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
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：
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}
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}
