hermite-function 1.1

A Hermite function series module.

Hermite Function Series

A Hermite function series package.

from hermitefunction import HermiteFunction
import numpy as np
import matplotlib.pyplot as plt

x = np.linspace(-4, +4, 1000)
for n in range(5):
    f = HermiteFunction(n)
    plt.plot(x, f(x), label=f'$h_{n}$')
plt.legend(loc='lower right')
plt.show()

Installation

pip install hermite-function

Usage

This package provides a single class, HermiteFunction, to handle Hermite function series. A series can be initialized in three ways:

• With the constructor HermiteFunction(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 factory HermiteFunction.random(deg) for a random Hermite series of a given degree.
• By fitting data with HermiteFunction.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()

The container interface is implemented so the coefficients can be

• accessed 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.

Methods for functions:

• evaluation with f(x),
• differentiation to an arbitrary degree f.der(n) &
• getting the degree of the series f.deg are 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 & subtraction f + g, f - g,
• scalar multiplication & division 2 * f, f / 2,
• inner product & norm f @ g, abs(f).

g = HermiteFunction(4)
h = f + g
i = 0.5 * f
plt.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.kin

Proofs

In 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 $H_k$:

$$ h_k(x) = \frac{e^{-\frac{x^2}{2}}}{\sqrt{2^kk!\sqrt{\pi}}} H_k(x) $$

from Wikipedia - 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 $h'_k=\sqrt{\frac{k}{2}}h_{k+1}-\sqrt{\frac{k+1}{2}}h_{k-1}$ from Wikipedia - Hermite functions.

Integration

With 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

$$ H_0(x) = \int_{-\infty}^xh_0(x')dx' = \int_{-\infty}^x\frac{e^{-\frac{x'^2}{2}}}{\sqrt[4]{\pi}}dx' = \sqrt{\frac{\sqrt{\pi}}{2}}\text{erf}\left(\frac{x}{\sqrt{2}}\right) \ \left(+\sqrt{\frac{\sqrt{\pi}}{2}}\right) $$

Kinetic energy

$$ \left\langle\frac{-\hat{P}^2}{2}\right\rangle = -\frac{1}{2}\int_{\mathbb{R}}f^*(x)\frac{d^2}{dx^2}f(x)dx = +\frac{1}{2}\int_{\mathbb{R}}|f'(x)|^2dx = \frac{1}{2}||f'||_{L_{\mathbb{R}}^2}^2 $$