\mathscr{H}$ can be decoupled if we use normal coordinates
$$
x_{\text{I},\text{II}}=\dfrac{x_{1}\pm x_{2}}{2^{1/2}}
$$
and the corresponding momenta
$$
p_{\text{I},\text{II}}=\dfrac{p_{1}\pm p_{2}}{2^{1/2}}
$$

(1) Rewrite $\mathscr{H}$ in terms of normal coordinates. Verify that the normal coordinates are also canonical, i.e., that
$$
\{x_{i},p_{j}\}=\delta_{ij}~\text{etc.;}\qquad i,j=\text{I},\text{II}
$$
Now quantize the system, promoting these variables to operators obeying
$$
[X_{i},P_{j}]=\mathrm{i}\hbar\delta_{ij}~\text{etc.;}\qquad i,j=\text{I},\text{II}
$$
Write the eigenvalue equation for $H$ in the simultaneous eigenbasis of $X_{\text{I}}$ and $X_{\text{II}}$.

(2) Quantize the system directly, by promoting $x_{1}$, $x_{2}$, $p_{1}$, and $p_{2}$ to quantum operators. Write the eigenvalue equation for $H$ in the simultaneous eigenbasis of $X_{1}$ and $X_{2}$. Now change from $x_{1}$, $x_{2}$ (and of course $\partial/\partial x_{1}$, $\partial/\partial x_{2}$) to $x_{\text{I}}$, $x_{\text{II}}$ (and $\partial/\partial x_{\text{I}}$, $\partial/\partial x_{\text{II}}$) in the differential equation. You should end up with the result form part (1).
## 10.2 More Particles in More Dimensions
**Exercise 10.2.1** (Particle in a Three-Dimensional Box)
Recall that a particle in a one-dimensional box extending from $x=0$ to $L$ is confined to the region $0 \leqslant x \leqslant L$; its wave function vanishes at the edges $x=0$ and $L$ and beyond (Exercise 5.2.5). Consider now a particle confined in a three-dimensional cubic box of volume $L^3$. Choosing as the origin one of its corners, and the $x, y$, and $z$ axes along the three edges meeting there, show that the normalized energy eigenfunctions are
$$
\psi_E(x, y, z)=\left(\frac{2}{L}\right)^{1 / 2} \sin \left(\frac{n_x \pi x}{L}\right)\left(\frac{2}{L}\right)^{1 / 2} \sin \left(\frac{n_y \pi y}{L}\right)\left(\frac{2}{L}\right)^{1 / 2} \sin \left(\frac{n_z \pi z}{L}\right)
$$
where
$$
E=\frac{\hbar^2 \pi^2}{2 M L^2}\left(n_x^2+n_y^2+n_z^2\right)
$$
and $n_i$ are positive integers.

**Exercise 10.2.2** Quantize the two-dimensional oscillator for which
$$
\mathscr{H}=\frac{p_x^2+p_y^2}{2 m}+\frac{1}{2} m \omega_x^2 x^2+\frac{1}{2} m \omega_y^2 y^2
$$
(1) Show that the allowed energies are
$$
E=\left(n_x+1 / 2\right) \hbar \omega_x+\left(n_y+1 / 2\right) \hbar \omega_y, \quad n_x, n_y=0,1,2, \ldots
$$
(2) Write down the corresponding wave functions in terms of single oscillator wave functions. Verify that they have definite parity (even/odd) number $x \rightarrow-x, y \rightarrow-y$ and that the parity depends only on $n=n_x+n_y$.

(3) Consider next the isotropic oscillator $\left(\omega_x=\omega_y\right)$. Write explicit, normalized eigenfunctions of the first three states (that is, for the cases $n=0$ and 1). Reexpress your results in terms of polar coordinates $\rho$ and $\phi$ (for later use). Show that the degeneracy of a level with $E=(n+1) \hbar \omega$ is $(n+1)$.

**Exercise 10.2.3** Quantize the three-dimensional isotropic oscillator for which
$$
\mathscr{H}=\frac{p_x^2+p_y^2+p_z^2}{2 m}+\frac{1}{2} m \omega^2\left(x^2+y^2+z^2\right)
$$
(1) Show that $E=(n+3 / 2) \hbar \omega$; $n=n_x+n_y+n_z$; $n_x, n_y, n_z=0,1,2, \ldots$.

(2) Write the corresponding eigenfunctions in terms of single-oscillator wave functions and verify that the parity of the level with a given $n$ is $(-1)^n$. Reexpress the first four states in terms of spherical coordinates. Show that the degeneracy of a level with energy $E=$ $(n+3 / 2) \hbar \omega$ is $(n+1)(n+2) / 2$.
## 10.3 Identical Particles
**Exercise 10.3.1** Two identical bosons are found to be in states $|\phi\rangle$ and $|\psi\rangle$. Write down the normalized state vector describing the system when $\langle\phi \mid \psi\rangle \neq 0$.

**Exercise 10.3.2** When an energy measurement is made on a system of three bosons in a box, the $n$ values obtained were $3$, $3$, and $4$. Write down a symmetrized, normalized state vector.

**Exercise 10.3.3** Imagine a situation in which there are three particles and only three states $a$, $b$, and $c$ available to them. Show that the total number of allowed, distinct configurations for this system is

(1) 27 if they are labeled

(2) 10 if they are bosons

(3) 1 if they are fermions

**Exercise 10.3.4** Two identical particles of mass $m$ are in a one-dimensional box of length $L$. Energy measurement of the system yields the value $E_{\text{sys}}=\hbar^2 \pi^2/mL^2$. Write down the state vector of the system. Repeat for $E_{\text{sys}}=5 \hbar^2 \pi^2 / 2 m L^2$. (There are two possible vectors in this case.) You are not told if they are bosons or fermions. You may assume that the only degrees of freedom are orbital.

**Exercise 10.3.5** Consider the exchange operator $P_{12}$ whose action on the $X$ basis is
$$
P_{12}\left|x_1, x_2\right\rangle=\left|x_2, x_1\right\rangle
$$
(1) Show that $P_{12}$ has eigenvalues $\pm 1$. (It is Hermitian and unitary.)

(2) Show that its action on the basis ket $\left|\omega_1, \omega_2\right\rangle$ is also to exchange the labels $1$ and $2$, and hence that $\mathbb{V}_{S/A}$ are its eigenspaces with eigenvalues $\pm 1$.

(3) Show that $P_{12} X_1 P_{12}=X_2$, $P_{12} X_2 P_{12}=X_1$ and similarly for $P_1$ and $P_2$. Then show that $P_{12} \Omega\left(X_1, P_1 ; X_2, P_2\right) P_{12}=\Omega\left(X_2, P_2 ; X_1, P_1\right)$. [Consider the action on $\left|x_1, x_2\right\rangle$ or $\left|p_1, p_2\right\rangle$. As for the functions of $X$ and $P$, assume they are given by power series and consider any term in the series. If you need help, peek into the discussion leading to Eq. (11.2.22).]

(4) Show that the Hamiltonian and propagator for two identical particles are left unaffected under $H \rightarrow P_{12} H P_{12}$ and $U \rightarrow P_{12} U P_{12}$. Given this, show that any eigenstate of $P_{12}$ continues to remain an eigenstate with the same eigenvalue as time passes, i.e., elements of $\mathbb{V}_{S/A}$ never leave the symmetric or antisymmetric subspaces they start in.

**Exercise 10.3.6** Consider a composite object such as the hydrogen atom. Will it behave as a boson or fermion? Argue in general that objects containing an even/odd number of fermions will behave as bosons/fermions.

§ QM-10 自由度为N的系统

§ QM-10 Systems with N Degrees of Freedom

# 第9章 Heisenberg 不确定关系

## 9.1 引言

## 9.2 不确定关系的推导

## 9.3 最小不确定度波包

## 9.4 不确定性原理的应用

**习题 9.4.1** 考虑处于状态 $|n=1\rangle$ 的谐振子，验证

$$
\left\langle\frac{1}{X^2}\right\rangle \simeq \frac{1}{\left\langle X^2\right\rangle} \simeq \frac{m \omega}{\hbar}
$$



**习题 9.4.2** (1) 参考附录 A.2 中的积分表，验证

$$
\psi=\frac{1}{\left(\pi a_0^3\right)^{1 / 2}} e^{-r / a_0}, \quad r=\left(x^2+y^2+z^2\right)^{1 / 2}
$$

是一个归一化波函数（氢原子的基态波函数）。注意，在三维中，对于只依赖于 $r$ 的函数，归一化条件为

$$
\begin{aligned}
\langle\psi \mid \psi\rangle & =\int \psi^*(r, \theta, \phi) \psi(r, \theta, \phi) r^2 \mathrm{d} r \mathrm{d}(\cos \theta) \mathrm{d} \phi \\
& =4 \pi \int \psi^*(r) \psi(r) r^2 \mathrm{d} r=1
\end{aligned}
$$

(2) 计算该状态中的 $(\Delta X)^2$ [说明 $(\Delta X)^2=\dfrac{1}{3}\left\langle r^2\right\rangle$]，并重新得到方程 (9.4.9) 中给出的结果。

(3) 证明在该状态中
$\langle 1 / r\rangle \simeq 1 /\langle r\rangle \simeq m e^2 / \hbar^2$。

**习题 9.4.3** 暂时忽略氢原子是一个三维系统这一事实，并假装

$$
H=\frac{P^2}{2 m}-\frac{e^2}{\left(R^2\right)^{1 / 2}} \quad\left(P^2=P_x^2+P_y^2+P_z^2, R^2=X^2+Y^2+Z^2\right)
$$

对应于一个一维问题。假设

$$
\Delta P \cdot \Delta R \geqslant \hbar / 2
$$

估算基态能量。

**习题 9.4.4** 计算 $\Delta T \cdot \Delta X$，其中 $T=P^2/2m$。为什么这一关系没有那么著名？

## 9.5 能量—时间不确定关系

# Chapter 9 The Heisenberg Uncertainty Relations
## 9.1 Introduction
## 9.2 Derivation of the Uncertainty Relations
## 9.3 The Minimum Uncertainty Packet
## 9.4 Applications of the Uncertainty Principle
**Exercise 9.4.1** Consider the oscillator in the state $|n=1\rangle$ and verify that
$$
\left\langle\frac{1}{X^2}\right\rangle \simeq \frac{1}{\left\langle X^2\right\rangle} \simeq \frac{m \omega}{\hbar}
$$



**Exercise 9.4.2** (1) By referring to the table of integrals in Appendix A.2, verify that
$$
\psi=\frac{1}{\left(\pi a_0^3\right)^{1 / 2}} e^{-r / a_0}, \quad r=\left(x^2+y^2+z^2\right)^{1 / 2}
$$
is a normalized wave function (of the ground state of hydrogen). Note that in three dimensions the normalization condition is
$$
\begin{aligned}
\langle\psi \mid \psi\rangle & =\int \psi^*(r, \theta, \phi) \psi(r, \theta, \phi) r^2 \mathrm{d} r \mathrm{d}(\cos \theta) \mathrm{d} \phi \\
& =4 \pi \int \psi^*(r) \psi(r) r^2 \mathrm{d} r=1
\end{aligned}
$$
for a function of just $r$.

(2) Calculate $(\Delta X)^2$ in this state [argue that $(\Delta X)^2=\dfrac{1}{3}\left\langle r^2\right\rangle$] and 
regain the result quoted in Eq. (9.4.9).

(3) Show that $\langle 1 / r\rangle \simeq 1 /\langle r\rangle \simeq m e^2 / \hbar^2$ in this state.

**Exercise 9.4.3** Ignore the fact that the hydrogen atom is a three-dimensional system and pretend that
$$
H=\frac{P^2}{2 m}-\frac{e^2}{\left(R^2\right)^{1 / 2}} \quad\left(P^2=P_x^2+P_y^2+P_z^2, R^2=X^2+Y^2+Z^2\right)
$$
corresponds to a one-dimensional problem. Assuming
$$
\Delta P \cdot \Delta R \geqslant \hbar / 2
$$
estimate the ground-state energy.

**Exercise 9.4.4** Compute $\Delta T \cdot \Delta X$, where $T=P^2/2m$. Why is this relation not so famous?

## 9.5 The Energy-Time Unvertainty Relation

第13章氢原子

13.1 本征值问题

第12章旋转不变性与角动量

12.1 二维中的平移

第11章对称性及其推论

11.1 概述

11.2 量子理论中的平移不变性

第10章具有 $N$ 个自由度的系统

10.1 一维中的 $N$ 个粒子

第9章 Heisenberg 不确定关系

9.1 引言

9.2 不确定关系的推导

9.3 最小不确定度波包

9.4 不确定性原理的应用

第13章 氢原子