Cheating-Paper

Basic Concepts

运算规则

1. $(|\psi ⟩{)}^{†}=⟨\psi |$
2. $(|{\psi }_i⟩⟨{\psi }_j|{)}^{†}=|{\psi }_j⟩⟨{\psi }_i|$
3. $(\hat{A}\hat{B}{)}^{†}={\hat{B}}^{†}{\hat{A}}^{†}$
4. $(⟨{\psi }_j|{\psi }_i⟩{)}^{\ast }=⟨{\psi }_i|{\psi }_j⟩$
5. $(⟨{\psi }_j|{\psi }_i⟩{)}^{†}=(|{\psi }_i⟩{)}^{†}(⟨{\psi }_j|{)}^{†}=⟨{\psi }_i|{\psi }_j⟩=(⟨{\psi }_j|{\psi }_i⟩{)}^{\ast }$
6. $⟨{\psi }_j|{\hat{A}}^T|{\psi }_i⟩=⟨{\psi }_i|\hat{A}|{\psi }_j⟩$
7. $⟨{\psi }_j|{\hat{A}}^{†}|{\psi }_i⟩=(⟨{\psi }_i|\hat{A}|{\psi }_j⟩{)}^{\ast }$
8. Hermitian operator: ${\hat{A}}^{†}=\hat{A}$
9. Hermitian operator qualities:
   • Real Eigenvalues:$⟨{\psi }_j|\hat{A}|{\psi }_i⟩=(⟨{\psi }_i|\hat{A}|{\psi }_j⟩{)}^{\ast }\Rightarrow ⟨\psi |\hat{A}|\psi ⟩\in \mathbb{R}$
   • 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:$⟨\psi |\hat{A}|\psi ⟩\in \mathbb{R}$
10. Unitary operator: ${\hat{U}}^{†}\hat{U}=\hat{U}{\hat{U}}^{†}=\hat{I}$
11. Matrix Elements: $A_{ij}=⟨i|\hat{A}|j⟩$
12. Projection operator: $\hat{P}=|\psi ⟩⟨\psi |$
13. Trace: $Tr(\hat{A})=\sum _i⟨i|\hat{A}|i⟩$
    • 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: $Tr(\hat{A})=Tr(\hat{U}\hat{A}{\hat{U}}^{†})\Rightarrow Tr(\hat{A})=\sum _i{\lambda }_i$

薛定谔表象和海森堡表象

1. $\hat{H}|\psi ⟩=i\hslash \frac{\partial }{\partial t}|\psi ⟩$
2. $⟨\psi |{\hat{H}}^{†}=-i\hslash \frac{\partial }{\partial t}⟨\psi |$
3. Ehrenfest’s theorem: $\frac{d⟨\hat{A}⟩}{dt}=\frac{i}{\hslash }⟨[\hat{H},\hat{A}]⟩+⟨\frac{\partial \hat{A}}{\partial t}⟩$
4. Heisenberg equation of motion: $\frac{d\hat{A}}{dt}=\frac{i}{\hslash }[\hat{H},\hat{A}]$

	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	$|{\alpha }_S(t)⟩$	$|{\alpha }_H(t)⟩$	$|{\alpha }_I(t)⟩$
薛定谔矢量方程	$i\hslash \frac{d}{dt}|{\alpha }_S(t)⟩={\hat{H}}_S|{\alpha }_S(t)⟩$	None	$i\hslash \frac{d}{dt}|{\alpha }_I(t)⟩={\hat{V}}_I(t)|{\alpha }_I(t)⟩$
薛定谔算符方程	None	$i\hslash \frac{d}{dt}{\hat{A}}_H(t)=[{\hat{A}}_H(t),{\hat{H}}_S]$	$i\hslash \frac{d}{dt}{\hat{A}}_I(t)=[{\hat{A}}_I(t),{\hat{H}}_0]$

算符的对易

1. $[\hat{A},\hat{B}]=\hat{A}\hat{B}-\hat{B}\hat{A}$
2. $[\hat{A},\hat{B}{]}^{†}=[{\hat{B}}^{†},{\hat{A}}^{†}]$
3. $[\hat{A},\hat{B}+\hat{C}]=[\hat{A},\hat{B}]+[\hat{A},\hat{C}]$
4. $[\hat{A},\hat{B}\hat{C}]=[\hat{A},\hat{B}]\hat{C}+\hat{B}[\hat{A},\hat{C}]$
5. $[\hat{A},\hat{B}\hat{C}]+[\hat{B},\hat{C}\hat{A}]+[\hat{C},\hat{A}\hat{B}]=0$
6. $[\hat{x},{\hat{p}}_x]=i\hslash$
7. $[\hat{x},k({\hat{p}}_x)]=i\hslash k^{\prime }({\hat{p}}_x)$
8. $[\lambda (\hat{x}),{\hat{p}}_x]=i\hslash {\lambda }^{\prime }(\hat{x})$
9. Expotential operator: $e^{\hat{A}}=\sum _{n=0}^{\mathrm{\infty }}\frac{{\hat{A}}^n}{n!}$
   • If $\hat{A}$ is Hermitian, $e^{\hat{A}}=\sum _n{e}^{a_n}|n⟩⟨n|$
   • Further, $e^{f(\hat{A})}=\sum _n{e}^{f(a_n)}|n⟩⟨n|$
   • Baker-Hausdorff formula: $e^{\xi \hat{A}}\hat{B}{e}^{-\xi \hat{A}}=\hat{B}+\xi [\hat{A},\hat{B}]+\frac{{\xi }^2}{2!}[\hat{A},[\hat{A},\hat{B}]]+\cdots$
   • Baker-Campbell-Hausdorff formula: $e^{\hat{A}+\hat{B}}=e^{\hat{A}}{e}^{\hat{B}}{e}^{-\frac{1}{2}[\hat{A},\hat{B}]}$
10. Uncertainty principle: $\mathrm{\Delta }A\mathrm{\Delta }B\ge \frac{1}{2}|⟨[\hat{A},\hat{B}]⟩|$ the equality holds when $\hat{A}=ic\hat{B}$
    • $\mathrm{\Delta }x\mathrm{\Delta }{p}_x\ge \frac{\hslash }{2}$
    • $\mathrm{\Delta }E\mathrm{\Delta }t\ge \frac{\hslash }{2}$

混合态和密度矩阵

• Density matrix: $\hat{\rho }=\sum _i{p}_i|{\psi }_i⟩⟨{\psi }_i|$
• Density matrix Quality:
  • Trace: $Tr(\hat{\rho })=1$
  • Idempotent: $\{\begin{array}{ll}{\hat{\rho }}^2=\hat{\rho }& purestate\\ {\hat{\rho }}^2\ne \hat{\rho }& mixedstate\end{array}$
  • Trace of Square: $\{\begin{array}{ll}Tr({\hat{\rho }}^2)=1& purestate\\ Tr({\hat{\rho }}^2)<1& mixedstate\end{array}$
  • Hermitian: ${\hat{\rho }}^{†}=\hat{\rho }$
  • Positive: $⟨\psi |\hat{\rho }|\psi ⟩\ge 0$
• Motion of density matrix: $\frac{d\hat{\rho }}{dt}=\frac{i}{\hslash }[\hat{H},\hat{\rho }]+\frac{\partial \hat{\rho }}{\partial t}$
• Average value: $⟨\hat{A}⟩=Tr(\hat{\rho }\hat{A})$
• Reduced density matrix: ${\hat{\rho }}_A=T{r}_B({\hat{\rho }}_{AB})$
  • Average value of partial system: $⟨\hat{A}⟩=T{r}_A({\hat{\rho }}_A\hat{A})=T{r}_A(T{r}_B({\hat{\rho }}_{AB})\hat{A})$

谐振子和升降算符

1. Hamiltonian: $\hat{H}=\frac{{\hat{p}}^2}{2m}+\frac{1}{2}m{\omega }^2{\hat{x}}^2$
2. Eigenvalue equation: $\hat{H}|n⟩=E_n|n⟩=\hslash \omega (n+\frac{1}{2})|n⟩$
3. Creation and Annihilation operator: $\hat{a}=\frac{1}{\sqrt{2\hslash m\omega }}(i\hat{p}+m\omega \hat{x}),{\hat{a}}^{†}=\frac{1}{\sqrt{2\hslash m\omega }}(-i\hat{p}+m\omega \hat{x})$
4. Commutation relation: $[\hat{a},{\hat{a}}^{†}]=1$
5. Hamiltonian in terms of $\hat{a}$ and ${\hat{a}}^{†}$: $\hat{H}=\hslash \omega ({\hat{a}}^{†}\hat{a}+\frac{1}{2})=\hslash \omega (\hat{a}{\hat{a}}^{†}-\frac{1}{2})$
6. Quality of $\hat{a}$ and ${\hat{a}}^{†}$:
   • $\hat{a}|n⟩=\sqrt{n}|n-1⟩$
   • ${\hat{a}}^{†}|n⟩=\sqrt{n+1}|n+1⟩$
   • $\hat{x}=\sqrt{\frac{\hslash }{2m\omega }}(\hat{a}+{\hat{a}}^{†})$
   • $\hat{p}=i\sqrt{\frac{\hslash m\omega }{2}}(\hat{a}-{\hat{a}}^{†})$
7. Ground state: $\hat{a}|0⟩=0\Rightarrow |0⟩=(\frac{m\omega }{\pi \hslash }{)}^{\frac{1}{4}}{e}^{-\frac{m\omega }{2\hslash }{x}^2}$

粒子数表象

1. Particle number operator: $\hat{N}={\hat{a}}^{†}\hat{a},\hat{H}=\hslash \omega (\hat{N}+\frac{1}{2})$
2. Eigenvalue equation: $\hat{N}|n⟩=n|n⟩$
3. Amplitude Operator: $\hat{B}=\sqrt{\hat{N}+1}=\sum _n\sqrt{n+1}|n⟩⟨n|$
4. Phase Operator: $e^{i\hat{\mathrm{\Phi }}}=\sum _{n=0}|n⟩⟨n+1|,e^{-i\hat{\mathrm{\Phi }}}=\sum _{n=0}|n+1⟩⟨n|$
5. Quality:
   • $\hat{a}=\hat{B}{e}^{i\hat{\mathrm{\Phi }}}=\sum _n\sqrt{n+1}|n⟩⟨n+1|$
   • ${\hat{a}}^{†}=e^{-i\hat{\mathrm{\Phi }}}\hat{B}=\sum _n\sqrt{n+1}|n+1⟩⟨n|$
   • $[\hat{N},\cos \hat{\mathrm{\Phi }}]=-i\sin \hat{\mathrm{\Phi }}$
   • $[\hat{N},\sin \hat{\mathrm{\Phi }}]=i\cos \hat{\mathrm{\Phi }}$

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：

$$\begin{array}{rl}{C}_m(t_0)& =C_m(0)\exp [-{\int }_0^{t_0}\frac{i}{\hslash }{E}_m(t)dt]\exp [-{\int }_0^{t_0}⟨m(t)|\dot{m}(t)⟩dt]\\ & =C_m(0)\exp [-\frac{i}{\hslash }{\int }_0^{t_0}{E}_m(t)dt]\exp [i\gamma (t_0)]\end{array}$$

其中

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

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：

$$\gamma ({\lambda }_0)=i{\int }_0^{{\lambda }_0}⟨m(\lambda (t))|\overrightarrow{{\mathrm{\nabla }}_{\lambda }}|m(\lambda (t))⟩dt$$

其中

$$⟨m(\lambda (t))|\overrightarrow{{\mathrm{\nabla }}_{\lambda }}|m(\lambda (t))⟩$$

被称为Berry联络。

角动量理论

轨道角动量

1. Definition: $[{\hat{L}}_i,{\hat{L}}_j]=i\hslash {ϵ}_{ijk}{\hat{L}}_k$
2. Commutation relation:
   • Scalar Operator: $[{\hat{L}}_i,\hat{S}]=0$
   • Vector Operator: $[{\hat{L}}_i,{\hat{V}}_j]=i\hslash {ϵ}_{ijk}{\hat{V}}_k$
   • $[{\hat{L}}^2,{\hat{L}}_i]=0$
   • Hamiltonian: $[{\hat{L}}^2,\hat{H}]=[{\hat{L}}_z,\hat{H}]=0$
3. Eigenvalue equation: ${\hat{L}}^2|l,m⟩=l(l+1){\hslash }^2|l,m⟩,{\hat{L}}_z|l,m⟩=m\hslash |l,m⟩$
4. Creation and Annihilation operator: ${\hat{L}}_{\pm }={\hat{L}}_x\pm i{\hat{L}}_y$
   • $[{\hat{L}}_z,{\hat{L}}_{\pm }]=\pm \hslash {\hat{L}}_{\pm }$
   • ${\hat{L}}_{\pm }|l,m⟩=\hslash \sqrt{l(l+1)-m(m\pm 1)}|l,m\pm 1⟩$

自旋角动量

1. Definition: $[{\hat{S}}_i,{\hat{S}}_j]=i\hslash {ϵ}_{ijk}{\hat{S}}_k$
2. Commutation relation is as same as orbital angular momentum
3. Pauli matrices: ${\sigma }_x=\left(\begin{array}{cc}0& 1\\ 1& 0\end{array}\right),{\sigma }_y=\left(\begin{array}{cc}0& -i\\ i& 0\end{array}\right),{\sigma }_z=\left(\begin{array}{cc}1& 0\\ 0& -1\end{array}\right)$
   • ${\sigma }_x^2={\sigma }_y^2={\sigma }_z^2=\hat{I}$
   • ${\sigma }_x{\sigma }_y=i{\sigma }_z,{\sigma }_y{\sigma }_z=i{\sigma }_x,{\sigma }_z{\sigma }_x=i{\sigma }_y$
   • ${\sigma }_x{\sigma }_y{\sigma }_z=i\hat{I}$
   • Commutation relation: $[{\sigma }_i,{\sigma }_j]=2i{ϵ}_{ijk}{\sigma }_k$
   • Inverse Commutation relation: $\{{\sigma }_i,{\sigma }_j\}=2{\delta }_{ij}$
4. Eigenvalue Vector of Pauli matrices:
   • ${\sigma }_x=\{\begin{array}{lll}& 1& \frac{1}{\sqrt{2}}\left(\begin{array}{c}1\\ 1\end{array}\right)\\ & -1& \frac{1}{\sqrt{2}}\left(\begin{array}{c}1\text{-}1\end{array}\right)\end{array}$
   • ${\sigma }_y=\{\begin{array}{lll}& 1& \frac{1}{\sqrt{2}}\left(\begin{array}{c}1\\ i\end{array}\right)\\ & -1& \frac{1}{\sqrt{2}}\left(\begin{array}{c}1\text{-}i\end{array}\right)\end{array}$
   • ${\sigma }_z=\{\begin{array}{lll}& 1& \left(\begin{array}{c}1\\ 0\end{array}\right)\\ & -1& \left(\begin{array}{c}0\\ 1\end{array}\right)\end{array}$
   • ${\sigma }_n=\{\begin{array}{lll}& 1& \left(\begin{array}{c}\cos \frac{\theta }{2}\\ \sin \frac{\theta }{2}{e}^{i\varphi }\end{array}\right)\\ & -1& \left(\begin{array}{c}\sin \frac{\theta }{2}{e}^{i\varphi }\text{-}\cos \frac{\theta }{2}\end{array}\right)\end{array}$

对称性

微扰论

不含时微扰

1. Perturbation Hamiltonian: $\hat{H}={\hat{H}}_0+\lambda \hat{V}$
2. Eigenvalue equation: $\hat{H}(\lambda )|n{⟩}_{\lambda }=E_n(\lambda )|n{⟩}_{\lambda }$
3. Modification of Eigenvalue and Vector: $|n{⟩}_{\lambda }=\sum _{i=0}^{\mathrm{\infty }}{\lambda }^i|n^i⟩,E_n(\lambda )=\sum _{i=0}^{\mathrm{\infty }}{\lambda }^i{E}_n^i$
4. Correction Equation: $$\begin{array}{rl}{\lambda }^0& ({\hat{H}}^0-E_n^0)|n^0⟩=0\\ {\lambda }^1& ({\hat{H}}^0-E_n^0)|n^1⟩=(E_n^1-\delta \hat{H})|n^0⟩\\ \cdots \\ {\lambda }^k& ({\hat{H}}^0-E_n^0)|n^k⟩=(E_n^1-\delta \hat{H})|n^{k-1}⟩+\sum _{i=2}^k{E}_n^i|n^{k-i}⟩\end{array}$$
5. First Order Correction: $E_n^1=⟨n^0|\hat{V}|n^0⟩,|n^1⟩=\sum _{m\ne n}\frac{⟨m^0|\hat{V}|n^0⟩}{E_n^0-E_m^0}|m^0⟩$
6. Second Order Correction: $E_n^2=⟨n^0|\hat{V}|n^1⟩-⟨n^0|E_n^1|n^1⟩$
7. Degenrate Case: Only to find “Good” basis

含时微扰

1. Perturbation Hamiltonian: $\hat{H}={\hat{H}}_0+\lambda \hat{V}(t)$
2. Expansion: $|n(t)⟩=\sum _{i=0}^{\mathrm{\infty }}{c}_i(t)|i⟩$
3. Accurate Equation: $i\hslash \frac{d}{dt}{c}_n(t)=\sum _m{V}_{nm}{c}_m(t)e^{i(E_n-E_m)t/\hslash }$$$i\hslash \frac{d}{dt}\left(\begin{array}{c}{c}_1(t)\\ c_2(t)\\ ⋮\end{array}\right)=\left(\begin{array}{ccc}{V}_{11}& V_{12}{e}^{i{\omega }_{12}t}& \cdots \\ V_{21}{e}^{i{\omega }_{21}t}& V_{22}& \cdots \\ ⋮& ⋮& \ddots \end{array}\right)\left(\begin{array}{c}{c}_1(t)\\ c_2(t)\\ ⋮\end{array}\right)$$
4. Dyson series: $c_n(t)=c_n^{(0)}(t)+c_n^{(1)}(t)+c_n^{(2)}(t)+\cdots$
5. Approximate Equation:$$U_I(t,t_0)=\sum _{n=0}^{\mathrm{\infty }}(-\frac{i}{\hslash }{)}^n{\int }_{t_0}^t\cdots {\int }_{t_0}^{t_{n-1}}{\hat{V}}_I(t_1)\cdots {\hat{V}}_I(t_n)d{t}_1\cdots d{t}_n$$
6. Zero Order Approximation: $c_n^{(0)}(t)=⟨n|i⟩$, $i$ is the initial state
7. First Order Approximation: $c_n^{(1)}(t)=-\frac{i}{\hslash }{\int }_{t_0}^t⟨n|\hat{V}(t^{\prime })|i⟩d{t}^{\prime }$
8. Second Order Approximation: $c_n^{(2)}(t)=\frac{-1}{{\hslash }^2}\sum _m{\int }_{t_0}^t{\int }_{t_0}^{t^{\prime }}⟨n|{\hat{V}}_I(t^{\prime })|m⟩⟨m|{\hat{V}}_I(t^{″})|i⟩d{t}^{″}d{t}^{\prime }$

变分法

$$\frac{⟨\tilde{0}|\hat{H}|\tilde{0}⟩}{⟨\tilde{0}|\tilde{0}⟩}\ge E_0$$

散射理论

• Scatter Wave Function: $\psi (\overrightarrow{r})=e^{i\overrightarrow{k}\cdot \overrightarrow{r}}+f(\theta ,\varphi )\frac{e^{ikr}}{r}$, $f(\theta ,\varphi )$ is the scattering amplitude
• Differential Cross Section: $D(\theta ,\varphi )=|f(\theta ,\varphi ){|}^2$
• Green Function Method: $\psi (\overrightarrow{r})=\varphi (\overrightarrow{r})+{\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}{G}_0(k_i,\overrightarrow{r}-\overrightarrow{r^{\prime }})V(\overrightarrow{r^{\prime }})\psi (\overrightarrow{r^{\prime }})d\overrightarrow{r^{\prime }}$
• Green Function: $G_0(k_i,\overrightarrow{r}-\overrightarrow{r^{\prime }})=-\frac{m}{2\pi {\hslash }^2}\frac{1}{|\overrightarrow{r}-\overrightarrow{r^{\prime }}|}{e}^{\pm ik|\overrightarrow{r}-\overrightarrow{r^{\prime }}|}$
• Operator Form: $|{\psi }_p^{+}⟩=|k_i⟩+{\hat{G}}_0^{+}\hat{V}|{\psi }_p^{+}⟩$
  • $|{\psi }_p^{+}⟩=|k_i⟩+{\hat{G}}_0^{+}\hat{V}|k_i⟩+{\hat{G}}_0^{+}\hat{V}{\hat{G}}_0^{+}\hat{V}|k_i⟩+\cdots =\sum _n({\hat{G}}_0^{+}\hat{V}{)}^n|k_i⟩$
  • $f(\theta ,\varphi )=-\frac{4{\pi }^{2}m}{{\hslash }^2}⟨\overrightarrow{k_f}|V|{\psi }_p^{+}⟩$
  • ${\hat{T}}^{+}|k_i⟩=\hat{V}|{\psi }_p^{+}⟩$
  • ${\hat{T}}^{+}=\hat{V}+\hat{V}{\hat{G}}_0^{+}\hat{V}+\hat{V}{\hat{G}}_0^{+}\hat{V}{\hat{G}}_0^{+}\hat{V}+\cdots$
• Optical Theorem: $\frac{k}{4\pi }{\sigma }_{tot}=\mathrm{\Im }[f(\theta =0,\varphi =0)]$