A Hermite function series module.
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()
pip install git+https://github.com/goessl/vector.git
pip install hermite-function
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
- 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 (
f.kin
In the following let
where
from Wikipedia - Hermite functions.
With
With the same relation as above we get
which can be applied from the highest to the lowest order. For
Copyright (c) 2022 Sebastian Gössl
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