量子物理整理

第一章 量子物理基础

1.1 光量子、玻尔原子

1.1.1 黑体辐射与普朗克的量子假设

基尔霍夫辐射定律

单色辐出度$M(\lambda,T)=\dfrac{dE_\lambda}{d\lambda}(w/m^3)$：单位时间物体表面单位面积内发射的波长在$\lambda$附近单位波长间隔内的辐射能

总辐出度$M(T)=\int_0^{+\infty}M(\lambda,T)d\lambda$

物体向周围发射辐射能的同时，也吸收周围物体的辐射能。辐射从外界入射到不透明物体时，一部分吸收，一部分反射（一部分透射）

单色吸收比$\alpha(\lambda,T)$在波长为$\lambda\sim\lambda+d\lambda$范围内的吸收比

任何温度下吸收比=1称为（绝对）黑体

$\dfrac{M_1(\lambda,T)}{\alpha_1(\lambda,T)}=\dfrac{M_2(\lambda,T)}{\alpha_2(\lambda,T)}=\cdots=M_0(\lambda,T),M_0(\lambda,T)是黑体单色辐出度$

带小孔孔腔视为绝对黑体的模型

黑体辐射基本规律：

1. 斯特藩-玻尔兹曼定律：$M_0(T)=\sigma T^4$，总辐出度$M_0(T)$
2. 维恩位移定律：$\lambda_mT=b$，单色辐出度的峰值波长$\lambda_m$

普朗克能量子假说

普朗克公式：$M_0(\lambda,T)=2\pi hc^2\lambda^{-5}\dfrac1{e^{hc/k_BT\lambda}-1}$

• 波长短/温度低的近似：

  维恩公式：$M_0(\lambda,T)=2\pi hc^2\lambda^{-5}e^{-hc/k_BT\lambda}$

• 波长高/温度高的近似：

  瑞利-金斯公式：$M_0(\lambda,T)=2\pi k_Bc\lambda^{-4}T$

普朗克假定：$谐振子能量值只取某个最小能量的整数倍\epsilon = h\nu$

1.1.2 光的波粒二象性

1. 光电效应

   1. 电势差足够大时，光电流饱和，饱和电流大小与入射光强成正比

   2. 遏制电压$U_0$，$U=-U_0$，光电流为0。光电子从表面逸出最大初速度$v_m$满足：$\dfrac12mv_m^2=eU_0$。最大初动能于入射光强度无关

   3. 红限频率

      遏制电势差与入射光频率有线性关系，斜率与金属种类无关

   4. 弛豫时间

      几乎瞬时

2. 爱因斯坦光电效应方程

   1. 光强：$nh\nu$
   2. 遏制电压：$\dfrac12mv^2=h\nu-A=eU_c,\ A:逸出功$
   3. 红限频率：$v_0=\dfrac Ah$

3. 康普顿散射

   $\Delta\lambda=\lambda-\lambda_0=\dfrac h{m_ec}(1-\cos\theta)$

1.1.3 玻尔原子理论

氢原子光谱来自电子跃迁，因此不是连续谱

里德伯公式：$\tilde{\nu}=\dfrac1\lambda=R_H(\dfrac1{m^2}-\dfrac1{n^2})$, $R_H=1.0967758\times10^7 m^{-1}$，单位是$1/m$

玻尔假设：

1. 定态条件：电子绕核圆周运动，不辐射能量，处于稳定状态称为定态，相应的能量称为能级
2. 频率条件：跃迁能量$h\nu=E_n-E_m$
3. 角动量量子化条件：$mv_nr_n=n\hbar=n\dfrac h{2\pi}$;$\dfrac {e^2}{4\pi\varepsilon_0 r^2}=m\dfrac{v^2}r$

$E_1=-13.6eV$单位是eV

1.2 物质波

1.2.1 物质波

实物粒子具有波动性

$E=h\nu=\hbar\omega$

$\vec p=\dfrac h\lambda\vec n=\hbar k\vec n$

1.3&1.4 波函数和态叠加原理&薛定谔方程

1.4.1 波函数

波函数用平面波描述：

1. 自由粒子：$\Psi(x,t)=Ae^{i(kx-\omega t)}=Ae^{\frac i\hbar(px-Et)}$
2. 势场中的粒子：$\Phi(\vec r,t)$

纹间距$\Delta\propto\lambda,\lambda=\dfrac h{mv}$

性质：单值、有限（全空间中模长积分为0）、连续

$\int_{-\infty}^{\infty} x^n e^{-\alpha x^2} dx = 0 \quad \text{（n 为奇数）}$

$\int_{-\infty}^{\infty} x^{2k} e^{-\alpha x^2} dx = \frac{(2k - 1)!!}{(2\alpha)^k} \sqrt{\frac{\pi}{\alpha}}$

特别一些，$\int_{-\infty}^\infty x^{2k}e^{-x^2}dx=(2k-1)!!\sqrt{\pi}$

再特别一些，$\int_{-\infty}^\infty e^{-x^2}dx=\sqrt{\pi}$

1.4.2波函数的应用

1. 电子绕核转动形成环形驻波：$2\pi r=n\lambda;\lambda=\dfrac h{mv}\Rightarrow L=mvr=n\hbar$
2. 一维无限深势阱驻波：$L=n\dfrac \lambda 2\Rightarrow p = \dfrac h\lambda=n\dfrac h{2L};E_n=\dfrac{p^2}{2m}=n^2\dfrac{h^2}{8mL^2}$

1.4.3波函数的统计解释

1. 亮度：德布罗意波强度的大小$\left |\Psi\right |^2=\Psi\Psi^*=\begin{cases}极大值 ，&亮条纹\0，极小值，&暗条纹\end{cases}$

   亮度的物理意义为概率密度

2. 波函数满足条件

   1. 单值，任何地方不可能有多个波函数（概率密度）

   2. 有限：$\int\limits_\Omega\left | \Psi(\vec r,t)\right |^2dV=1（\Omega为全空间）$

      （但不排除有些孤立奇点趋于无穷

   3. 连续：二阶导存在

1.3.1 薛定谔方程

1. 自由粒子的薛定谔方程

   $i\hbar \dfrac{\partial}{\partial t}\Psi(x,t)=-\dfrac{\hbar^2}{2m}\dfrac{\partial}{\partial x ^2}\Psi(x,t)$

   1. 推广到势场$U(x,t)$

      $i\hbar \dfrac{\partial}{\partial t}\Psi(x,t)=\left[-\dfrac{\hbar^2}{2m}\dfrac{\partial}{\partial x ^2}+U(x,t)\right]\Psi(x,t)$

   2. 推广到三维情况

      $\nabla^2=\dfrac{\partial^2}{\partial x^2}+\dfrac{\partial^2}{\partial y^2}+\dfrac{\partial^2}{\partial z^2}$

      $\hat H=-\dfrac{\hbar^2}{2m}\nabla^2+U(\vec r,t)$

      $i\hbar\dfrac{\partial}{\partial t}\Psi(\vec r,t)=\hat H\Psi(\vec r, t)$

2. 定态薛定谔方程(粒子在空间中出现的概率是稳定不变的)

   若$\dfrac{\partial\hat H}{\partial t}=0$或者$U$与时间无关，则薛定谔方程可以分离变量

   1. 分离变量

      1. $i\hbar\dfrac{dT(t)}{dt}=ET(t)$
      2. $\hat H\Phi(\vec r)=E\Phi(\vec r)$

   2. 振动因子

      分离变量方程1的解为$T(t)=Ce^{-\frac ihEt}$

   3. 定态薛定谔方程（能量本征方程）就是方程2

      $\left(-\dfrac{\hbar^2}{2m}\nabla^2+U(\vec r)\right)\Phi(\vec r)=E\Phi(\vec r)$

      通解：$\Psi(x,t)=\sum\limits_{n=1}^{\infty}c_ne^{-\frac{iE_nt}\hbar}\Phi_n(x)$

   定态：能量取特定值的状态，薛定谔方程的特解

   判断是否是定态：$\left | \Psi(x,t)\right |^2=\Psi(x,t)^*\cdot\Psi(x,t)$是否与$t$相关，不相关则定态

1.3.2 不确定度关系

$\Delta x\cdot\Delta p_x\geq\dfrac{\hbar}2$

$\Delta y\cdot\Delta p_y\geq\dfrac{\hbar}2$

$\Delta z\cdot\Delta p_z\geq\dfrac{\hbar}2$

$\Delta E\cdot \Delta t\geq \dfrac \hbar 2$

1.3.3 几率流密度矢量

$i\hbar(\Psi^\Psi)=\dfrac{\hbar^2}{2m}\nabla\cdot[\Psi\nabla\Psi^-\Psi^*\nabla\Psi]$

几率密度：$\rho=\Psi^*\Psi$

几率流密度矢量：$\vec J=\dfrac{i\hbar}{2m}[\Psi\nabla\Psi^-\Psi^\nabla\Psi]$

几率守恒方程：

微分形式：$\dfrac \partial{\partial t}\rho+\nabla\cdot\vec J=0$

积分形式：$\frac{d}{dt}\int_\tau\rho(\vec r,t)d\tau=-\oint_S \vec J\cdot d\vec S$

左边：闭区域$\tau$内找到粒子的总几率（或粒子数）在单位时间内的增量

右边：单位时间内通过$\tau$的封闭表面 $S$流入（面积分前面的负号）$\tau$内的几率（或粒子数）

1.3.3 叠加态

1. 若$\Psi_1,\Psi_2,\cdots,\Psi_n$是粒子的可能状态，则粒子也可处在它们的线性叠加态$\Psi=\sum\limits_{i=1}^nc_i\Psi_i$

2. 处于$\Psi态时，部分地处于每种态，处于\Psi_k态的几率是\left|c_k\right|^2$，且$\sum\limits_{k=1}^n\left|c_k\right|^2=1$

   条件：

   1. $\Psi_1,\cdots,\Psi_n$正交归一
   2. $\Psi$满足归一化条件$\int\Psi^*\Psi_ldx=\delta_{kl}=\begin{cases}1&k=l\0&k\neq l\end{cases}$

1.4.4 无限深阱势中的粒子

$\Phi(x)=\begin{cases}\sqrt{\dfrac{2}{a}}\sin\dfrac{n\pi}ax &0<x<a&阱内\0&x\leq0,x\geq a&阱外\end{cases}$

性质：

1. 正交性：$\int\limits_{-\infty}^{+\infty}\Phi^*m(x)\Phi_n(x)dx=\delta{mn}=\begin{cases}1&m=n\0&m\ne n\end{cases}$
2. 通过初始状态确定每一个本征波函数的系数
3. 基态是$E_1$，第n激发态是$E_{n+1}$，$E_n=\dfrac{n^2\pi^2\hbar^2}{2ma^2}$
4. 边界条件：函数连续且一阶导连续

1.5 物理量与算符

1. 位置算符：$\hat x\rightleftharpoons x$

2. 动量算符：$\hat p \rightleftharpoons -i\hbar\nabla$

3. 总能量算符：$\hat H\rightleftharpoons-\dfrac{\hbar^2}{2m}\nabla^2+U(\vec r)$

4. 设算符为$\hat A$，则以下物理量的平均值为$\int_{-\infty}^\infty \Psi^*\hat A\Psi dx$

   物理量	坐标空间表示（$x$ 表象）	动量空间表示（$p$ 表象）	说明
   位置算符$\hat{x}$	$\hat{x} = x$	$\hat{x} = i\hbar \frac{d}{dp}$	乘法/导数互为傅里叶对
   动量算符$\hat{p}$	$\hat{p} = -i\hbar \frac{d}{dx}$	$\hat{p} = p$
   动能算符$\hat{T}$	$\hat{T} = -\frac{\hbar^2}{2m} \frac{d^2}{dx^2}$	$\hat{T} = \frac{p^2}{2m}$
   势能算符$\hat{U}(x)$	$\hat{U}(x) = U(x)$	$\hat{U}(x) = U(i\hbar \frac{d}{dp})$	非平移不变势变成微分算符
   哈密顿量$\hat{H}$	$\hat{H} = -\frac{\hbar^2}{2m} \nabla^2 + U(x)$	$\hat{H} = \frac{p^2}{2m} + U(i\hbar \frac{d}{dp})$	总能量
   力矩$\hat{L}_z$	$\hat{L}_z = -i\hbar \left( x \frac{\partial}{\partial y} - y \frac{\partial}{\partial x} \right)$	$\hat{L}_z = -i\hbar \left( p_x \frac{\partial}{\partial p_y} - p_y \frac{\partial}{\partial p_x} \right)$	平面角动量（z分量）
   总角动量$\hat{\vec{L}}$	$\hat{\vec{L}} = -i\hbar \vec{r} \times \nabla$	$\hat{\vec{L}} = -i\hbar \vec{p} \times \nabla_p$	三维角动量向量
   角动量平方$\hat{L}^2$	$\hat{L}^2 = -\hbar^2 \left( \frac{1}{\sin \theta} \frac{\partial}{\partial \theta} \left( \sin \theta \frac{\partial}{\partial \theta} \right) + \frac{1}{\sin^2 \theta} \frac{\partial^2}{\partial \phi^2} \right)$	同左（在$p$ 空间极坐标中结构相同）	球坐标形式，常见于中心势问题
   $\hat{L}_x$	$-i\hbar (y \dfrac \partial{\partial z} - z \dfrac{\partial}{\partial y})$	同形式	角动量 x 分量
   $\hat{L}_y$	$-i\hbar (z \dfrac\partial{\partial x} - x \dfrac\partial{\partial z})$	同形式	角动量 y 分量
   $\hat{L}_z$	$-i\hbar (x \dfrac\partial{\partial y} - y \dfrac\partial{\partial x})$	同形式	角动量 z 分量
   自旋$\hat{S}_x$	$\hat{S}_x = \frac{\hbar}{2} \begin{pmatrix} 0 & 1 \ 1 & 0 \end{pmatrix}$	同左	Pauli 矩阵$\sigma_x$
   自旋$\hat{S}_y$	$\hat{S}_y = \frac{\hbar}{2} \begin{pmatrix} 0 & -i \ i & 0 \end{pmatrix}$	同左	Pauli 矩阵$\sigma_y$
   自旋$\hat{S}_z$	$\hat{S}_z = \frac{\hbar}{2} \begin{pmatrix} 1 & 0 \ 0 & -1 \end{pmatrix}$	同左	Pauli 矩阵$\sigma_z$
   粒子数算符$\hat{N}$	$\hat{N} = \hat{a}^\dagger \hat{a}$	同左	通常用于谐振子/量子场理论

5. 算符的运算规则

   1. 线性算符

   2. 算符之和：$(\hat A+\hat B)\psi=\hat A\psi+\hat B\psi$

   3. 算符之积：$(\hat A\hat B)\phi=\hat A(\hat B\phi)$

   4. $\left[\hat A\hat B-\hat B\hat A\right ]=\hat A\hat B-\hat B\hat A为方程\hat A和\hat B的对易式，若为0，则两者对易$

   5. 算符的本征方程（类似矩阵的特征值）$\hat A\varphi=\lambda\phi$

   6. 算符的复共轭：所有复量换成共轭复量

   7. 厄密算符（厄密算符的共轭转置等于本身）

      若两个任意的波函数$\psi$和$\varphi$，满足$\int\psi^\hat F\varphi d\tau=\int(\hat F\psi)^\varphi d\tau,d\tau\equiv dxdydz$

      常见厄密算符：位置算符、动量算符、能量算符

      量子力学中的力学量算符都是厄密的

6. 厄密算符

   1. 实数性：本征值为实数

   2. 正交性：任意两个本征函数总是正交的

      函数的正交：$\int\psi^*_1(\vec r)\psi_2(\vec r)d\tau=0$

   3. 完备性：

   4. 即对任一模平方可积函数$\psi$，可表示为$\psi(\vec r)=\sum\limits_nc_n\psi_n(\vec r)$，这也可称为广义傅立展开

1.6 力学的测量值 测量与量子态的坍缩

1. 粒子处于本征态$\psi _n$时，测量值是确定的，为本征值$\lambda_n$
2. 如果体系 $\psi(\vec{r})=\sum\limits_n c_n \psi_{n}(\vec{r})$ 不处于 F 的本征态，它的力学量的测量值不是确定的：测量值可能为任意一个本征态 $\psi_{n}$ 对应的本征值 $\lambda_{n}(n=1,2,…)$，对应的概率分布为 $|c_n|^2$。
   $c_{n}=\int \psi_{n}^{*}(\vec{r}) \psi(\vec{r}) d\tau$
   $c_{n}$ 代表了两个态的“相似程度”→ 测量坍缩到本征态 $\psi_{n}$ 的概率
3. 第一次测量之后，量子态会坍缩为一个$\psi_n$，第二次测量时，体系的态就是$\psi_n$了
4. 物理量的偏差为方均根偏差，即$\Delta x = <x^2>-^2$

1.7 不同力学量能同时确定的条件

两个力学量能同时确定（即具有共同本征态）的充分必要条件是它们的算符对易。

角动量算符互相不对易

角动量平方算符与其分量对应

角动量算符与坐标算符对应

角动量算符本征值：$l=0,1,2,\cdots,m=l,l-1,\cdots,-l+1,-l$，对应的本征函数（波函数）是$\Psi_m(\varphi)=\dfrac1{\sqrt{2\pi}}e^{im\varphi}$

球谐函数$Y_{lm}$是$\hat L^2$和$\hat L_z$共同的本征函数

$\hat L^2 Y_{lm}(\theta,\varphi)=l(l+1)\hbar^2Y_{lm}(\theta,\varphi)$

$\hat L_zY_{lm}(\theta,\varphi)=m\hbar Y_{lm}(\theta,\varphi)$

角动量大小：$\sqrt{l(l+1)}\hbar$

$l=0,1,\cdots,n-1;m=-l,-l+1,\cdots,l-1,l$

谐振子能量是量子化的$E_n=(n+\dfrac12)\hbar\omega$

波函数$\Psi_n(x)=\left(\dfrac{\alpha}{2^n\sqrt\pi n!}\right)^{1/2}H_n(\alpha x)e^{-\dfrac12\alpha^2x^2},\alpha=\sqrt{\dfrac{m\omega}{\hbar}}$

$H_n(\xi)=(-1)^ne^{\xi}\dfrac{d^n}{d\xi^n}(e^{-\xi^2})$

$H_0(\xi)=1,H_1(\xi)=2\xi,H_2(\xi)=4\xi^2-2,H_3(\xi)=8\xi^3-12\xi$

n偶：偶宇称，n奇：奇宇称

参数	取值范围	描述	公式
主量子数 (n)	$n = 1, 2, 3, …$	能量大小	$E_n = -13.6 \dfrac1{ n^2} eV$
轨道角量子数 (l)	$l = 0, 1, 2, …, (n-1)$	角动量的大小	$L = \sqrt{l(l+1)} ħ$
轨道磁量子数 (ml)	$m_l = 0, ±1, ±2, …, ±l$	L 的 z 分量	$L_z = m_l ħ$
自旋磁量子数 (ms)	$m_s = ±1/2$	S 的 z 分量	$S_z = m_s ħ=\pm\dfrac12\hbar$

第二章

2.1 有限深势阱

2.2 方势垒

$\begin{cases}\varPsi_1=Ae^{ikx}+A’e^{-ikx}&k<0\\varPsi_2=Be^{k’x}+B’e^{-k’x}&0<x<a\\varPsi_3=Ce^{ikx}&x>a\end{cases}$

反射系数：$R=\dfrac{\left |A’\right |^2}{\left | A \right |^2}$

透射系数：$T=\dfrac{\left |C\right |^2}{\left | A \right |^2}$

R+T=1

2.3 氢原子

类氢原子：$E_n=-\dfrac{m_eZ^2e^4}{2(4\pi\epsilon_0)^2\hbar^2n^2}$

$\varPsi(r,\theta,\varphi)=R_{nl}(\theta,\varphi)Y_{lm}(\theta,\varphi)$，前者径向，后者角向（球谐函数）

2.4 双态系统

态空间是二维的，于是哈密顿算符变成哈密顿矩阵，薛定谔方程的矩阵形式变为：

$i\hbar\dfrac{\partial}{\partial t}\begin{pmatrix}C_1\C_2\end{pmatrix}=\begin{pmatrix}H_{11}&H_{12}\H_{21}&H_{22}\end{pmatrix}\begin{pmatrix}C_1\C_2\end{pmatrix}$

取基矢，使$\hat H$为本征态，则有

$\begin{pmatrix}H_{11}&H_{12}\H_{21}&H_{22}\end{pmatrix}=\begin{pmatrix}E_I&0\0&E_{II}\end{pmatrix}$，进一步得到$i\hbar\dfrac{\partial}{\partial t}\begin{pmatrix}C_I\C_{II}\end{pmatrix}=\begin{pmatrix}E_I&0\0&E_{II}\end{pmatrix}\begin{pmatrix}C_I\C_{II}\end{pmatrix}$

常用互易关系

$[\hat{A} \hat{B}, \hat{C}] = \hat{A} [\hat{B}, \hat{C}] + [\hat{A}, \hat{C}] \hat{B}$

$[\hat x,\hat y]=0$

$[\hat p_x,\hat p_y]=0$

$[\hat x,\hat p_x]=i\hbar$

$[\hat x,\hat L_y]=i\hbar z$

$[\hat x,\hat L_z]=-i\hbar y$

$[x,\hat p_y]=[x,\hat p_z]=0$

$\left[\hat{L_x},\hat{L_y}\right]=i\hbar\hat{L_z}$

$\left[\hat{L}^2,\hat{L_x}\right]=0$

$[\hat H,\hat L_z]=0$

$[\hat H,\hat p]=0$

Chapter 1: Foundations of Quantum Physics

1.1 Light Quanta, Bohr Atom

1.1.1 Blackbody Radiation and Planck’s Quantum Hypothesis

Kirchhoff’s Law of Radiation

Monochromatic radiant exitance $M(\lambda,T)=\dfrac{dE_\lambda}{d\lambda}(w/m^3)$: the radiant energy emitted per unit time per unit area of an object’s surface, within a unit wavelength interval near wavelength $\lambda$

Total radiant exitance $M(T)=\int_0^{+\infty}M(\lambda,T)d\lambda$

While an object emits radiant energy into its surroundings, it also absorbs radiant energy from surrounding objects. When radiation is incident on an opaque object from the outside, part is absorbed, part is reflected (and part is transmitted)

Monochromatic absorptance $\alpha(\lambda,T)$: the absorptance within the wavelength range $\lambda\sim\lambda+d\lambda$

An object whose absorptance equals 1 at any temperature is called an (absolute) blackbody

$\dfrac{M_1(\lambda,T)}{\alpha_1(\lambda,T)}=\dfrac{M_2(\lambda,T)}{\alpha_2(\lambda,T)}=\cdots=M_0(\lambda,T)$, where $M_0(\lambda,T)$ is the monochromatic radiant exitance of a blackbody

A cavity with a small hole is regarded as a model of an absolute blackbody

Basic laws of blackbody radiation:

1. Stefan-Boltzmann law: ==$M_0(T)=\sigma T^4$==, total radiant exitance $M_0(T)$
2. Wien displacement law: ==$\lambda_mT=b$==, peak wavelength $\lambda_m$ of the monochromatic radiant exitance

Planck’s Energy Quantum Hypothesis

Planck’s formula: ==$M_0(\lambda,T)=2\pi hc^2\lambda^{-5}\dfrac1{e^{hc/k_BT\lambda}-1}$==

• Approximation for short wavelengths / low temperatures:

  Wien’s formula: $M_0(\lambda,T)=2\pi hc^2\lambda^{-5}e^{-hc/k_BT\lambda}$

• Approximation for long wavelengths / high temperatures:

  Rayleigh-Jeans formula: $M_0(\lambda,T)=2\pi k_Bc\lambda^{-4}T$

Planck’s assumption: $the energy values of a harmonic oscillator can only be integer multiples of some minimum energy \epsilon = h\nu$

1.1.2 Wave-Particle Duality of Light

1. Photoelectric effect

   1. When the potential difference is large enough, the photocurrent saturates, and the saturation current is proportional to the incident light intensity

   2. Stopping voltage $U_0$, $U=-U_0$, photocurrent is 0. The maximum initial velocity $v_m$ of photoelectrons escaping from the surface satisfies: $\dfrac12mv_m^2=eU_0$. The maximum initial kinetic energy is independent of the incident light intensity

   3. Threshold frequency

      The stopping potential difference has a linear relationship with the incident light frequency, and the slope is independent of the metal type

   4. Relaxation time

      Nearly instantaneous

2. Einstein’s photoelectric equation

   1. Light intensity: $nh\nu$
   2. Stopping voltage: $\dfrac12mv^2=h\nu-A=eU_c,\ A: work function$
   3. Threshold frequency: $v_0=\dfrac Ah$

3. Compton scattering

   ==$\Delta\lambda=\lambda-\lambda_0=\dfrac h{m_ec}(1-\cos\theta)$==

1.1.3 Bohr’s Atomic Theory

The hydrogen atom spectrum comes from electron transitions, so it is not a continuous spectrum

Rydberg formula: $\tilde{\nu}=\dfrac1\lambda=R_H(\dfrac1{m^2}-\dfrac1{n^2})$, $R_H=1.0967758\times10^7 m^{-1}$, unit is $1/m$

Bohr’s postulates:

1. Stationary state condition: an electron moves in a circular orbit around the nucleus without radiating energy, in a stable state called a stationary state, and the corresponding energy is called an energy level
2. Frequency condition: transition energy $h\nu=E_n-E_m$
3. Angular momentum quantization condition: ==$mv_nr_n=n\hbar=n\dfrac h{2\pi}$;$\dfrac {e^2}{4\pi\varepsilon_0 r^2}=m\dfrac{v^2}r$==

==$E_1=-13.6eV$== unit is eV

1.2 Matter Waves

1.2.1 Matter Waves

Material particles possess wave nature

$E=h\nu=\hbar\omega$

$\vec p=\dfrac h\lambda\vec n=\hbar k\vec n$

1.3 & 1.4 Wave Function and Superposition Principle & Schrödinger Equation

1.4.1 Wave Function

The wave function is described by a plane wave:

1. ==Free particle: $\Psi(x,t)=Ae^{i(kx-\omega t)}=Ae^{\frac i\hbar(px-Et)}$==
2. Particle in a potential field: $\Phi(\vec r,t)$

Fringe spacing $\Delta\propto\lambda,\lambda=\dfrac h{mv}$

Properties: single-valued, finite (the integral of the modulus squared over all space is 0), continuous

$\int_{-\infty}^{\infty} x^n e^{-\alpha x^2} dx = 0 \quad \text{（n is odd）}$

$\int_{-\infty}^{\infty} x^{2k} e^{-\alpha x^2} dx = \frac{(2k - 1)!!}{(2\alpha)^k} \sqrt{\frac{\pi}{\alpha}}$

==In particular, $\int_{-\infty}^\infty x^{2k}e^{-x^2}dx=(2k-1)!!\sqrt{\pi}$==

==Even more particularly, $\int_{-\infty}^\infty e^{-x^2}dx=\sqrt{\pi}$==

1.4.2 Applications of the Wave Function

1. An electron revolving around the nucleus forms a circular standing wave: $2\pi r=n\lambda;\lambda=\dfrac h{mv}\Rightarrow L=mvr=n\hbar$
2. One-dimensional infinite square well standing wave: $L=n\dfrac \lambda 2\Rightarrow p = \dfrac h\lambda=n\dfrac h{2L};E_n=\dfrac{p^2}{2m}=n^2\dfrac{h^2}{8mL^2}$

1.4.3 Statistical Interpretation of the Wave Function

1. Brightness: the magnitude of the de Broglie wave intensity $\left |\Psi\right |^2=\Psi\Psi^*=\begin{cases}maximum, &bright fringes\0, minimum, &dark fringes\end{cases}$

   The physical meaning of brightness is probability density

2. Conditions satisfied by the wave function

   1. Single-valued: there cannot be multiple wave function values (probability densities) at any place

   2. Finite: $\int\limits_\Omega\left | \Psi(\vec r,t)\right |^2dV=1（\Omega is all space）$

      (but some isolated singular points tending to infinity are not excluded

   3. Continuous: the second derivative exists

1.3.1 Schrödinger Equation

1. Schrödinger equation for a free particle

   ==$i\hbar \dfrac{\partial}{\partial t}\Psi(x,t)=-\dfrac{\hbar^2}{2m}\dfrac{\partial}{\partial x ^2}\Psi(x,t)$==

   1. Generalized to a potential field $U(x,t)$

      ==$i\hbar \dfrac{\partial}{\partial t}\Psi(x,t)=\left[-\dfrac{\hbar^2}{2m}\dfrac{\partial}{\partial x ^2}+U(x,t)\right]\Psi(x,t)$==

   2. Generalized to the three-dimensional case

      $\nabla^2=\dfrac{\partial^2}{\partial x^2}+\dfrac{\partial^2}{\partial y^2}+\dfrac{\partial^2}{\partial z^2}$

      ==$\hat H=-\dfrac{\hbar^2}{2m}\nabla^2+U(\vec r,t)$==

      $i\hbar\dfrac{\partial}{\partial t}\Psi(\vec r,t)=\hat H\Psi(\vec r, t)$

2. Time-independent Schrödinger equation (the probability of the particle appearing in space is stable and unchanging)

   If $\dfrac{\partial\hat H}{\partial t}=0$ or $U$ is independent of time, then the Schrödinger equation can be separated into variables

   1. Separation of variables

      1. ==$i\hbar\dfrac{dT(t)}{dt}=ET(t)$==
      2. ==$\hat H\Phi(\vec r)=E\Phi(\vec r)$==

   2. Oscillation factor

      The solution of separated equation 1 is $T(t)=Ce^{-\frac ihEt}$

   3. The time-independent Schrödinger equation (energy eigenvalue equation) is equation 2

      ==$\left(-\dfrac{\hbar^2}{2m}\nabla^2+U(\vec r)\right)\Phi(\vec r)=E\Phi(\vec r)$==

      General solution: $\Psi(x,t)=\sum\limits_{n=1}^{\infty}c_ne^{-\frac{iE_nt}\hbar}\Phi_n(x)$

   Stationary state: a state in which energy takes a specific value, a particular solution of the Schrödinger equation

   ==To determine whether it is a stationary state: whether $\left | \Psi(x,t)\right |^2=\Psi(x,t)^*\cdot\Psi(x,t)$ is related to $t$; if not, it is a stationary state==

1.3.2 Uncertainty Relation

$\Delta x\cdot\Delta p_x\geq\dfrac{\hbar}2$

$\Delta y\cdot\Delta p_y\geq\dfrac{\hbar}2$

$\Delta z\cdot\Delta p_z\geq\dfrac{\hbar}2$

$\Delta E\cdot \Delta t\geq \dfrac \hbar 2$

1.3.3 Probability Current Density Vector

==$i\hbar(\Psi^\Psi)=\dfrac{\hbar^2}{2m}\nabla\cdot[\Psi\nabla\Psi^-\Psi^*\nabla\Psi]$==

Probability density: $\rho=\Psi^*\Psi$

Probability current density vector: ==$\vec J=\dfrac{i\hbar}{2m}[\Psi\nabla\Psi^-\Psi^\nabla\Psi]$==

Probability conservation equation:

Differential form: $\dfrac \partial{\partial t}\rho+\nabla\cdot\vec J=0$

Integral form: $\frac{d}{dt}\int_\tau\rho(\vec r,t)d\tau=-\oint_S \vec J\cdot d\vec S$

Left side: the increment per unit time of the total probability (or number of particles) of finding the particle in the closed region $\tau$

Right side: the probability (or number of particles) flowing into $\tau$ per unit time through the closed surface $S$ of $\tau$ (the negative sign in front of the surface integral)

1.3.3 Superposition States

1. If $\Psi_1,\Psi_2,\cdots,\Psi_n$ are possible states of a particle, then the particle can also be in their linear superposition state $\Psi=\sum\limits_{i=1}^nc_i\Psi_i$

2. When in the $\Psi state, it is partly in each state, and the probability of being in the \Psi_k state is \left|c_k\right|^2$, and $\sum\limits_{k=1}^n\left|c_k\right|^2=1$

   Conditions:

   1. $\Psi_1,\cdots,\Psi_n$ are orthonormal
   2. $\Psi$ satisfies the normalization condition $\int\Psi^*\Psi_ldx=\delta_{kl}=\begin{cases}1&k=l\0&k\neq l\end{cases}$

1.4.4 Particle in an Infinite Square Well

==$\Phi(x)=\begin{cases}\sqrt{\dfrac{2}{a}}\sin\dfrac{n\pi}ax &0<x<a&inside the well\0&x\leq0,x\geq a&outside the well\end{cases}$==

Properties:

1. Orthogonality: $\int\limits_{-\infty}^{+\infty}\Phi^*m(x)\Phi_n(x)dx=\delta{mn}=\begin{cases}1&m=n\0&m\ne n\end{cases}$
2. Determine the coefficient of each eigenwave function from the initial state
3. The ground state is $E_1$, the nth excited state is $E_{n+1}$, ==$E_n=\dfrac{n^2\pi^2\hbar^2}{2ma^2}$==
4. Boundary conditions: the function is continuous and its first derivative is continuous

1.5 Physical Quantities and Operators

1. Position operator: $\hat x\rightleftharpoons x$

2. Momentum operator: $\hat p \rightleftharpoons -i\hbar\nabla$

3. Total energy operator: $\hat H\rightleftharpoons-\dfrac{\hbar^2}{2m}\nabla^2+U(\vec r)$

4. Let the operator be $\hat A$, then the ==average value of the following physical quantities is $\int_{-\infty}^\infty \Psi^*\hat A\Psi dx$==

   Physical quantity	Coordinate-space representation ($x$ representation)	Momentum-space representation ($p$ representation)	Notes
   Position operator $\hat{x}$	$\hat{x} = x$	$\hat{x} = i\hbar \frac{d}{dp}$	Multiplication/derivative are Fourier pairs
   Momentum operator $\hat{p}$	==$\hat{p} = -i\hbar \frac{d}{dx}$==	$\hat{p} = p$
   Kinetic energy operator $\hat{T}$	==$\hat{T} = -\frac{\hbar^2}{2m} \frac{d^2}{dx^2}$==	$\hat{T} = \frac{p^2}{2m}$
   Potential energy operator $\hat{U}(x)$	==$\hat{U}(x) = U(x)$==	$\hat{U}(x) = U(i\hbar \frac{d}{dp})$	A non-translation-invariant potential becomes a differential operator
   Hamiltonian $\hat{H}$	==$\hat{H} = -\frac{\hbar^2}{2m} \nabla^2 + U(x)$==	$\hat{H} = \frac{p^2}{2m} + U(i\hbar \frac{d}{dp})$	Total energy
   Torque $\hat{L}_z$	$\hat{L}_z = -i\hbar \left( x \frac{\partial}{\partial y} - y \frac{\partial}{\partial x} \right)$	$\hat{L}_z = -i\hbar \left( p_x \frac{\partial}{\partial p_y} - p_y \frac{\partial}{\partial p_x} \right)$	Planar angular momentum (z component)
   Total angular momentum $\hat{\vec{L}}$	$\hat{\vec{L}} = -i\hbar \vec{r} \times \nabla$	$\hat{\vec{L}} = -i\hbar \vec{p} \times \nabla_p$	Three-dimensional angular momentum vector
   Angular momentum squared $\hat{L}^2$	$\hat{L}^2 = -\hbar^2 \left( \frac{1}{\sin \theta} \frac{\partial}{\partial \theta} \left( \sin \theta \frac{\partial}{\partial \theta} \right) + \frac{1}{\sin^2 \theta} \frac{\partial^2}{\partial \phi^2} \right)$	Same as left (same structure in $p$-space polar coordinates)	Spherical coordinate form, common in central potential problems
   $\hat{L}_x$	$-i\hbar (y \dfrac \partial{\partial z} - z \dfrac{\partial}{\partial y})$	Same form	Angular momentum x component
   $\hat{L}_y$	$-i\hbar (z \dfrac\partial{\partial x} - x \dfrac\partial{\partial z})$	Same form	Angular momentum y component
   $\hat{L}_z$	$-i\hbar (x \dfrac\partial{\partial y} - y \dfrac\partial{\partial x})$	Same form	Angular momentum z component
   Spin $\hat{S}_x$	$\hat{S}_x = \frac{\hbar}{2} \begin{pmatrix} 0 & 1 \ 1 & 0 \end{pmatrix}$	Same as left	Pauli matrix $\sigma_x$
   Spin $\hat{S}_y$	$\hat{S}_y = \frac{\hbar}{2} \begin{pmatrix} 0 & -i \ i & 0 \end{pmatrix}$	Same as left	Pauli matrix $\sigma_y$
   Spin $\hat{S}_z$	$\hat{S}_z = \frac{\hbar}{2} \begin{pmatrix} 1 & 0 \ 0 & -1 \end{pmatrix}$	Same as left	Pauli matrix $\sigma_z$
   Particle number operator $\hat{N}$	$\hat{N} = \hat{a}^\dagger \hat{a}$	Same as left	Usually used in harmonic oscillators/quantum field theory

5. Operator operation rules

   1. Linear operator

   2. Sum of operators: $(\hat A+\hat B)\psi=\hat A\psi+\hat B\psi$

   3. Product of operators: $(\hat A\hat B)\phi=\hat A(\hat B\phi)$

   4. $\left[\hat A\hat B-\hat B\hat A\right ]=\hat A\hat B-\hat B\hat A is the commutator of operators \hat A and \hat B; if it is 0, then the two commute$

   5. Eigenvalue equation of an operator (similar to the eigenvalues of a matrix) $\hat A\varphi=\lambda\phi$

   6. Complex conjugate of an operator: replace all complex quantities with their complex conjugates

   7. Hermitian operator (the conjugate transpose of a Hermitian operator equals itself)

      If for any two wave functions $\psi$ and $\varphi$, $\int\psi^\hat F\varphi d\tau=\int(\hat F\psi)^\varphi d\tau,d\tau\equiv dxdydz$

      Common Hermitian operators: position operator, momentum operator, energy operator

      All mechanical quantity operators in quantum mechanics are Hermitian

6. Hermitian operators

   1. Realness: eigenvalues are real

   2. Orthogonality: any two eigenfunctions are always orthogonal

      Orthogonality of functions: $\int\psi^*_1(\vec r)\psi_2(\vec r)d\tau=0$

   3. Completeness:

   4. That is, for any square-integrable function $\psi$, it can be expressed as $\psi(\vec r)=\sum\limits_nc_n\psi_n(\vec r)$, which can also be called a generalized Fourier expansion

1.6 Measured Values of Mechanical Quantities, Measurement, and Collapse of Quantum States

1. When a particle is in an eigenstate $\psi _n$, the measured value is definite and is the eigenvalue $\lambda_n$
2. If the system $\psi(\vec{r})=\sum\limits_n c_n \psi_{n}(\vec{r})$ is not in an eigenstate of F, the measured value of its mechanical quantity is not definite: the measured value may be the eigenvalue $\lambda_{n}(n=1,2,…)$ corresponding to any eigenstate $\psi_{n}$, and the corresponding probability distribution is $|c_n|^2$.
   $c_{n}=\int \psi_{n}^{*}(\vec{r}) \psi(\vec{r}) d\tau$
   $c_{n}$ represents the “degree of similarity” between the two states → the probability that the measurement collapses to the eigenstate $\psi_{n}$
3. After the first measurement, the quantum state collapses to some $\psi_n$; at the second measurement, the state of the system is $\psi_n$
4. The deviation of a physical quantity is the root-mean-square deviation, that is, $\Delta x = <x^2>-^2$

1.7 Conditions for Different Mechanical Quantities to Be Determined Simultaneously

==The necessary and sufficient condition for two mechanical quantities to be determined simultaneously (that is, to have common eigenstates) is that their operators commute.==

Angular momentum operators do not commute with each other

The angular momentum squared operator corresponds to its components

Angular momentum operators correspond to coordinate operators

==Eigenvalues of the angular momentum operator: $l=0,1,2,\cdots,m=l,l-1,\cdots,-l+1,-l$, and the corresponding eigenfunction (wave function) is $\Psi_m(\varphi)=\dfrac1{\sqrt{2\pi}}e^{im\varphi}$==

Spherical harmonics $Y_{lm}$ are common eigenfunctions of $\hat L^2$ and $\hat L_z$

$\hat L^2 Y_{lm}(\theta,\varphi)=l(l+1)\hbar^2Y_{lm}(\theta,\varphi)$

$\hat L_zY_{lm}(\theta,\varphi)=m\hbar Y_{lm}(\theta,\varphi)$

Magnitude of angular momentum: ==$\sqrt{l(l+1)}\hbar$==

$l=0,1,\cdots,n-1;m=-l,-l+1,\cdots,l-1,l$

Harmonic oscillator energy is quantized ==$E_n=(n+\dfrac12)\hbar\omega$==

Wave function $\Psi_n(x)=\left(\dfrac{\alpha}{2^n\sqrt\pi n!}\right)^{1/2}H_n(\alpha x)e^{-\dfrac12\alpha^2x^2},\alpha=\sqrt{\dfrac{m\omega}{\hbar}}$

$H_n(\xi)=(-1)^ne^{\xi}\dfrac{d^n}{d\xi^n}(e^{-\xi^2})$

$H_0(\xi)=1,H_1(\xi)=2\xi,H_2(\xi)=4\xi^2-2,H_3(\xi)=8\xi^3-12\xi$

n even: even parity, n odd: odd parity

Parameter	Range	Description	Formula
Principal quantum number (n)	$n = 1, 2, 3, …$	Energy magnitude	$E_n = -13.6 \dfrac1{ n^2} eV$
Orbital angular quantum number (l)	$l = 0, 1, 2, …, (n-1)$	Magnitude of angular momentum	$L = \sqrt{l(l+1)} ħ$
Orbital magnetic quantum number (ml)	$m_l = 0, ±1, ±2, …, ±l$	z component of L	$L_z = m_l ħ$
Spin magnetic quantum number (ms)	$m_s = ±1/2$	z component of S	$S_z = m_s ħ=\pm\dfrac12\hbar$

Chapter 2

2.1 Finite Square Well

2.2 Rectangular Potential Barrier

$\begin{cases}\varPsi_1=Ae^{ikx}+A’e^{-ikx}&k<0\\varPsi_2=Be^{k’x}+B’e^{-k’x}&0<x<a\\varPsi_3=Ce^{ikx}&x>a\end{cases}$

Reflection coefficient: $R=\dfrac{\left |A’\right |^2}{\left | A \right |^2}$

Transmission coefficient: $T=\dfrac{\left |C\right |^2}{\left | A \right |^2}$

R+T=1

2.3 Hydrogen Atom

Hydrogen-like atom: $E_n=-\dfrac{m_eZ^2e^4}{2(4\pi\epsilon_0)^2\hbar^2n^2}$

$\varPsi(r,\theta,\varphi)=R_{nl}(\theta,\varphi)Y_{lm}(\theta,\varphi)$, the former is radial, the latter is angular (spherical harmonics)

2.4 Two-State System

The state space is two-dimensional, so the Hamiltonian operator becomes a Hamiltonian matrix, and the matrix form of the Schrödinger equation becomes:

$i\hbar\dfrac{\partial}{\partial t}\begin{pmatrix}C_1\C_2\end{pmatrix}=\begin{pmatrix}H_{11}&H_{12}\H_{21}&H_{22}\end{pmatrix}\begin{pmatrix}C_1\C_2\end{pmatrix}$

Choosing basis vectors so that $\hat H$ is an eigenstate, we have

$\begin{pmatrix}H_{11}&H_{12}\H_{21}&H_{22}\end{pmatrix}=\begin{pmatrix}E_I&0\0&E_{II}\end{pmatrix}$, and further obtain $i\hbar\dfrac{\partial}{\partial t}\begin{pmatrix}C_I\C_{II}\end{pmatrix}=\begin{pmatrix}E_I&0\0&E_{II}\end{pmatrix}\begin{pmatrix}C_I\C_{II}\end{pmatrix}$

Common Commutation Relations

$[\hat{A} \hat{B}, \hat{C}] = \hat{A} [\hat{B}, \hat{C}] + [\hat{A}, \hat{C}] \hat{B}$

==$[\hat x,\hat y]=0$==

==$[\hat p_x,\hat p_y]=0$==

==$[\hat x,\hat p_x]=i\hbar$==

==$[\hat x,\hat L_y]=i\hbar z$==

==$[\hat x,\hat L_z]=-i\hbar y$==

==$[x,\hat p_y]=[x,\hat p_z]=0$==

$\left[\hat{L_x},\hat{L_y}\right]=i\hbar\hat{L_z}$

$\left[\hat{L}^2,\hat{L_x}\right]=0$

$[\hat H,\hat L_z]=0$

$[\hat H,\hat p]=0$