简洁地列出几个数学恒等式

简洁地列出几个数学恒等式

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你好,

我想紧凑地列出几个数学恒等式。目前,我使用 enumerate 环境与 align 结合使用,但这占用了太多空间。首先,枚举符号(我也想将其更改为 i、ii、iii、iv 等)后的换行符似乎没有必要,只会不必要地增加大小。我还想在

a   b
  c

但是,如果不对枚举符号进行硬编码,我就无法做到这一点。

梅威瑟:

\documentclass{scrartcl}

\usepackage{amsmath,amssymb} 
\usepackage{physics}
\usepackage{braket} 

\usepackage[ngerman]{babel} 
\usepackage[T1]{fontenc} 
\usepackage[utf8]{inputenc} 
\usepackage{lmodern} 

\newcommand{\an}{\hat{a}} 
\newcommand{\cre}{\hat{a}^\dagger} 

\begin{document}
\begin{enumerate}
    \item   \begin{align}
                \comm{\an}{\cre} \ket{n}    &= \an \cre \ket{n} - \cre \an \ket{n} \nonumber \\ 
                                                    &= \left( n + 1 \right) \ket{n} - n \ket{n} \nonumber \\ 
                                                    &= \ket{n} \nonumber \\ 
                \Rightarrow \comm{\an}{\cre}= 1
            \end{align}
    \item   \begin{align}
                \comm{\hat{N}}{\an} \ket{n}         &= \hat{N} \an \ket{n} - \an \hat{N} \ket{N} \nonumber \\ 
                                                    &= \left( n - 1 \right) \sqrt{n} \ket{n-1} - n \sqrt{n} \ket{n-1} \nonumber \\ 
                                                    &= - \sqrt{n} \ket{n-1} \nonumber \\ 
                                                    &= - \an \ket{n} \nonumber \\ 
                \Rightarrow \comm{\hat{N}}{\an}     &= - \an
            \end{align}
    \item   \begin{align}
                \comm{\hat{N}}{\cre} \ket{n}        &= \hat{N} \cre \ket{n} - \cre \hat{N} \ket{n} \nonumber \\ 
                                                    &= \left( n+1 \right) \sqrt{n+1} \ket{n+1} - n \sqrt{n+1} \ket{n+1} \nonumber \\ 
                                                    &= \sqrt{n+1} \ket{n+1} \nonumber \\ 
                                                    & \cre \ket{n} \nonumber \\ 
                \Rightarrow \comm{\hat{N}}{\cre}    &= \cre 
            \end{align}
\end{enumerate}
\end{document}

编辑:

通过使用对齐的环境,我得到了这个:

使用对齐环境更新示例

\begin{enumerate}
    \item   $\begin{aligned}[t]
                \comm{\an}{\cre} \ket{n}        &= \an \cre \ket{n} - \cre \an \ket{n} \\ 
                                                &= \left( n + 1 \right) \ket{n} - n \ket{n} \\ 
                                                &= \ket{n} \\ 
                \Rightarrow \comm{\an}{\cre}    &= 1
            \end{aligned}$ \label{one}
    \item   $\begin{aligned}[t] 
                \comm{\hat{N}}{\an} \ket{n}     &= \hat{N} \an \ket{n} - \an \hat{N} \ket{N}  \\ 
                                                &= \left( n - 1 \right) \sqrt{n} \ket{n-1} - n \sqrt{n} \ket{n-1} \\ 
                                                &= - \sqrt{n} \ket{n-1} \\ 
                                                &= - \an \ket{n} \\ 
                \Rightarrow \comm{\hat{N}}{\an} &= - \an
            \end{aligned}$
    \item   $\begin{aligned}[t]
                \comm{\hat{N}}{\cre} \ket{n}        &= \hat{N} \cre \ket{n} - \cre \hat{N} \ket{n} \\ 
                                                    &= \left( n+1 \right) \sqrt{n+1} \ket{n+1} - n \sqrt{n+1} \ket{n+1}\\ 
                                                    &= \sqrt{n+1} \ket{n+1} \\ 
                                                    & \cre \ket{n} \\ 
                \Rightarrow \comm{\hat{N}}{\cre}    &= \cre 
            \end{aligned}$
\end{enumerate}

答案1

我建议用包来解决tasks

t正如 Zarko 所建议的,通过环境选项aligned,您还可以在顶部进行编号。

\documentclass{scrartcl}

\usepackage{amsmath,amssymb} 
\usepackage{physics}
\usepackage{braket} 

\usepackage[ngerman]{babel} 
\usepackage[T1]{fontenc} 
\usepackage[utf8]{inputenc} 
\usepackage{lmodern} 

\newcommand{\an}{\hat{a}} 
\newcommand{\cre}{\hat{a}^\dagger} 


\usepackage{tasks}
\settasks{
    counter-format={tsk[r])},% lowcase roman number
    label-offset={0pt},% distance between label and item
    label-width={1.1em},%width of the item label
    item-indent={1.1em},% horizontal space available for both label and label-offset
    column-sep={0em},
    %debug=true
}
%\usepackage{showframe}

\begin{document}
    The \verb|tasks| package
    creates lists in columns counting horizontally rather than vertically.
    Here you can find the first \ref{eq:first} and the second \ref{eq:second} equations side by side, and the third \ref{eq:third} centered 
    with a \verb|minipage|. 
\begin{center}
    \footnotesize
    %\scriptsize% otherwise you go out of the page margins
\begin{tasks}(2)
    \task\label{eq:first}
    $\begin{aligned}
    \comm{\an}{\cre} \ket{n}    &= \an \cre \ket{n} - \cre \an \ket{n} \nonumber \\ 
    &= \left( n + 1 \right) \ket{n} - n \ket{n} \nonumber \\ 
    &= \ket{n} \nonumber \\ 
    \Rightarrow \comm{\an}{\cre} &= 1
    \end{aligned}$ 
    \task\label{eq:second}  
    $\begin{aligned}
        \comm{\hat{N}}{\an} \ket{n}         &= \hat{N} \an \ket{n} - \an \hat{N} \ket{N} \nonumber \\ 
        &= \left( n - 1 \right) \sqrt{n} \ket{n-1} - n \sqrt{n} \ket{n-1} \nonumber \\ 
        &= - \sqrt{n} \ket{n-1} \nonumber \\ 
        &= - \an \ket{n} \nonumber \\ 
        \Rightarrow \comm{\hat{N}}{\an}     &= - \an
    \end{aligned}$  
\end{tasks}
\begin{minipage}{.56\linewidth}%.56 to avoid overfull hbox
\begin{tasks}[resume=true]% resume=true to continue the enumeration from the previous task 
    \task\label{eq:third}
        $\begin{aligned}  
            \comm{\hat{N}}{\cre} \ket{n}        &= \hat{N} \cre \ket{n} - \cre \hat{N} \ket{n} \nonumber \\ 
            &= \left( n+1 \right) \sqrt{n+1} \ket{n+1} - n \sqrt{n+1} \ket{n+1} \nonumber \\ 
            &= \sqrt{n+1} \ket{n+1} \nonumber \\ 
            & \cre \ket{n} \nonumber \\ 
            \Rightarrow \comm{\hat{N}}{\cre}    &= \cre 
    \end{aligned}$
\end{tasks}
\end{minipage}
\end{center}


    Or with option \verb|t|, as Zarko suggested, with \ref{eq:zfirst}, \ref{eq:zsecond} and the third \ref{eq:zthird} at the top: 
\begin{center}
    \footnotesize
    %\scriptsize% otherwise you go out of the page margins
\begin{tasks}(2)
    \task\label{eq:zfirst}
    $\begin{aligned}[t]
    \comm{\an}{\cre} \ket{n}    &= \an \cre \ket{n} - \cre \an \ket{n} \nonumber \\ 
    &= \left( n + 1 \right) \ket{n} - n \ket{n} \nonumber \\ 
    &= \ket{n} \nonumber \\ 
    \Rightarrow \comm{\an}{\cre} &= 1
    \end{aligned}$ 
    \task\label{eq:zsecond}  
    $\begin{aligned}[t]
        \comm{\hat{N}}{\an} \ket{n}         &= \hat{N} \an \ket{n} - \an \hat{N} \ket{N} \nonumber \\ 
        &= \left( n - 1 \right) \sqrt{n} \ket{n-1} - n \sqrt{n} \ket{n-1} \nonumber \\ 
        &= - \sqrt{n} \ket{n-1} \nonumber \\ 
        &= - \an \ket{n} \nonumber \\ 
        \Rightarrow \comm{\hat{N}}{\an}     &= - \an
    \end{aligned}$  
\end{tasks}
\begin{minipage}{.56\linewidth}%.56 to avoid overfull hbox
\begin{tasks}[resume=true]% resume=true to continue the enumeration from the previous task 
    \task\label{eq:zthird}
        $\begin{aligned}[t]  
            \comm{\hat{N}}{\cre} \ket{n}        &= \hat{N} \cre \ket{n} - \cre \hat{N} \ket{n} \nonumber \\ 
            &= \left( n+1 \right) \sqrt{n+1} \ket{n+1} - n \sqrt{n+1} \ket{n+1} \nonumber \\ 
            &= \sqrt{n+1} \ket{n+1} \nonumber \\ 
            & \cre \ket{n} \nonumber \\ 
            \Rightarrow \comm{\hat{N}}{\cre}    &= \cre 
    \end{aligned}$
\end{tasks}
\end{minipage}
\end{center}
\end{document}

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