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编辑1

tag-icon • tag-icon tikz-pgf pgfplots physics
编辑1

有几种方法可以绘制 Morse 势（参见这个问题）和其他智能解决方案，在抛物线上绘制量子谐振子波函数（参见另一个问题）。

到目前为止，我复制了莫尔斯势的解决方案，因为它看起来非常好（我更喜欢机器执行循环，而不是我手动处理 100 个波函数）：

 % ... %
\usepackage{tikz}
    \usetikzlibrary{intersections}
\usepackage{pgfplots}
    \usepgfplotslibrary{fillbetween}

% ... %

\begin{tikzpicture}
%%%%%%% Define Potential Function %%%%%%%
    \pgfmathsetmacro{\De}{6}
    \pgfmathsetmacro{\Ro}{1}
    \pgfmathsetmacro{\alpha}{1}
    \pgfmathdeclarefunction{V}{1}{%
        \pgfmathparse{%
            \De*((1-exp(-\alpha*(#1-\Ro)))^2-1)%
        }%
    }%
%%%%%%% Energy Levels %%%%%%%
    \pgfmathdeclarefunction{energy}{1}{%
        \pgfmathparse{%
            -\De+(#1+.5) - (#1+.5)^2/(4*\De)
        }%
    }%

    \begin{axis}[
            axis lines=none,
            smooth,
            no markers,
            domain=0:8,
            ymax=6,
            scale=1.5
        ]
        \addplot [red, samples=50, name path global=MorseCurve] {V(x)};
        \pgfplotsinvokeforeach{0,1,2,3,4,5,6, 7, 8, 9, 10, 11, 12}{
            \path [name path global=HelperLine-#1] (axis cs: 0,{energy(#1)}) -- (axis cs: 10, {energy(#1)});
            \draw[name intersections={of=MorseCurve and HelperLine-#1}] (intersection-1) -- (intersection-2);
        }
    \end{axis}
\end{tikzpicture}
 % ... %

因为我想代表弗兰克-康登原理我正在寻找一种将量子谐振子和莫尔斯势结合起来的方法，这样我就可以实现类似

在此处输入图片描述

我确实希望将所有内容都保留在类似 for 循环的结构中，但我知道使用 LaTeX 计算 Hermite 多项式是不可能的（不是吗？），因此也许用数字进行标记\pgfplotsinvokeforeach甚至会是一大进步。有什么想法吗？

编辑1

前 4 个功能是（根据要求）

H_1(x) = exp(-x^2/2)
H_2(x) = 2*x * exp(-x^2/2)
H_3(x) = (4*x^2-2) * exp(-x^2/2)
H_4(x) = (-12*x+8*x^3) * exp(-x^2/2)
...

多项式部分是 Hermite 多项式，而指数部分保持不变。这里我用 Mathematica 计算了它们，最高为 10：

在此处输入图片描述

答案1

至少对我来说，使用 pgfplots 来做到这一点相当困难，因为存在各种扩展问题。您只提供了 Hermite 多项式的前四个，并且没有对它们进行规范化。因此，我提出了一个 Ti钾Z“唯一”的解决方案，我在其中进行了规范化。但是，一旦您提供包括规范化在内的所有相关多项式，我将很乐意改进此解决方案。

\documentclass[tikz,border=3.14mm]{standalone}
\usetikzlibrary{intersections,calc}
\begin{document}
\begin{tikzpicture}[yscale=0.5]
\tikzset{declare function={H0(\x)=pi^(-1/4)*exp(-\x*\x); 
H1(\x)=(sqrt(2)*\x)/pi^(1/4)*exp(-\x*\x); 
H2(\x)=(-(1/(sqrt(2)*pi^(1/4))) +  (sqrt(2)*abs(\x^2))/pi^(1/4))*exp(-\x*\x); 
H3(\x)=(-((sqrt(3)*\x)/pi^(1/4)) + 
   (2*\x^3)/(sqrt(3)*pi^(1/4)))*exp(-\x*\x); 
H4(\x)=(sqrt(3)/(2*sqrt(2)*pi^(1/4)) - 
   (sqrt(6)*abs(\x^2))/pi^(1/4) + (sqrt(2/3)*abs(\x^4))/pi^(1/4))*exp(-\x*\x); 
H5(\x)=((sqrt(15)*\x)/(2*pi^(1/4)) - 
   (2*sqrt(5)*\x^3)/(sqrt(3)*pi^(1/4)) + 
   (2*\x^5)/(sqrt(15)*pi^(1/4)))*exp(-\x*\x); 
H6(\x)=(-sqrt(5)/(4*pi^(1/4)) + (3*sqrt(5)*abs(\x^2))/
    (2*pi^(1/4)) - (sqrt(5)*abs(\x^4))/pi^(1/4) + 
   (2*abs(\x^6))/(3*sqrt(5)*pi^(1/4)))*exp(-\x*\x); 
H7(\x)=(-(sqrt(35)*\x)/(2*sqrt(2)*pi^(1/4)) + 
   (sqrt(35/2)*\x^3)/pi^(1/4) - (sqrt(14)*\x^5)/
    (sqrt(5)*pi^(1/4)) + (2*sqrt(2/35)*\x^7)/(3*pi^(1/4)))*exp(-\x*\x); 
H8(\x)=(sqrt(35)/(8*sqrt(2)*pi^(1/4)) - 
   (sqrt(35)*abs(\x^2))/(sqrt(2)*pi^(1/4)) + 
   (sqrt(35/2)*abs(\x^4))/pi^(1/4) - (2*sqrt(14)*abs(\x^6))/
    (3*sqrt(5)*pi^(1/4)) + (sqrt(2/35)*abs(\x^8))/(3*pi^(1/4)))*exp(-\x*\x); 
H9(\x)=((3*sqrt(35)*\x)/(8*pi^(1/4)) - 
   (sqrt(35)*\x^3)/pi^(1/4) + (3*sqrt(7)*\x^5)/
    (sqrt(5)*pi^(1/4)) - (4*\x^7)/(sqrt(35)*pi^(1/4)) + 
   (2*\x^9)/(9*sqrt(35)*pi^(1/4)))*exp(-\x*\x); 
H10(\x)=((-3*sqrt(7))/(16*pi^(1/4)) + 
   (15*sqrt(7)*abs(\x^2))/(8*pi^(1/4)) - (5*sqrt(7)*abs(\x^4))/
    (2*pi^(1/4)) + (sqrt(7)*abs(\x^6))/pi^(1/4) - 
   abs(\x^8)/(sqrt(7)*pi^(1/4)) + (2*abs(\x^10))/(45*sqrt(7)*pi^(1/4)))*exp(-\x*\x);}
}

%%%%%%% Define Potential Function %%%%%%%
    \pgfmathsetmacro{\De}{6}
    \pgfmathsetmacro{\Ro}{1}
    \pgfmathsetmacro{\alpha}{1}
    \pgfmathdeclarefunction{V}{1}{%
        \pgfmathparse{%
            \De*((1-exp(-\alpha*(#1-\Ro)))^2-1)%
        }%
    }%
%%%%%%% Energy Levels %%%%%%%
    \pgfmathdeclarefunction{energy}{1}{%
        \pgfmathparse{%
            -\De+(#1+.5) - (#1+.5)^2/(4*\De)
        }%
    }%

        \draw[red, name path global=MorseCurve] 
        plot[samples=100,variable=\x,domain=0.25:8] 
        ({\x},{V(\x)});
        \foreach \X in {0,...,5}{
            \path [name path global=HelperLine-\X] (0,{energy(\X)}) -- 
            (10, {energy(\X)});
           \path[name intersections={of=MorseCurve and HelperLine-\X}] (intersection-1) -- (intersection-2);
            \draw let \p1=(intersection-1),\p2=(intersection-2) in 
            \pgfextra{\pgfmathsetmacro{\h}{\y1/1cm}\xdef\h{\h}
            \pgfmathsetmacro{\xleft}{(\x1-1cm)/1cm}\xdef\xleft{\xleft}
            \pgfmathsetmacro{\xright}{(\x2+1cm)/1cm}\xdef\xright{\xright}}
            (\xleft,\h) -- (\xright,\h);
            \pgfmathsetmacro{\xmid}{(\xleft+\xright)/2}
            \filldraw[fill=orange] plot[domain=\xleft:\xright,variable=\x,samples=100] 
            ({\x},{H\X(2.5*(\x-\xmid))/2+\h}) -- (\xright,\h) -- (\xleft,\h);
        }
\end{tikzpicture}
\end{document}

在此处输入图片描述

更新\x^2：修复了与 相比的问题\x*\x，使图居中并添加了所有规范化因子（使用 Mathematica）。不幸的是，由于“维度太大错误”，我无法绘制更高级别的函数……

答案2

我想这很接近您想要实现的目标。因为这里做了很多事情，请查看代码中的注释以找出哪些适合哪些。

（为了快速编译，我建议使用 LuaLaTeX。）

% used PGFPlots v1.16
\documentclass[border=5pt]{standalone}
\usepackage{pgfplots}
    \usepgfplotslibrary{fillbetween}
    \pgfplotsset{
        % use this `compat' level or higher to use Lua as calculation engine
        % when compiling/TeXing with LuaLaTeX
        compat=1.12,
    }
\begin{document}
\begin{tikzpicture}[
    % redeclared the variables/functions to a bit simpler form (as I think)
    /pgf/declare function={
        % Define Potential Function
        De = 6;
        Ro = 1;
        alpha = 1;
        V(\x) = De * ( (1 - exp(-alpha*(\x-Ro)))^2 - 1 );
        %
        % inverse function(s) of `V(x)'
        M(\V) = Ro - 1/alpha * ln( 1 + sqrt(\V/De +1) );
        N(\V) = Ro - 1/alpha * ln( 1 - sqrt(\V/De +1) );
%        xVmin = Ro;         % which is the same as: M(-De); or N(-De);
        %
        % Energy Levels
        energy(\z) = -De + (\z+0.5) - (\z+0.5)^2 / (4*De);
        %
        % how much should the horizontal lines be enlarged (in axis units)?
        xshift = 1;
        % ---------------------------------------------------------------------
        % number of samples that should be used to plot the function
        Samples(\n) = (1+\n)*5;
        % common factor for `Hn'
        EE(\x)  = exp(-\x^2/2);
        % (just "copied" from the image in the question)
        H0(\x)  = EE(\x);
        H1(\x)  = EE(\x) *   2*\x;
        H2(\x)  = EE(\x) * (-2 + 4*\x^2);
        H3(\x)  = EE(\x) * (-12*\x + 8*\x^3);
        H4(\x)  = EE(\x) * (+12 - 48*\x^2 + 16*\x^4);
        H5(\x)  = EE(\x) * (+120*\x - 160*\x^3 + 32*\x^5);
        H6(\x)  = EE(\x) * (-120 + 720*\x^2 - 480*\x^4 + 64*\x^6);
        H7(\x)  = EE(\x) * (-1680*\x + 3360*\x^3 - 1344*\x^5 + 128*\x^7);
        H8(\x)  = EE(\x) * (+1680 - 13440*\x^2 + 13440*\x^4 - 3584*\x^6 + 256*\x^8);
        H9(\x)  = EE(\x) * (+30240*\x - 80640*\x^3 + 48384*\x^5 - 9216*\x^7 + 512*\x^9);
        H10(\x) = EE(\x) * (-30240 + 302400*\x^2 - 403200*\x^4 + 161280*\x^6 - 23040*\x^8 + 1024*\x^10);
        %
        % scaling factors to "normalize" the `Hn' functions to around 1 (= max(Hn) - min(Hn))
        % ("guessed" by manual inspection of the resulting graph/plot)
        G0  = 1;
        G1  = 2.4;
        G2  = 4.3;
        G3  = 4.6+5.4;
        G4  = 12.5+15;
        G5  = 38+46;
        G6  = 130+160;
        G7  = 480+600;
        G8  = 1900+2300;
        G9  = (0.8+0.95)*1e4;
        G10 = (3.4+4.4)*1e4;
        % additional scaling factor to further "shrink" the height
        F  = 1.5;
    },
]
    \begin{axis}[
        smooth,
        domain=0:8,
        axis lines=none,
        ymax=6,
%        % ---------------------------------------------------------------------
%        % for debugging purposes only
%        % (comment the previous two lines and uncomment the following lines to
%        %  determine `Gn')
%        minor y tick num=4,
%        xminorgrids,
%        % ---------------------------------------------------------------------
    ]
        % adapted `samples' so one of the sample points is near `xVmin'
        \addplot [red,samples=49] {V(x)};

        \pgfplotsinvokeforeach {0,...,10} {
            % draw the horizontal lines at the different energy levels
            % enlarge the lines by `xshift'
            \addplot [very thin,name path=h#1] coordinates {
                ({M(energy(#1)) - xshift},{energy(#1)})
                ({N(energy(#1)) + xshift},{energy(#1)})
            }
                % add the nodes to the right of the lines
                node (m#1) [
                    node font=\footnotesize,
                    at end,
                    right,
                    xshift=6ex,
                ] {$#1$}
            ;
        }

        \pgfplotsinvokeforeach {0,...,6} {
            % add the "wave" lines
            \addplot [
                name path=H#1,
                very thin,
                % adapt the `domain' to consider the shift in `Ro' and `xshift'
                % for the x values
                domain={-Ro + M(energy(#1)) - xshift}:{-Ro + N(energy(#1)) + xshift},
                % use a dynamic sample rate depending on the `Hn'
                samples={Samples(#1)},
            ] (
                % shift `x' by `Ro'
                {x + Ro},
                % scale `Hn' by `Gn' and `F', and shift the result by `energy(n)'
                {H#1(x)/G#1/F + energy(#1)}
            );

            % color the area between `hn' and `Hn'
            \addplot [
                fill=orange,
            ] fill between [
                of=h#1 and H#1,
            ];
        }

        % add the missing label part of the `m0'
        \node [
            node font=\footnotesize,
            anchor=base east,
        ] at (m0.base east) {$v' = \phantom{0}$};

    \end{axis}
\end{tikzpicture}
\end{document}

该图显示了上述代码的结果

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