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Description:

hermitefunction 2.0

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 git+https://github.com/goessl/vector.git
pip install hermite-function

Usage
This package provides a single class, HermiteFunction, to handle Hermite function series.
HermiteFunction extends Vector from the [https://github.com/goessl/vector](vector module) and therefore provides the same functionality.
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()

Container and sequence interfaces are 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 &
trimmed: f.trim() (trailing non-zero coefficients get removed).

Methods for functions:

evaluation with f(x),
differentiation to an arbitrary degree f.der(n),
integration f.antider(),
Fourier transformation f.fourier() &
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 LR2 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 (⟨P^2⟩=12∫Rf∗(x)f″(x)dx, natural units):
f.kin

Proofs
In the following let
f=∑k=0∞fkhk, g=∑k=0∞gkhk.
where hk are the Hermite functions, defined by the Hermite polynomials Hk:
hk(x)=e−x222kk!πHk(x)
from Wikipedia - Hermite functions.
Differentiation
f′=∑kfkhk′ ∣hk′=k2hk−1−k+12hk+1 =∑kfk(k2hk−1−k+12hk+1) =∑k=0∞fkk2hk−1−∑k=0∞fkk+12hk+1 ∣k−1→k, k+1→k =∑k=−1∞k+12fk+1hk−∑k=1∞k2fk−1hk ∣−0+0=−−1+12f−1+1h−1+02f0−1h0 =∑k=0∞k+12fk+1hk−∑k=0∞k2fk−1hk =∑k(k+12fk+1−k2fk−1)hk
With hk′=k2hk+1−k+12hk−1 from Wikipedia - Hermite functions.
Integration
With the same relation as above we get
hk′=k2hk−1−k+12hk+1 ∣+k+12hk+1−hk′ k+12hk+1=k2hk−1−hk′ ∣⋅2k+1 hk+1=kk+1hk−1−2k+1hk′ ∣k+1→k hk=k−1khk−2−2khk−1′ ∣∫ Hk=k−1kHk−2−2khk−1
which can be applied from the highest to the lowest order. For h0 we then get
H0(x)=∫−∞xh0(x′)dx′=∫−∞xe−x′22π4dx′=π2erf(x2) (+π2)
Fourier transformation
Wikipedia - Hermite functions
Kinetic energy
⟨−P^22⟩=−12∫Rf∗(x)d2dx2f(x)dx=+12∫R|f′(x)|2dx=12||f′||LR22
License (MIT)
Copyright (c) 2022 Sebastian GÃ¶ssl
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