hermite-function 2.0
A Hermite function series module.
License: MIT
Requires Python: >=3.8

# Hermite Function Series

A Hermite function series package.
```python
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()
```
![png](https://raw.githubusercontent.com/goessl/hermite-function/main/readme/hermite_functions.png)

## 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`](https://github.com/goessl/vector/blob/main/vector.py) 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).
```python
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()
```
![png](https://raw.githubusercontent.com/goessl/hermite-function/main/readme/initialization.png)
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.
```python
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()
```
![png](https://raw.githubusercontent.com/goessl/hermite-function/main/readme/differentiation.png)
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)`.
```python
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()
```
![png](https://raw.githubusercontent.com/goessl/hermite-function/main/readme/arithmetic.png)
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):
```python
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](https://en.wikipedia.org/wiki/Hermite_polynomials\#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](https://en.wikipedia.org/wiki/Hermite_polynomials\#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

$$
    H_0(x) = \int_{-\infty}^xh_0(x')dx' = \int_{-\infty}^x\frac{e^{-\frac{x'^2}{2}}}{\sqrt[4]{\pi}}dx' = \sqrt{\frac{\sqrt{\pi}}{2}}\text{erf}\left(\frac{x}{\sqrt{2}}\right) \ \left(+\sqrt{\frac{\sqrt{\pi}}{2}}\right)
$$

### Fourier transformation

[Wikipedia - Hermite functions](https://en.wikipedia.org/wiki/Hermite_polynomials#Hermite_functions_as_eigenfunctions_of_the_Fourier_transform)

### Kinetic energy

$$
    \left\langle\frac{-\hat{P}^2}{2}\right\rangle = -\frac{1}{2}\int_{\mathbb{R}}f^*(x)\frac{d^2}{dx^2}f(x)dx = +\frac{1}{2}\int_{\mathbb{R}}|f'(x)|^2dx = \frac{1}{2}||f'||\_{L_{\mathbb{R}}^2}^2
$$

## License (MIT)

Copyright (c) 2022 Sebastian GÃ¶ssl

Permission is hereby granted, free of charge, to any person obtaining a copy
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