Package: hermite-function
Hermite Function SeriesA Hermite function series package.fromhermitefunctionimportHermiteFunctionimportnumpyasnpimportmatplotlib.pyplotaspltx=np.linspace(-4,+4,1000)forninrange(5):f=HermiteFunction(n)plt.plot(x,f(x),label=f'$h_{n}$')plt.legend(loc='lower right')plt.show()Installationpip install git+https://github.com/goessl/vector.git
pip install hermite-functionUsageThis package provides a single class,HermiteFunction, to handle Hermite function series.HermiteFunctionextendsVectorfrom the [https://github.com/goessl/vector](vector module) and therefore provides the same functionality.A series can be initialized in three ways:With the constructorHermiteFunction(coef), that takes a non-negative integer to create a pure Hermite function with the given index, or an iterable of coefficients to create a Hermite function series.With the random factoryHermiteFunction.random(deg)for a random Hermite series of a given degree.By fitting data withHermiteFunction.fit(x, y, deg).
The objects are immutable (coefficients are internally stored in a tuple).f=HermiteFunction((1,2,3))g=HermiteFunction.random(15)h=HermiteFunction.fit(x,g(x),10)plt.plot(x,f(x),label='$f$')plt.plot(x,g(x),'--',label='$g$')plt.plot(x,h(x),':',label='$h$')plt.legend()plt.show()Container and sequence interfaces are implemented so the coefficients can beaccessed by indexing:f[2](coefficients not set return to 0),iterated over:for c in f(stops at last set coefficient),counted:len(f)(number of set coefficients),compared:f == g(tuple of coefficients get compared),shifted:f >> 1, f << 2&trimmed:f.trim()(trailing non-zero coefficients get removed).Methods for functions:evaluation withf(x),differentiation to an arbitrary degreef.der(n),integrationf.antider(),Fourier transformationf.fourier()&getting the degree of the seriesf.degare implemented.f_p=f.der()f_pp=f.der(2)plt.plot(x,f(x),label=rf"$f \ (\deg f={f.deg})$")plt.plot(x,f_p(x),'--',label=rf"$f' \ (\deg f'={f_p.deg})$")plt.plot(x,f_pp(x),':',label=rf"$f'' \ (\deg f''={f_pp.deg})$")plt.legend()plt.show()Hilbert space operations are also provided, where the Hermite functions are used as an orthonormal basis of the $L_\mathbb{R}^2$ space:Vector addition & subtractionf + g, f - g,scalar multiplication & division2 * f, f / 2,inner product & normf @ g, abs(f).g=HermiteFunction(4)h=f+gi=0.5*fplt.plot(x,f(x),label='$f$')plt.plot(x,g(x),'--',label='$g$')plt.plot(x,h(x),':',label='$h$')plt.plot(x,i(x),'-.',label='$i$')plt.legend()plt.show()Because this package was intended as a tool to work with quantum mechanical wavefunctions, the expectation value for the kinetic energy is also implemented ($\langle\hat{P}^2\rangle=\frac{1}{2}\int_\mathbb{R}f^*(x)f''(x)dx$, natural units):f.kinProofsIn the following let$$
f=\sum_{k=0}^\infty f_k h_k, \ g=\sum_{k=0}^\infty g_k h_k.
$$where $h_k$ are the Hermite functions, defined by the Hermite polynomials $H_k$:$$
h_k(x) = \frac{e^{-\frac{x^2}{2}}}{\sqrt{2^kk!\sqrt{\pi}}} H_k(x)
$$fromWikipedia - Hermite functions.Differentiation$$
\begin{aligned}
f' &= \sum_k f_k h_k' \
&\qquad\mid h'_k = \sqrt{\frac{k}{2}}h_{k-1} - \sqrt{\frac{k+1}{2}}h_{k+1} \
&= \sum_k f_k \left( \sqrt{\frac{k}{2}}h_{k-1} - \sqrt{\frac{k+1}{2}}h_{k+1} \right) \
&= \sum_{k=0}^\infty f_k\sqrt{\frac{k}{2}} h_{k-1} - \sum_{k=0}^\infty f_k\sqrt{\frac{k+1}{2}} h_{k+1} \
&\qquad\mid k-1 \to k, \ k+1 \to k \
&= \sum_{k=-1}^\infty \sqrt{\frac{k+1}{2}}f_{k+1} h_k - \sum_{k=1}^\infty \sqrt{\frac{k}{2}}f_{k-1} h_k \
&\qquad\mid -0+0 = -\sqrt{\frac{-1+1}{2}}f_{-1+1}h_{-1} + \sqrt{\frac{0}{2}}f_{0-1} h_0 \
&= \sum_{k=0}^\infty \sqrt{\frac{k+1}{2}}f_{k+1} h_k - \sum_{k=0}^\infty \sqrt{\frac{k}{2}}f_{k-1} h_k \
&= \sum_k \left( \sqrt{\frac{k+1}{2}}f_{k+1} - \sqrt{\frac{k}{2}}f_{k-1} \right) h_k
\end{aligned}
$$With $h'_k=\sqrt{\frac{k}{2}}h_{k+1}-\sqrt{\frac{k+1}{2}}h_{k-1}$ fromWikipedia - Hermite functions.IntegrationWith the same relation as above we get$$
\begin{aligned}
h_k' &= \sqrt{\frac{k}{2}}h_{k-1} - \sqrt{\frac{k+1}{2}}h_{k+1} \
&\qquad\mid +\sqrt{\frac{k+1}{2}}h_{k+1} - h_k' \
\sqrt{\frac{k+1}{2}}h_{k+1} &= \sqrt{\frac{k}{2}}h_{k-1} - h_k' \
&\qquad\mid \cdot\sqrt{\frac{2}{k+1}} \
h_{k+1} &= \sqrt{\frac{k}{k+1}}h_{k-1} - \sqrt{\frac{2}{k+1}}h_k' \
&\qquad\mid k+1 \to k \
h_k &= \sqrt{\frac{k-1}{k}}h_{k-2} - \sqrt{\frac{2}{k}}h_{k-1}' \
&\qquad\mid \int \
H_k &= \sqrt{\frac{k-1}{k}}H_{k-2} - \sqrt{\frac{2}{k}}h_{k-1}
\end{aligned}
$$which can be applied from the highest to the lowest order. For $h_0$ we then get$$
H_0(x) = \int_{-\infty}^xh_0(x')dx' = \int_{-\infty}^x\frac{e^{-\frac{x'^2}{2}}}{\sqrt[4]{\pi}}dx' = \sqrt{\frac{\sqrt{\pi}}{2}}\text{erf}\left(\frac{x}{\sqrt{2}}\right) \ \left(+\sqrt{\frac{\sqrt{\pi}}{2}}\right)
$$Fourier transformationWikipedia - Hermite functionsKinetic energy$$
\left\langle\frac{-\hat{P}^2}{2}\right\rangle = -\frac{1}{2}\int_{\mathbb{R}}f^*(x)\frac{d^2}{dx^2}f(x)dx = +\frac{1}{2}\int_{\mathbb{R}}|f'(x)|^2dx = \frac{1}{2}||f'||_{L_{\mathbb{R}}^2}^2
$$License (MIT)Copyright (c) 2022 Sebastian GÃ¶sslPermission is hereby granted, free of charge, to any person obtaining a copy
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