%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 新命令定义
% 命令
%======== quantum mechanics ===========% \Yang
% use \overline instead of \Bar
%==============================%
\renewcommand{\bra}[1]{{\left< #1 \right|}}
\renewcommand{\ket}[1]{{\left| #1 \right>}}
\renewcommand{\braket}[2]{{\left< #1 \middle| #2 \right>}}
\newcommand{\bracket}[1]{\left(#1\right)}
\newcommand{\abracket}[1]{\left\langle#1\right\rangle}
%\newcommand{\bp}{{\bra\psi}}
\newcommand{\kp}{{\ket\psi}}
\newcommand{\h}[1]{{\hat{#1}}}
\newcommand{\V}{\mathbb{V}}
%\newcommand{\hH}{{\hat H}}
% use \HH and \hH
\newcommand{\KK}{{K}}
\newcommand{\NN}{{N}}
\newcommand{\rk}{\operatorname{rank}}
\renewcommand{\O}{\mathcal{O}}
\renewcommand{\P}{{P}}
\newcommand{\op}{\operatorname}
%\newcommand{\formal}{{[\![\hbar]\!]}}
\newcommand{\cl}{{_{cl}}}
\newcommand{\bp}{(\xp)}
\newcommand{\abs}[1]{{ \left| #1 \right|}}
\newcommand{\im}{\op{im}}
\newcommand{\intx}{{\int d{x}\,}}
\newcommand{\intk}{{\int d{k}\,}}
%==============================%
%\renewcommand{\epsilon}{\varepsilon}
\newcommand{\qw}[1]{\subsection{#1.}\indent}
\newcommand{\A}{\mathbb{A} }
\newcommand{\cA}{\mathcal A}
% \cA : 1-form on P, A: potential 1-form on R^{3,1}
\renewcommand{\R}{\mathbb{R} }
\newcommand{\C}{\mathbb{C} }
\renewcommand{\Z}{\mathbb{Z} }
\newcommand{\Q}{\mathbb{Q} }
\newcommand{\U}{\mathcal{U} }
\newcommand{\g}{{\mathfrak{g}}}
\newcommand{\LL}{\mathscr{L} }
\newcommand{\Ua}{{\U_\alpha }}
\newcommand{\Ub}{{\U_\beta }}
\newcommand{\Uc}{{\U_\c }}
\newcommand{\Uab}{{\U_{\alpha\beta} }}
\newcommand{\Uabc}{{\U_{\alpha\beta\c} }}
\newcommand{\s}{\sigma}
\newcommand{\sa}{{\sigma_\a }}
\newcommand{\sb}{{\sigma_\b }}
\newcommand{\pa}{{\varphi_\alpha }}
\newcommand{\pai}{{\varphi_\alpha\inv }}
\newcommand{\pb}{{\varphi_\beta }}
\newcommand{\pc}{{\varphi_\c }}
\newcommand{\pbi}{{\varphi_\beta\inv }}
\newcommand{\pab}{{\varphi_{\alpha\beta} }}
\newcommand{\pabi}{{\varphi_{\alpha\beta}\inv }}
\newcommand{\pac}{{\varphi_{\a\c} }}
\newcommand{\pcb}{{\varphi_{\c\b} }}
\newcommand{\pba}{{\varphi_{\beta\alpha} }}
\newcommand{\pbai}{{\varphi_{\beta\alpha}\inv }}
\newcommand{\pf}{\textbf{证明}}
\newcommand{\E}{{\mathbb E}} % 1-form
\newcommand{\bE}{{\mathbf E}}
\newcommand{\cE}{{\mathcal E}}
\newcommand{\B}{\mathbb B} % 1-form
\newcommand{\bA}{{\mathbf A}}
\newcommand{\bB}{{\mathbf B}}
\renewcommand{\j}{{\bf{j}}}
\newcommand{\bj}{\mathbbm j} % 1-form
\newcommand{\vE}{{\v\bE}}
\newcommand{\vA}{{\v\bA}}
\newcommand{\vB}{{\v\bB}}
\newcommand{\vj}{{\v\j}}
\newcommand{\vk}{{\v k}}
%\newcommand{\Ab}{\mathbb A}
\renewcommand{\L}{\mathscr{L} }
\renewcommand{\H}[1]{{H_{\text{dR}}^{#1}}}
\newcommand{\K}{\mathscr{K} }
\newcommand{\HH}{\mathscr{H} }
\newcommand{\ut}[1]{$\underline{\text{#1}}$}
\newcommand{\t}[1]{{\text{#1}}}
%\newcommand{\intt}[1]{\int _{t_0}^{t_1} #1 \,dt}
%\newcommand{\inte}{[t_0,t_1]}
\newcommand{\ddt}[1]{\frac {d #1}{dt}}
\newcommand{\pp}[2]{\frac{\partial #1}{\partial #2}} %\pp{a}{b} not \pp ab
\newcommand{\ppt}[1]{\frac{\partial #1}{\partial t}}
\newcommand{\pppp}[4]{\pp{#1}{#3}\pp{#2}{#4}-\pp{#1}{#4}\pp{#2}{#3}}
\newcommand{\ppo}[2]{\LR{\pp{P}{#2}\pp{#1}{P}+\pp{Q}{#2}\pp{#1}{Q}}}
\newcommand{\LR}[1]{\left ( #1 \right )}
\newcommand{\Lr}[1]{\left [ #1 \right ]}
\newcommand{\lr}[1]{\left\{ #1\right \}}
\newcommand{\lR}[1]{\left< #1\right >}
\newcommand{\set}[2]{{\left\{ #1 \, \middle | \, #2 \right\}}} %{{\left\{ \left. #1 \,\right|\, #2 \right\}}} % {...|...}
\newcommand{\ali}[1]{$$\begin{aligned} #1 \end{aligned}$$}
\newcommand{\align}[1]{\begin{align*} #1 \end{align*}}
\newcommand{\a}{\alpha}
\newcommand{\b}{\beta}
\renewcommand{\c}{\gamma}
\newcommand{\Lv}{\LL_V}
\newcommand{\lv}{\iota_V}
\newcommand{\vh}{{{\tilde V}_{\HH}}}
\newcommand{\oh}{{\omega_\HH}}
\newcommand{\p}{\varphi}
\newcommand{\e}{\varepsilon }
\newcommand{\nt}[1]{{#1}_1,\cdots,{#1}_n}
\newcommand{\mt}[1]{{#1}_1,\cdots,{#1}_m}
\newcommand{\kt}[1]{{#1}_1,\cdots,{#1}_k}
\newcommand{\dt}[1]{{#1}_1,\cdots,{#1}_d}
%\renewcommand{\Rn}{\R^n}
\newcommand{\Rn}{\R^n}
\newcommand{\inv}{^{-1}}
\newcommand{\dij}{\delta_{ij}}
\newcommand{\sumi}{{\sum_i}}
\newcommand{\sumin}{{\sum_{i=1}^n}}
\newcommand{\w}{\wedge}
\renewcommand{\l}{\lambda}
\newcommand{\m}{\mu}
%\newcommand{\tp}{\tilde{p}}
\newcommand{\tq}{\tilde{q}}
\newcommand{\HJ}{Hamilton-Jacobi }
\newcommand{\Tl}{Toda lattice }
\newcommand{\He}{Hamilton's equation }
\newcommand{\vv}{{\vec v} }
\newcommand{\vw}{{\vec w} }
\newcommand{\sm}{symplectomorphism}
\newcommand{\cinf}{C^\infty}
\renewcommand{\v}{\vec }
\newcommand{\dimc}{\dim_{\mathbb C}}
\newcommand{\n}{\notag}
\newcommand{\lra}{ \longrightarrow}
\newcommand{\lmt}{ \longmapsto}
\newcommand{\zeroform}[1]{{#1_x +#1_y +#1_z}}
\newcommand{\zero}[1]{{#1_x ,#1_y ,#1_z}}
\newcommand{\oneform}[1]{{#1_x dx+#1_y dy+#1_zdz}}
\newcommand{\twoform}[1]{{#1_x dy\wedge dz+#1_y dz\wedge dx+#1_z dx\wedge dy}}
\newcommand{\dvol}{{dx\w dy \w dz}}
\newcommand{\Ms}{\R^{3,1}}
\newcommand{\rr}{{\v r-{{\v r}\,}'}}
\newcommand{\rra}{{\v r-{{\v r}_{\alpha}\!}'}}
\newcommand{\rrr}{{\v r_1-\v r_2}}
%\newcommand{\iff}{\Longleftrightarrow }
\newcommand{\rp}{{{{\v r}\,}'}}
\newcommand{\vr}{{{\v r}\,}}
\newcommand{\ds}{{d^*_3}}
\newcommand{\st}{{*_3}}
\newcommand{\fp}{{4\pi\e_0}}
\newcommand{\tc}[1]{{\raisebox{.5pt}{\textcircled{\raisebox{-.9pt}{#1}}}}}
%\renewcommand{\G}{{G(\v r,\v r\,') }}
\newcommand{\GD}{{G_D(\v r,\v r\,') }}
\newcommand{\GN}{{G_N(\v r,\v r\,') }}
\newcommand{\ns}{{\nabla^2{}' }}
\newcommand{\pn}{{\partial_{\hat n}}}
\newcommand{\hn}{{\hat n}}
\newcommand{\hr}{{\hat{ r}}}
\newcommand{\hrp}{{\hat{r}\,'}}
\newcommand{\hS}{{\widehat S}}
\newcommand{\vP}{{\v P}}
\newcommand{\Eext}{{\vE_{\text{ext}}}}
\newcommand{\Etot}{{\vE_{\text{tot}}}}
\newcommand{\Eint}{{\vE_{\text{int}}}}
\newcommand{\rf}{{\rho_{\text{free}}}}
\newcommand{\rb}{{\rho_{\text{bound}}}}
\newcommand{\vD}{{\v D}}
\newcommand{\xx}{\mathbf{x}}
\newcommand{\xy}{\mathbf{y}}
\newcommand{\xp}{\mathbf{p}}
\newcommand{\xk}{\mathbf{k}}
\newcommand{\xxi}{\boldsymbol{\xi}}
\newcommand{\xa}{\mathbf{a}}
\newcommand{\xb}{\mathbf{b}}
\newcommand{\bx}{(\xx)}
\newcommand{\bxt}{(\xx,t)}
\newcommand{\br}{{(\v r\,)}}
\newcommand{\brp}{{(\rp)}}
\newcommand{\bt}{{(t)}}
\newcommand{\btp}{{(t')}}
\newcommand{\mf}{{\frac{\mu_0}{4\pi}}}
\newcommand{\mb}[1]{\mathbf{#1}}
\newcommand{\mfr}{{\frac{\mu_0}{4\pi r}}}
\newcommand{\of}{{\frac{1}{4\pi\e_0}}}
\newcommand{\intV}{{\int_V d^3{r}\,}}
\newcommand{\intVp}{{\int_V d^3{r}'\,}}
\newcommand{\intr}{{\int d^3{r}\,}}
%\newcommand{\intk}{{\int d^3{k}\,}}
\newcommand{\intp}{{\int d^3{r}'\,}}
\newcommand{\intt}{{\int_{t_1}^{t_2}dt\,}}
\newcommand{\inttp}{{\int_{t_1}^{t_2}dt'\,}}
\newcommand{\intpV}{{\int_{\partial V}d\v \sigma\,}}
\newcommand{\dl}{{d\v l}}
\newcommand{\dr}{{d\v r}}
\newcommand{\ddr}{d^3r}
\newcommand{\dS}{{d\v \sigma}}
\newcommand{\Fe}[1]{{{\v F_{#1}}^e}}
\newcommand{\Fm}[1]{{{\v F_{#1}}^m}}
%\newcommand{\rt}{{\frac{\rr}{|\rr|^3}}}
\newcommand{\he}{{\hat e}}
\newcommand{\hk}{{\hat k}}
\newcommand{\ha}{{\hat a}}
\newcommand{\hb}{{\hat b}}
\newcommand{\hc}{{\hat c}}
\newcommand{\hx}{{\hat x}}
\newcommand{\hp}{{\hat p}}
\newcommand{\hy}{{\hat y}}
\newcommand{\hz}{{\hat z}}
\newcommand{\hth}{{\hat \theta}}
\newcommand{\cc}{{\text{c.c.} }}
\newcommand{\GG}{{G(\v r,t;\rp,t')}}
\newcommand{\rt}{{(\v r,t)}}
\newcommand{\rpt}{{(\rp,t)}}
\newcommand{\rtp}{{(\rp,t')}}
\newcommand{\tp}{{(t-|\rr|/c)}}
\newcommand{\ret}{_{\text{ret}}}
\newcommand{\ei}[1]{{e^{i #1}}}
\newcommand{\Yang}{{}}
\newcommand{\blue}[1]{{\color{blue}{#1}}}
\newcommand{\red}[1]{{\color{red}{#1}}}
\newcommand{\green}[1]{{\color{green}{#1}}}
\newcommand{\bl}{\blue}
\newcommand{\0}{\mathbf{0}}
\newcommand{\diag}{\operatorname{diag}}
%==============================%
%==============================%
% 私货，定义了一些小命令 %
%------------------------------%
\newcommand{\Area}{\operatorname{Area}}
\newcommand{\Arg}{\operatorname{Arg}}
\newcommand{\Aut}{\operatorname{Aut}}
\newcommand{\Conn}{\operatorname{Conn}}
\newcommand{\dd}{\operatorname{d\!}}
\newcommand{\Diff}{\operatorname{Diff}}
\newcommand{\Dom}{\operatorname{Dom}}
\newcommand{\dR}{\operatorname{dR}}
\newcommand{\End}{\operatorname{End}}
\newcommand{\GL}{\operatorname{GL}}
\newcommand{\gl}{\operatorname{gl}}
\newcommand{\Hol}{\operatorname{Hol}}
\newcommand{\img}{\operatorname{Im}}
\newcommand{\Map}{\operatorname{Map}}
\newcommand{\Obs}{\operatorname{Obs}}
\newcommand{\Op}{\operatorname{Op}}
\newcommand{\ord}{\operatorname{ord}}
\newcommand{\ran}{\operatorname{ran}}
\newcommand{\rank}{\operatorname{rank}}
\newcommand{\Res}{\operatorname{Res}}
\newcommand{\sign}{\operatorname{sign}}
\newcommand{\so}{\operatorname{so}}
\newcommand{\SO}{\operatorname{SO}}
\newcommand{\Span}{\operatorname{Span}}
\newcommand{\Spec}{\operatorname{spec}}
\newcommand{\Sym}{\operatorname{Sym}}
\newcommand{\Tr}{\operatorname{Tr}}
\newcommand{\Vect}{\operatorname{Vect}}
\newcommand{\Vol}{\operatorname{Vol}}
\renewcommand{\Re}{\operatorname{Re}}
\renewcommand{\Im}{\operatorname{Im}}
\newcommand{\Ai}{\operatorname{Ai}}
\newcommand{\Bi}{\operatorname{Bi}}
\newcommand{\OO}{\operatorname{O}}
%\newcommand{\SO}{\operatorname{SO}}
\newcommand{\UU}{\operatorname{U}}
\newcommand{\SU}{\operatorname{SU}}
\newcommand{\card}{\texttt{\#}}
\newcommand{\ie}{\emph{i.e.}}
\newcommand{\eps}{\varepsilon}
\newcommand{\vphi}{\varphi}
\newcommand{\vp}{\varphi}
\newcommand{\vthe}{\vartheta}
\newcommand{\II}{I\!I}
\renewcommand{\emptyset}{\varnothing}
\theoremstyle{plain}
\newtheorem{thm}{定理}
\newtheorem*{thm*}{定理}
\newtheorem{coro}{推论}
\newtheorem{lemma}{引理}
\newtheorem{proposition}{命题}
\newtheorem{conj}{猜想}
\newtheorem{ques}{问题}
\theoremstyle{definition}
\newtheorem{defn}{定义}
\newtheorem{const}{构造}
\newtheorem{notation}{记号}
\theoremstyle{remark}
\newtheorem*{app}{应用}
\newtheorem{example}[thm]{例子}
\newtheorem*{exer}{练习}
\newtheorem*{remark}{注记}
\newtheorem*{note}{附注}
\newcommand{\noindent}{}
\pdef\pf:{\textbf{证明:}}
\nosectionnumbers