Basic Concepts

运算规则

$(|\psi\rangle)^\dagger=\langle\psi|$
$(|\psi_i\rangle\langle\psi_j|)^\dagger=|\psi_j\rangle\langle\psi_i|$
$(\hat A\hat B)^\dagger=\hat B^\dagger\hat A^\dagger$
$(\langle\psi_j|\psi_i\rangle)^*=\langle\psi_i|\psi_j\rangle$
$(\langle\psi_j|\psi_i\rangle)^\dagger=(|\psi_i\rangle)^\dagger(\langle\psi_j|)^\dagger=\langle\psi_i|\psi_j\rangle=(\langle\psi_j|\psi_i\rangle)^*$
$\langle\psi_j|\hat A^T|\psi_i\rangle=\langle\psi_i|\hat A|\psi_j\rangle$
$\langle\psi_j|\hat A^\dagger|\psi_i\rangle=(\langle\psi_i|\hat A|\psi_j\rangle)^*$
Hermitian operator: $\hat A^\dagger=\hat A$
Hermitian operator qualities:
- Real Eigenvalues:$\langle\psi_j|\hat A|\psi_i\rangle=(\langle\psi_i|\hat A|\psi_j\rangle)^*\Rightarrow\langle\psi|\hat A|\psi\rangle\in\mathbb R$
- Orthogonal Eigenvectors of different Eigenvalues:$\langle\psi_j|\hat A|\psi_i\rangle=a_i\langle\psi_j|\psi_i\rangle=a_j^\langle\psi_j|\psi_i\rangle=(\langle\psi_i|\hat A|\psi_j\rangle)^\Rightarrow\langle\psi_i|\psi_j\rangle=0,a_i\neq a_j$
- Observable:$\langle\psi|\hat A|\psi\rangle\in \mathbb R$
Unitary operator: $\hat U^\dagger\hat U=\hat U\hat U^\dagger=\hat I$
Matrix Elements: $A_{ij}=\langle i|\hat A|j\rangle$
Projection operator: $\hat P=|\psi\rangle\langle\psi|$
Trace: $Tr(\hat A)=\sum_i\langle i|\hat A|i\rangle$
- Linearity: $Tr(\hat A+\hat B)=Tr(\hat A)+Tr(\hat B)$
- Cyclic property: $Tr(\hat A\hat B\hat C)=Tr(\hat B\hat C\hat A)=Tr(\hat C\hat A\hat B)$
- Similarity: $Tr(\hat A)=Tr(\hat U\hat A\hat U^\dagger)\Rightarrow Tr(\hat A)=\sum_i \lambda_i$

薛定谔表象和海森堡表象

$\hat H|\psi\rangle=i\hbar\frac{\partial}{\partial t}|\psi\rangle$
$\langle\psi|\hat H^\dagger=-i\hbar\frac{\partial}{\partial t}\langle\psi|$
Ehrenfest’s theorem: $\frac{d\langle\hat A\rangle}{dt}=\frac{i}{\hbar}\langle[\hat H,\hat A]\rangle+\langle\frac{\partial\hat A}{\partial t}\rangle$
Heisenberg equation of motion: $\frac{d\hat A}{dt}=\frac{i}{\hbar}[\hat H,\hat A]$

	薛定谔绘景	海森堡绘景	相互作用绘景
算符	$\hat A_S$	$\hat A_H(t)$	$\hat A_I(t)$
矢量	$\lvert\alpha_S(t)\rangle$	$\lvert\alpha_H(t)\rangle$	$\lvert\alpha_I(t)\rangle$
薛定谔矢量方程	$i\hbar\frac{d}{dt}\lvert\alpha_S(t)\rangle=\hat H_S\lvert\alpha_S(t)\rangle$	None	$i\hbar\frac{d}{dt}\lvert\alpha_I(t)\rangle=\hat V_I(t)\lvert\alpha_I(t)\rangle$
薛定谔算符方程	None	$i\hbar\frac{d}{dt}\hat A_H(t)=[\hat A_H(t),\hat H_S]$	$i\hbar\frac{d}{dt}\hat A_I(t)=[\hat A_I(t),\hat H_0]$

算符的对易

$[\hat A,\hat B]=\hat A\hat B-\hat B\hat A$
$[\hat A,\hat B]^\dagger=[\hat B^\dagger,\hat A^\dagger]$
$[\hat A,\hat B+\hat C]=[\hat A,\hat B]+[\hat A,\hat C]$
$[\hat A,\hat B\hat C]=[\hat A,\hat B]\hat C+\hat B[\hat A,\hat C]$
$[\hat A,\hat B\hat C]+[\hat B,\hat C\hat A]+[\hat C,\hat A\hat B]=0$
$[\hat x,\hat p_x]=i\hbar$
$[\hat x,k(\hat p_x)]=i\hbar k’(\hat p_x)$
$[\lambda(\hat x),\hat p_x]=i\hbar\lambda’(\hat x)$
Expotential operator: $e^{\hat A}=\sum_{n=0}^\infty\frac{\hat A^n}{n!}$
- If $\hat A$ is Hermitian, $e^{\hat A}=\sum_n e^{a_n}|n\rangle\langle n|$
- Further, $e^{f(\hat A)}=\sum_n e^{f(a_n)}|n\rangle\langle n|$
- Baker-Hausdorff formula: $e^{\xi\hat A}\hat Be^{-\xi\hat A}=\hat B+\xi[\hat A,\hat B]+\frac{\xi^2}{2!}[\hat A,[\hat A,\hat B]]+\cdots$
- Baker-Campbell-Hausdorff formula: $e^{\hat A+\hat B}=e^{\hat A}e^{\hat B}e^{-\frac{1}{2}[\hat A,\hat B]}$
Uncertainty principle: $\Delta A\Delta B\geq\frac{1}{2}|\langle[\hat A,\hat B]\rangle|$ the equality holds when $\hat A=ic \hat B$
- $\Delta x\Delta p_x\geq\frac{\hbar}{2}$
- $\Delta E\Delta t\geq\frac{\hbar}{2}$

混合态和密度矩阵

Density matrix: $\hat\rho=\sum_i p_i|\psi_i\rangle\langle\psi_i|$
Density matrix Quality:
- Trace: $Tr(\hat\rho)=1$
- Idempotent: $\begin{cases}\hat\rho^2=\hat\rho&pure~state\\\hat\rho^2\neq\hat\rho&mixed~state\end{cases}$
- Trace of Square: $\begin{cases}Tr(\hat\rho^2)=1&pure~state\\Tr(\hat\rho^2)<1&mixed~state\end{cases}$
- Hermitian: $\hat\rho^\dagger=\hat\rho$
- Positive: $\langle\psi|\hat\rho|\psi\rangle\geq 0$
Motion of density matrix: $\frac{d\hat\rho}{dt}=\frac{i}{\hbar}[\hat H,\hat\rho]+\frac{\partial\hat\rho}{\partial t}$
Average value: $\langle\hat A\rangle=Tr(\hat\rho\hat A)$
Reduced density matrix: $\hat\rho_A=Tr_B(\hat\rho_{AB})$
- Average value of partial system: $\langle\hat A\rangle=Tr_A(\hat\rho_{A}\hat A)=Tr_A(Tr_B(\hat\rho_{AB})\hat A)$

谐振子和升降算符

Hamiltonian: $\hat H=\frac{\hat p^2}{2m}+\frac{1}{2}m\omega^2\hat x^2$
Eigenvalue equation: $\hat H|n\rangle=E_n|n\rangle=\hbar\omega(n+\frac{1}{2})|n\rangle$
Creation and Annihilation operator: $\hat a=\frac{1}{\sqrt{2\hbar m\omega}}(i\hat p+m\omega \hat x),\hat a^\dagger=\frac{1}{\sqrt{2\hbar m\omega}}(-i\hat p+m\omega \hat x)$
Commutation relation: $[\hat a,\hat a^\dagger]=1$
Hamiltonian in terms of $\hat a$ and $\hat a^\dagger$: $\hat H=\hbar\omega(\hat a^\dagger\hat a+\frac{1}{2})=\hbar\omega(\hat a\hat a^\dagger-\frac{1}{2})$
Quality of $\hat a$ and $\hat a^\dagger$:
- $\hat a|n\rangle=\sqrt{n}|n-1\rangle$
- $\hat a^\dagger|n\rangle=\sqrt{n+1}|n+1\rangle$
- $\hat x=\sqrt{\frac{\hbar}{2m\omega}}(\hat a+\hat a^\dagger)$
- $\hat p=i\sqrt{\frac{\hbar m\omega}{2}}(\hat a-\hat a^\dagger)$
Ground state: $\hat a|0\rangle=0\Rightarrow|0\rangle=(\frac{m\omega}{\pi\hbar})^{\frac14}e^{-\frac{m\omega}{2\hbar}x^2}$

粒子数表象

Particle number operator: $\hat N=\hat a^\dagger\hat a,\hat H=\hbar\omega(\hat N+\frac12)$
Eigenvalue equation: $\hat N|n\rangle=n|n\rangle$
Amplitude Operator: $\hat B=\sqrt{\hat N+1}=\sum_n \sqrt{n+1}|n\rangle\langle n|$
Phase Operator: $e^{i\hat \Phi}=\sum_{n=0} |n\rangle\langle n+1|,e^{-i\hat \Phi}=\sum_{n=0} |n+1\rangle\langle n|$
Quality:
- $\hat a=\hat B e^{i\hat \Phi}=\sum_n\sqrt{n+1}|n\rangle\langle n+1|$
- $\hat a^\dagger=e^{-i\hat \Phi}\hat B=\sum_n\sqrt{n+1}|n+1\rangle\langle n|$
- $[\hat N,\cos{\hat\Phi}]=-i\sin{\hat\Phi}$
- $[\hat N,\sin{\hat\Phi}]=i\cos{\hat\Phi}$

量子几何

哈密顿算符变化一周的时候，本征态的变化分为动力学相位和几何相位：

$\begin{aligned}C_m(t_0)&=C_m(0)\exp{[-\int^{t_0}_0 \frac{i}{\hbar}E_m(t) dt]}\exp{[-\int^{t_0}_0\langle m(t)|\dot{m}(t)\rangle dt]}\\&=C_m(0)\exp{[-\frac{i}{\hbar}\int^{t_0}_0 E_m(t) dt]}\exp{[i\gamma(t_0)]}\end{aligned}$

其中

$\gamma(t_0)=i\int^{t_0}_0\langle m(t)|\dot{m}(t)\rangle dt$

当然，如果引起本征态或哈密顿算符变化的参数不直接是时间而是$\lambda(t)$，原定义推广为：

$\gamma(\lambda_0)=i\int^{\lambda_0}_0\langle m(\lambda(t))|\vec{\nabla_\lambda}|m(\lambda(t))\rangle dt$

其中

$\langle m(\lambda(t))|\vec{\nabla_\lambda}|m(\lambda(t))\rangle$

被称为Berry联络。

角动量理论

轨道角动量

Definition: $[\hat L_i,\hat L_j]=i\hbar\epsilon_{ijk}\hat L_k$
Commutation relation:
- Scalar Operator: $[\hat L_i,\hat S]=0$
- Vector Operator: $[\hat L_i,\hat V_j]=i\hbar\epsilon_{ijk}\hat V_k$
- $[\hat L^2,\hat L_i]=0$
- Hamiltonian: $[\hat L^2,\hat H]=[\hat L_z,\hat H]=0$
Eigenvalue equation: $\hat L^2|l,m\rangle=l(l+1)\hbar^2|l,m\rangle,\hat L_z|l,m\rangle=m\hbar|l,m\rangle$
Creation and Annihilation operator: $\hat L_\pm=\hat L_x\pm i\hat L_y$
- $[\hat L_z,\hat L_\pm]=\pm\hbar\hat L_\pm$
- $\hat L_\pm|l,m\rangle=\hbar\sqrt{l(l+1)-m(m\pm1)}|l,m\pm1\rangle$

自旋角动量

Definition: $[\hat S_i,\hat S_j]=i\hbar\epsilon_{ijk}\hat S_k$
Commutation relation is as same as orbital angular momentum
Pauli matrices: $\sigma_x=\begin{pmatrix}0&1\\1&0\end{pmatrix},\sigma_y=\begin{pmatrix}0&-i\\i&0\end{pmatrix},\sigma_z=\begin{pmatrix}1&0\\0&-1\end{pmatrix}$
- $\sigma_x^2=\sigma_y^2=\sigma_z^2=\hat I$
- $\sigma_x\sigma_y=i\sigma_z,\sigma_y\sigma_z=i\sigma_x,\sigma_z\sigma_x=i\sigma_y$
- $\sigma_x\sigma_y\sigma_z=i\hat I$
- Commutation relation: $[\sigma_i,\sigma_j]=2i\epsilon_{ijk}\sigma_k$
- Inverse Commutation relation: $\{\sigma_i,\sigma_j\}=2\delta_{ij}$
Eigenvalue Vector of Pauli matrices:
- $\sigma_x=\begin{cases}&1&\frac{1}{\sqrt{2}}\begin{pmatrix}1\\1\end{pmatrix}\\&-1&\frac{1}{\sqrt{2}}\begin{pmatrix}1\-1\end{pmatrix}\end{cases}$
- $\sigma_y=\begin{cases}&1&\frac{1}{\sqrt{2}}\begin{pmatrix}1\\i\end{pmatrix}\\&-1&\frac{1}{\sqrt{2}}\begin{pmatrix}1\-i\end{pmatrix}\end{cases}$
- $\sigma_z=\begin{cases}&1&\begin{pmatrix}1\\0\end{pmatrix}\\&-1&\begin{pmatrix}0\\1\end{pmatrix}\end{cases}$
- $\sigma_n=\begin{cases}&1&\begin{pmatrix}\cos{\frac\theta2}\\\sin{\frac\theta2}e^{i\phi}\end{pmatrix}\\&-1&\begin{pmatrix}\sin{\frac\theta2}e^{i\phi}\-\cos{\frac\theta2}\end{pmatrix}\end{cases}$

对称性

微扰论

不含时微扰

Perturbation Hamiltonian: $\hat H=\hat H_0+\lambda\hat V$
Eigenvalue equation: $\hat H(\lambda)|n\rangle_\lambda=E_n(\lambda)|n\rangle_\lambda$
Modification of Eigenvalue and Vector: $|n\rangle_\lambda=\sum_{i=0}^\infty \lambda^i|n^i\rangle,E_n(\lambda)=\sum_{i=0}^\infty \lambda^iE^i_n$
Correction Equation: $\begin{aligned} \lambda^0\quad&(\hat H^0-E^0_n)|n^0\rangle=0\\ \lambda^1\quad&(\hat H^0-E^0_n)|n^1\rangle=(E^1_n-\delta \hat H)|n^0\rangle\\\cdots\quad\\ \lambda^k\quad&(\hat H^0-E^0_n)|n^k\rangle=(E^1_n-\delta \hat H)|n^{k-1}\rangle+\sum_{i=2}^k E^i_n|n^{k-i}\rangle \end{aligned}$
First Order Correction: $E^1_n=\langle n^0|\hat V|n^0\rangle,|n^1\rangle=\sum_{m\neq n}\frac{\langle m^0|\hat V|n^0\rangle}{E^0_n-E^0_m}|m^0\rangle$
Second Order Correction: $E^2_n=\langle n^0| \hat V|n^{1}\rangle- \langle n^0|E^1_n|n^{1}\rangle$
Degenrate Case: Only to find “Good” basis

含时微扰

Perturbation Hamiltonian: $\hat H=\hat H_0+\lambda\hat V(t)$
Expansion: $|n(t)\rangle=\sum_{i=0}^\infty c_i(t)|i\rangle$
Accurate Equation: $i\hbar\frac{d}{dt}c_n(t)=\sum_mV_{nm} c_m(t)e^{i(E_n-E_m)t/\hbar}$ $i\hbar\frac{d}{dt}\begin{pmatrix}c_1(t)\\c_2(t)\\\vdots\end{pmatrix}=\begin{pmatrix}V_{11}&V_{12}e^{i\omega_{12}t}&\cdots\\V_{21}e^{i\omega_{21}t}&V_{22}&\cdots\\\vdots&\vdots&\ddots\end{pmatrix}\begin{pmatrix}c_1(t)\\c_2(t)\\\vdots\end{pmatrix}$
Dyson series: $c_n(t)=c_n^{(0)}(t)+c_n^{(1)}(t)+c_n^{(2)}(t)+\cdots$
Approximate Equation: $U_I(t,t_0) =\sum_{n=0}^\infty(-\frac{i}{\hbar})^n\int_{t_0}^t\cdots\int_{t_0}^{t_{n-1}}\hat V_I(t_1)\cdots\hat V_I(t_n)dt_1\cdots dt_n$
Zero Order Approximation: $c_n^{(0)}(t)=\langle n|i\rangle$, $i$ is the initial state
First Order Approximation: $c_n^{(1)}(t)=-\frac{i}{\hbar}\int_{t_0}^t\langle n|\hat V(t’)|i\rangle dt’$
Second Order Approximation: $c_n^{(2)}(t)=\frac{-1}{\hbar^2}\sum_m\int_{t_0}^t\int_{t_0}^{t’}\langle n|\hat V_I(t’)|m\rangle\langle m|\hat V_I(t’’)|i\rangle dt’’dt’$

变分法

$\frac{\langle \tilde{0}|\hat H|\tilde0\rangle}{\langle \tilde{0}|\tilde0\rangle}\geq E_0$

散射理论

Scatter Wave Function: $\psi(\vec{r})=e^{i\vec{k}\cdot\vec{r}}+f(\theta,\phi)\frac{e^{ikr}}{r}$, $f(\theta,\phi)$ is the scattering amplitude
Differential Cross Section: $D(\theta,\phi)=|f(\theta,\phi)|^2$
Green Function Method: $\psi(\vec{r})=\phi(\vec{r})+\int_{-\infty}^\infty G_0(k_i,\vec{r}-\vec{r’})V(\vec{r’})\psi(\vec{r’})d\vec{r’}$
Green Function: $G_0(k_i,\vec{r}-\vec{r’})=-\frac{m}{2\pi\hbar^2}\frac{1}{|\vec{r}-\vec{r’}|}e^{\pm ik|\vec{r}-\vec{r’}|}$
Operator Form: $|\psi^+_p\rangle=|k_i\rangle+\hat G_0^+\hat V|\psi^+_p\rangle$
- $|\psi^+_p\rangle=|k_i\rangle+\hat G_0^+\hat V|k_i\rangle+\hat G_0^+\hat V\hat G_0^+\hat V|k_i\rangle+\cdots=\sum_n (\hat G_0^+\hat V)^n|k_i\rangle$
- $f(\theta,\phi)=-\frac{4\pi^2m}{\hbar^2}\langle \vec{k_f}|V|\psi^+_p\rangle$
- $\hat T^+|k_i\rangle=\hat V|\psi^+_p\rangle$
- $\hat T^+=\hat V+\hat V\hat G_0^+\hat V+\hat V\hat G_0^+\hat V\hat G_0^+\hat V+\cdots$
Optical Theorem: $\frac{k}{4\pi}\sigma_{tot}=\Im [f(\theta=0,\phi=0)]$